Again Use a Least Squares Approach but Now to Estimate Both the Initial Velocity and Gravity

Initial Velocity

Applications Related to Ordinary and Partial Differential Equations

Martha L. Abell , James P. Braselton , in Mathematica by Instance (Fifth Edition), 2017

vi.v.1 The One-Dimensional Wave Equation

Suppose that nosotros pluck a string (similar a guitar or violin cord) of length p and constant mass density that is fixed at each end. A question that we might inquire is: What is the position of the string at a particular instance of time? We answer this question by modeling the physical state of affairs with a partial differential equation, namely the moving ridge equation in i spatial variable:

(half-dozen.41) $c^2\frac{{\partial }^{2}u}{\partial {\mathrm{ten}}^2}=\frac{{\partial }^{2}u}{\partial t^2}\text{or}{c}^{\mathrm{two}}{u}_{\mathrm{10}x}=u_{tt}.$

In equation (6.41), $c^2=T/\rho$ , where T is the tension of the string and ρ is the abiding mass of the cord per unit of measurement length. The solution $u(\mathrm{ten},t)$ represents the displacement of the string from the x-axis at time t. To decide u we must describe the boundary and initial weather that model the physical situation. At the ends of the string, the displacement from the x-axis is stock-still at zero, so we use the homogeneous purlieus atmospheric condition $u(0,t)=u(p,t)=0$ for $t>0$ . The movement of the cord besides depends on the deportation and the velocity at each point of the cord at $t=0$ . If the initial displacement is given past $f(x)$ and the initial velocity past $\mathrm{chiliad}(x)$ , we have the initial weather condition $u(x,0)=f(x)$ and $u_t(x,0)=\mathrm{yard}(x)$ for $0⩽\mathrm{ten}⩽p$ . Therefore, we determine the deportation of the string with the initial-purlieus value problem

(6.42) $\{\begin{array}{c}{c}^{\mathrm{two}}\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{\mathrm{ii}}u}{\partial t^2},0<x<p,t>0\hfill \\ u(0,t)=u(p,t)=0,t>0\hfill \\ u(\mathrm{10},0)=f(x),u_t(x,0)=g(x),0<\mathrm{ten}<p.\hfill \end{array}$

This trouble is solved through separation of variables past assuming that $u(x,t)=X(x)T(t)$ . Substitution into equation (6.41) yields

λ is a constant.

$c^2{X}^{″}T=X{T}^{″}\text{or}\frac{{\mathrm{10}}^{″}}{X}=\frac{T^{″}}{c^{2}T}=-\lambda$

so we obtain the two 2d-order ordinary differential equations ${\mathrm{10}}^{″}+\lambda \mathrm{Ten}=0$ and $T^{″}+c^2\lambda T=0$ . At this signal, nosotros solve the equation that involves the homogeneous boundary conditions. The boundary conditions in terms of $u(x,t)=\mathrm{Ten}(x)T(t)$ are $u(0,t)=X(0)T(t)=0$ and $u(p,t)=\mathrm{10}(p)T(t)=0$ , so we have $X(0)=0$ and $\mathrm{10}(p)=0$ . Therefore, we determine $X(x)$ past solving the eigenvalue problem

$\{\begin{array}{c}{X}^{″}+\lambda X=0,0<\mathrm{10}<p\hfill \\ \mathrm{Ten}(0)=X(p)=0.\hfill \end{array}$

The eigenvalues of this problem are ${\lambda }_n={(n\pi /p)}^{\mathrm{two}}$ , $\mathrm{northward}=1$ , three, … with corresponding eigenfunctions ${\mathrm{Ten}}_{\mathrm{northward}}(x)=\sin {(\mathrm{due\; north}\pi x/p)}^{\mathrm{ii}}$ , $n=\mathrm{i}$ , 3, …. Next, we solve the equation $T^{″}+c^2{\lambda }_{\mathrm{northward}}T=0$ . A full general solution is

$T_{\mathrm{due\; north}}(t)=a_n\cos \left(c\sqrt{{\lambda }_n}t\right)+b_n\sin \left(c\sqrt{{\lambda }_{\mathrm{northward}}}t\right)=a_n\cos \left(\frac{c\mathrm{due\; north}\pi t}{p}\right)+b_n\sin \left(\frac{cn\pi t}{p}\right),$

where the coefficients $a_{\mathrm{northward}}$ and $b_{\mathrm{northward}}$ must be determined. Putting this information together, we obtain

$u_n(x,t)=(a_n\cos \left(\frac{cn\pi t}{p}\right)+b_n\sin \left(\frac{cn\pi t}{p}\right))\sin \left(\frac{n\pi x}{p}\right),$

so by the Principle of Superposition, we have

$u(x,t)=\sum _{n=\mathrm{one}}^{\infty }(a_n\cos \left(\frac{cn\pi t}{p}\right)+b_n\sin \left(\frac{cn\pi t}{p}\right))\sin \left(\frac{n\pi x}{p}\right).$

Applying the initial displacement $u(\mathrm{ten},0)=f(x)$ yields

$u(\mathrm{ten},0)=\sum _{n=1}^{\infty }{a}_{\mathrm{north}}\sin \left(\frac{n\pi x}{p}\right)=f(\mathrm{10}),$

then $a_n$ is the Fourier sine series coefficient for $f(x)$ , which is given by

$a_n=\frac{\mathrm{ii}}{p}{\int }_0^{p}f(x)\sin \left(\frac{n\pi \mathrm{ten}}{p}\right)d\mathrm{ten},n=1,\mathrm{two},\dots .$

In order to determine $b_n$ , we must use the initial velocity. Therefore, we compute

$\frac{\partial u}{\partial t}(x,t)=\sum _{n=1}^{\infty }(-a_n\frac{c\mathrm{due\; north}\pi }{p}\sin \left(\frac{cn\pi t}{p}\right)+b_n\frac{c\mathrm{due\; north}\pi }{p}\cos \left(\frac{cn\pi t}{p}\right))\sin \left(\frac{\mathrm{north}\pi x}{p}\right).$

Then,

$\frac{\partial u}{\partial t}(x,0)=\sum _{n=\mathrm{i}}^{\infty }{b}_n\frac{cn\pi }{p}\sin \left(\frac{n\pi x}{p}\right)=g(x)$

so $b_{\mathrm{north}}\frac{c\mathrm{northward}\pi }{p}$ represents the Fourier sine serial coefficient for $g(x)$ which means that

$b_n=\frac{p}{c\mathrm{due\; north}\pi }{\int }_0^{p}g(x)\sin \left(\frac{n\pi \mathrm{10}}{p}\right)dx,n=1,2,\dots .$

Instance half-dozen.34

Solve $\{\begin{array}{c}{u}_{xx}=u_{tt},0<\mathrm{10}<\mathrm{ane},t>0\hfill \\ u(0,t)=u(1,t)=0,t>0\hfill \\ u(x,0)=\sin \pi x,u_t(x,0)=\mathrm{iii}x+\mathrm{i},0<\mathrm{10}<1.\hfill \end{array}$

Solution

The initial displacement and velocity functions are defined first.

$\mathit{f}\mathbf{[}\text{<mi mathvariant="bold" is="true">x</mi>}\mathbf{\_}\mathbf{]}\mathbf{=}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{\pi }\mathit{x}\mathbf{]}\mathbf{;}$

$\mathit{chiliad}\mathbf{[}\text{<mi mathvariant="bold" is="true">ten</mi>}\mathbf{\_}\mathbf{]}\mathbf{=}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{;}$

Next, the functions to determine the coefficients $a_{\mathrm{northward}}$ and $b_n$ in the series approximation of the solution $u(x,t)$ are defined. Here, $p=c=\mathrm{one}$ .

${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{2}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\mathit{f}\mathbf{[}\mathit{10}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{\pi }\mathit{x}\mathbf{]}\mathit{d}\mathit{10}$

ane

${\mathit{a}}_{\text{<mi mathvariant="bold" is="true">n</mi>}\mathbf{\_}}\mathbf{=}\mathbf{ii}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\mathit{f}\mathbf{[}\mathit{x}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathit{x}\mathbf{]}\mathit{d}\mathit{x}$

$\frac{\mathrm{two}\text{Sin}[n\pi ]}{\pi -n^{\mathrm{ii}}\pi }$

${\mathit{b}}_{\text{<mi mathvariant="bold" is="true">northward</mi>}\mathbf{\_}}\mathbf{=}\frac{\mathbf{ii}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\mathit{g}\mathbf{[}\mathit{10}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathit{x}\mathbf{]}\mathit{d}\mathit{x}}{\mathit{due\; north}\mathit{\pi }}\text{<mi mathvariant="bold" is="true">//</mi>}\text{<mi mathvariant="bold" is="true">Simplify</mi>}$

$\frac{\mathrm{two}\mathrm{northward}\pi -8\mathrm{north}\pi \text{Cos}[n\pi ]+6\text{Sin}[n\pi ]}{n^3{\pi }^3}$

Considering north represents an integer, these results betoken that $a_n=0$ for all $\mathrm{north}⩾2$ , which nosotros confirm with Simplify together with the Assumptions by instructing Mathematica to assume that due north is an integer.

$\text{<mi mathvariant="bold" is="true">Simplify</mi>}\mathbf{[}\frac{\mathbf{ii}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathbf{]}}{\mathit{\pi }\mathbf{-}{\mathit{due\; north}}^{\mathbf{2}}\mathit{\pi }}\mathbf{,}\text{<mi mathvariant="bold" is="true">Assumptions</mi>}\mathbf{\to }\text{<mi mathvariant="bold" is="true">Element</mi>}\mathbf{[}\mathit{n}\mathbf{,}\text{<mi mathvariant="bold" is="true">Integers</mi>}\mathbf{]}\mathbf{]}$

0

$\text{<mi mathvariant="bold" is="true">Simplify</mi>}\mathbf{[}\frac{\mathbf{2}\mathit{n}\mathit{\pi }\mathbf{-}\mathbf{8}\mathit{n}\mathit{\pi }\text{<mi mathvariant="bold" is="true">Cos</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathbf{]}\mathbf{+}\mathbf{6}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathbf{]}}{{\mathit{n}}^{\mathbf{3}}{\mathit{\pi }}^{\mathbf{three}}}\mathbf{,}$

$\text{<mi mathvariant="bold" is="true">Assumptions</mi>}\mathbf{\to }\text{<mi mathvariant="bold" is="true">Element</mi>}\mathbf{[}\mathit{north}\mathbf{,}\text{<mi mathvariant="bold" is="true">Integers</mi>}\mathbf{]}\mathbf{]}$

$\frac{\mathrm{ii}-8{(-\mathrm{one})}^{\mathrm{due\; north}}}{n^{\mathrm{two}}{\pi }^{\mathrm{two}}}$

Nosotros use Tabular array to calculate the first ten values of $b_n$ .

$\text{<mi mathvariant="bold" is="true">Table</mi>}\mathbf{[}\mathbf{\{}\mathit{n}\mathbf{,}{\mathit{b}}_{\mathit{n}}\mathbf{,}{\mathit{b}}_{\mathit{n}}\text{<mi mathvariant="bold" is="true">//</mi>}\mathit{N}\mathbf{\}}\mathbf{,}\mathbf{\{}\mathit{n}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{ten}\mathbf{\}}\mathbf{]}\text{<mi mathvariant="bold" is="true">//</mi>}\text{<mi mathvariant="bold" is="true">TableForm</mi>}$

$\begin{array}{ccc}1\hfill & \frac{10}{{\pi }^2}\hfill & 1.01321\hfill \\ 2\hfill & -\frac{\mathrm{iii}}{\mathrm{ii}{\pi }^2}\hfill & -0.151982\hfill \\ 3\hfill & \frac{\mathrm{ten}}{9{\pi }^{\mathrm{ii}}}\hfill & 0.112579\hfill \\ \mathrm{iv}\hfill & -\frac{\mathrm{iii}}{8{\pi }^2}\hfill & -0.0379954\hfill \\ 5\hfill & \frac{2}{\mathrm{v}{\pi }^2}\hfill & 0.0405285\hfill \\ \mathrm{vi}\hfill & -\frac{1}{\mathrm{six}{\pi }^{\mathrm{ii}}}\hfill & -0.0168869\hfill \\ 7\hfill & \frac{10}{49{\pi }^2}\hfill & 0.0206778\hfill \\ 8\hfill & -\frac{\mathrm{three}}{32{\pi }^2}\hfill & -0.00949886\hfill \\ 9\hfill & \frac{10}{81{\pi }^{\mathrm{ii}}}\hfill & 0.0125088\hfill \\ 10\hfill & -\frac{3}{50{\pi }^2}\hfill & -0.00607927\hfill \end{array}$

Discover that we ascertain uapprox[n] so that Mathematica "remembers" the terms uapprox that are computed. That is, Mathematica does not demand to recompute uapprox[due north-i] to compute uapprox[n] provided that uapprox[n-1] has already been computed.

The office u defined next computes the due northth term in the series expansion. Thus, uapprox determines the approximation of order k by summing the showtime k terms of the expansion, as illustrated with approx[10].

$\text{<mi mathvariant="bold" is="true">Clear</mi>}\mathbf{[}\mathit{u}\mathbf{,}\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{]}$

$\mathit{u}\mathbf{[}\text{<mi mathvariant="bold" is="true">n</mi>}\mathbf{\_}\mathbf{]}\mathbf{=}{\mathit{b}}_{\mathit{n}}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{n}\mathit{\pi }\mathit{ten}\mathbf{]}\mathbf{;}$

$\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">k</mi>}\mathbf{\_}\mathbf{]}\text{<mi mathvariant="bold" is="true">:=</mi>}\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\mathit{g}\mathbf{]}\mathbf{=}\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\mathit{k}\mathbf{-}\mathbf{1}\mathbf{]}\mathbf{+}\mathit{u}\mathbf{[}\mathit{k}\mathbf{]}\mathbf{;}$

$\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\mathbf{0}\mathbf{]}\mathbf{=}\text{<mi mathvariant="bold" is="true">Cos</mi>}\mathbf{[}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{\pi }\mathit{10}\mathbf{]}\mathbf{;}$

$\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\mathbf{10}\mathbf{]}$

$\text{Cos}[\pi t]\text{Sin}[\pi x]+\frac{10\text{Sin}[\pi t]\text{Sin}[\pi \mathrm{10}]}{{\pi }^{\mathrm{ii}}}-\frac{3\text{Sin}[2\pi t]\text{Sin}[2\pi x]}{\mathrm{ii}{\pi }^2}+\frac{\mathrm{x}\text{Sin}[\mathrm{iii}\pi t]\text{Sin}[3\pi x]}{9{\pi }^{\mathrm{ii}}}-\frac{3\text{Sin}[\mathrm{four}\pi t]\text{Sin}[\mathrm{four}\pi x]}{8{\pi }^2}+\frac{2\text{Sin}[\mathrm{five}\pi t]\text{Sin}[5\pi \mathrm{10}]}{\mathrm{v}{\pi }^2}-\frac{\text{Sin}[\mathrm{half-dozen}\pi t]\text{Sin}[6\pi x]}{6{\pi }^2}+\frac{10\text{Sin}[7\pi t]\text{Sin}[7\pi \mathrm{10}]}{49{\pi }^2}-\frac{\mathrm{iii}\text{Sin}[\mathrm{viii}\pi t]\text{Sin}[\mathrm{eight}\pi x]}{32{\pi }^2}+\frac{10\text{Sin}[\mathrm{ix}\pi t]\text{Sin}[9\pi x]}{81{\pi }^2}-\frac{3\text{Sin}[10\pi t]\text{Sin}[10\pi \mathrm{10}]}{50{\pi }^{\mathrm{two}}}$

To illustrate the motion of the string, we graph uapprox[ten], the tenth partial sum of the series, on the interval $[0,\mathrm{one}]$ for sixteen equally spaced values of t between 0 and 2 in Fig. 6.52.

Figure 6.52. The motion of the string for 16 equally spaced values of t between 0 and two. (University of Wyoming colors)

$\text{<mi mathvariant="bold" is="true">somegraphs</mi>}\mathbf{=}$

$\text{<mi mathvariant="bold" is="true">Table</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">Plot</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">Evaluate</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\mathbf{10}\mathbf{]}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{10}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{i}\mathbf{\}}\mathbf{,}\text{<mi mathvariant="bold" is="true">PlotRange</mi>}\mathbf{\to }\mathbf{\{}\mathbf{-}\frac{\mathbf{3}}{\mathbf{ii}}\mathbf{,}\frac{\mathbf{iii}}{\mathbf{two}}\mathbf{\}}\mathbf{,}$

$\text{<mi mathvariant="bold" is="true">Ticks</mi>}\mathbf{\to }\mathbf{\{}\mathbf{\{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{\}}\mathbf{,}\mathbf{\{}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{i}\mathbf{\}}\mathbf{\}}\mathbf{,}\text{<mi mathvariant="bold" is="true">PlotStyle</mi>}\mathbf{\to }\text{<mi mathvariant="bold" is="true">CMYKColor</mi>}\mathbf{[}\mathbf{.}\mathbf{53}\mathbf{,}\mathbf{.}\mathbf{72}\mathbf{,}\mathbf{.}\mathbf{77}\mathbf{,}\mathbf{.}\mathbf{57}\mathbf{]}\mathbf{]}\mathbf{,}$

$\mathbf{\{}\mathit{t}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{,}\frac{\mathbf{2}}{\mathbf{15}}\mathbf{\}}\mathbf{]}\mathbf{;}$

$\text{<mi mathvariant="bold" is="true">toshow</mi>}\mathbf{=}\text{<mi mathvariant="bold" is="true">Sectionalization</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">somegraphs</mi>}\mathbf{,}\mathbf{iv}\mathbf{]}\mathbf{;}$

$\text{<mi mathvariant="bold" is="true">Evidence</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">GraphicsGrid</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">toshow</mi>}\mathbf{]}\mathbf{]}$

If instead we wished to see the move of the string, we tin can use Animate. We evidence a frame from the resulting animation.

$\text{<mi mathvariant="bold" is="true">uapprox</mi>}\mathbf{[}\mathbf{10}\mathbf{]}$

$\text{<mi mathvariant="bold" is="true">Animate</mi>}\mathbf{[}$

$\text{<mi mathvariant="bold" is="true">Plot</mi>}\mathbf{[}\text{<mi mathvariant="bold" is="true">Cos</mi>}\mathbf{[}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{\pi }\mathit{x}\mathbf{]}\mathbf{+}\frac{\mathbf{x}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathit{\pi }\mathit{ten}\mathbf{]}}{{\mathit{\pi }}^{\mathbf{two}}}\mathbf{-}\frac{\mathbf{iii}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{2}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{2}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{2}{\mathit{\pi }}^{\mathbf{2}}}\mathbf{+}\mathbf{.}$

$\frac{\mathbf{x}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{3}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{3}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{nine}{\mathit{\pi }}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{iii}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{4}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{4}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{8}{\mathit{\pi }}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{ii}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{five}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{5}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{v}{\mathit{\pi }}^{\mathbf{two}}}\mathbf{-}$

$\frac{\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{6}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{6}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{six}{\mathit{\pi }}^{\mathbf{two}}}\mathbf{+}\frac{\mathbf{10}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{vii}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{7}\mathit{\pi }\mathit{ten}\mathbf{]}}{\mathbf{49}{\mathit{\pi }}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{3}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{eight}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{8}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{32}{\mathit{\pi }}^{\mathbf{2}}}\mathbf{+}$

$\frac{\mathbf{10}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{9}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{9}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{81}{\mathit{\pi }}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{3}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{10}\mathit{\pi }\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">Sin</mi>}\mathbf{[}\mathbf{x}\mathit{\pi }\mathit{x}\mathbf{]}}{\mathbf{l}{\mathit{\pi }}^{\mathbf{two}}}\mathbf{,}\mathbf{\{}\mathit{10}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{i}\mathbf{\}}\mathbf{,}$

$\text{<mi mathvariant="bold" is="true">PlotRange</mi>}\mathbf{\to }\mathbf{\{}\mathbf{-}\mathbf{3}\mathbf{/}\mathbf{ii}\mathbf{,}\mathbf{3}\mathbf{/}\mathbf{two}\mathbf{\}}\mathbf{,}\text{<mi mathvariant="bold" is="true">Ticks</mi>}\mathbf{\to }\mathbf{\{}\mathbf{\{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{\}}\mathbf{,}\mathbf{\{}\mathbf{-}\mathbf{one}\mathbf{,}\mathbf{one}\mathbf{\}}\mathbf{\}}\mathbf{,}$

$\text{<mi mathvariant="bold" is="true">PlotStyle</mi>}\text{<mi mathvariant="bold" is="true">-</mi>}\mathbf{>}\text{<mi mathvariant="bold" is="true">CMYKColor</mi>}\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{24}\mathbf{,}\mathbf{.}\mathbf{94}\mathbf{,}\mathbf{0}\mathbf{]}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{t}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{\}}\mathbf{]}$

Finally, nosotros remark that DSolve can find D'Alembert's solution to the wave equation.

$\text{<mi mathvariant="bold" is="true">Articulate</mi>}\mathbf{[}\mathit{u}\mathbf{,}\mathit{c}\mathbf{]}$

$\text{<mi mathvariant="bold" is="true">DSolve</mi>}\mathbf{[}{\mathit{c}}^{\mathbf{\wedge }}\mathbf{2}\mathit{D}\mathbf{[}\mathit{u}\mathbf{[}\mathit{x}\mathbf{,}\mathit{t}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{ten}\mathbf{,}\mathbf{2}\mathbf{\}}\mathbf{]}\text{<mi mathvariant="bold" is="true">==</mi>}\mathit{D}\mathbf{[}\mathit{u}\mathbf{[}\mathit{ten}\mathbf{,}\mathit{t}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{t}\mathbf{,}\mathbf{2}\mathbf{\}}\mathbf{]}\mathbf{,}$

$\mathit{u}\mathbf{[}\mathit{x}\mathbf{,}\mathit{t}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{ten}\mathbf{,}\mathit{t}\mathbf{\}}\mathbf{]}$

$\{\{u[x,t]\to C[\mathrm{one}][t-\frac{x}{\sqrt{c^2}}]+C[2][t+\frac{\mathrm{10}}{\sqrt{c^{\mathrm{ii}}}}]\}\}$

$\text{<mi mathvariant="bold" is="true">DSolve</mi>}\mathbf{[}{\mathit{c}}^{\mathbf{2}}{\mathbf{\partial }}_{\mathbf{\{}\mathit{x}\mathbf{,}\mathbf{2}\mathbf{\}}}\mathit{u}\mathbf{[}\mathit{x}\mathbf{,}\mathit{t}\mathbf{]}\text{<mi mathvariant="bold" is="true">==</mi>}{\mathbf{\partial }}_{\mathbf{\{}\mathit{t}\mathbf{,}\mathbf{2}\mathbf{\}}}\mathit{u}\mathbf{[}\mathit{x}\mathbf{,}\mathit{t}\mathbf{]}\mathbf{,}\mathit{u}\mathbf{[}\mathit{x}\mathbf{,}\mathit{t}\mathbf{]}\mathbf{,}\mathbf{\{}\mathit{x}\mathbf{,}\mathit{t}\mathbf{\}}\mathbf{]}$

$\{\{u[x,t]\to C[1][t-\frac{\mathrm{ten}}{\sqrt{c^{\mathrm{ii}}}}]+C[2][t+\frac{x}{\sqrt{c^2}}]\}\}$  □

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Calculator Simulations

R.K. Pathria , Paul D. Beale , in Statistical Mechanics (Tertiary Edition), 2011

16.iii.A Molecular dynamics algorithm

Showtime, start the system by choosing an initial country by setting the initial positions and velocities of all the particles. The initial velocities are usually set past choosing each component of the velocity vector of each particle from the Maxwell distribution. In reduced units, this is

(6) $P_{\text{maxwell}}({\upsilon }_x(0))=\frac{\mathrm{one}}{\sqrt{\mathrm{ii}\pi T}}\exp \left(-\frac{{\upsilon }_x^{\mathrm{two}}(0)}{\mathrm{ii}T}\right);$

\due northencounter Appendix I to run into how to apply a uniform pseudorandom number generator to select from a Gaussian distribution. The initial velocities tin then exist used to set the positions of the particles afterwards the first time-step, namely

(7) $r_i(\Delta t)=r_i(0)+{\upsilon }_i(0)\Delta t+\frac{1}{2}\frac{{(\Delta t)}^2}{m_i}{F}_i(0).$

i.

Side by side, use equation (2) to motility the system forward in time through M _eq = τ_eq/Δt time steps. The equilibration time τ_eq must be chosen big enough for the organisation to equilibrate; see the Monte Carlo give-and-take in Section 16.two. A thermostat is often used to evolve the system to a state with the desired temperature; see Frenkel and Smit (2002).

2.

Now, apply equation (ii) to move the system frontwards in fourth dimension through M = τ_acg/Δr time steps while keeping track of all the thermodynamic variables {A(q+)} ane wants to measure. Finally, employ equation (16.1.2) to determine the equilibrium averages and uncertainties.

To decide averages at a different set of parameters (temperature, density, etc.), change the parameters by a small amount and repeat steps 1 and 2. Using the last configuration of the previous run as the first configuration of the new run can often reduce the equilibration time.

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COMPENDIUM OF THE FOUNDATIONS OF CLASSICAL STATISTICAL PHYSICS

Jos Uffink , in Philosophy of Physics, 2007

3.3 Maxwell (1867)

Whatever the claim and problems of his first paper, Maxwell'due south side by side newspaper on gas theory of 1867 rejected his previous try to derive the distribution function from the assumptions (24, 25) as "precarious" and proposed a completely different argument. This time, he considered a model of indicate particles with equal masses interacting by means of a repulsive central force, proportional to the fifth power of their mutual distance. What is more important, this time the collisions are used in the statement.

Maxwell considers an rubberband collision between a pair of particles such that the initial velocities are ${\overrightarrow{\upsilon }}_1$ , ${\overrightarrow{\upsilon }}_2$ and final velocities ${\overrightarrow{\upsilon }}_1$ , ${\overrightarrow{\upsilon }}_{\mathrm{two}})$ . ¹³ These quantities are related by the conservation laws of momentum and energy, yielding four equations, and ii parameters depending on the geometrical factors of the collision process.

Information technology is convenient to consider a coordinate frame such that particle ane is at remainder in the origin, and the relative velocity ${\overrightarrow{\upsilon }}_2-{\overrightarrow{\upsilon }}_{\mathrm{ane}}$ is directed forth the negative z axis, and to use cylindrical coordinates. If (b, ϕ, z) denote the coordinates of the trajectory of the centre of particle 2, we and so have b = const., ϕ = const, $z\left(t\right)=z_0-\Vert {\overrightarrow{\upsilon }}_2-{\overrightarrow{\upsilon }}_1\Vert t$ before the collision. In the case where the particles are elastic difficult spheres, a collision will take place only if the impact parameter b is less than the bore d of the spheres. The velocities afterwards the collision are so determined past $\Vert {\overrightarrow{\upsilon }}_1-{\overrightarrow{\upsilon }}_{\mathrm{ii}}\Vert$ , b and ϕ. Transformed back to the laboratory frame, the final velocities ${\overrightarrow{\upsilon }}_{\mathrm{ane}}$ , ${\overrightarrow{\upsilon }}_2$ can then be expressed equally functions of ${\overrightarrow{\upsilon }}_{\mathrm{ane}}$ , ${\overrightarrow{\upsilon }}_2$ , b and ϕ.

Maxwell now assumes what the Ehrenfests afterward called the Stoßzahlansatz:the number of collisions during a fourth dimension dt, say $N\left({\overrightarrow{\upsilon }}_{\mathrm{ane}},{\overrightarrow{\upsilon }}_{\mathrm{two}}\right)$ , in which the initial velocities ${\overrightarrow{\upsilon }}_1$ , ${\overrightarrow{\upsilon }}_2$ inside an chemical element $d^{\mathrm{three}}{\overrightarrow{\upsilon }}_{\mathrm{i}}{d}^3{\overrightarrow{\upsilon }}_{\mathrm{ii}}$ are changed into final velocities ${\overrightarrow{\upsilon }}_1$ , ${\overrightarrow{\upsilon }}_2$ in an chemical element $d^3{\overrightarrow{\upsilon }}_1{d}^{\mathrm{iii}}{\overrightarrow{\upsilon }}_2$ within a spatial volume element $dV=bdbd\varphi dz=\Vert {\overrightarrow{\upsilon }}_{\mathrm{i}}-{\overrightarrow{\upsilon }}_2\Vert bdbd\varphi dt$ is proportional to the production of the number of particles with velocity ${\overrightarrow{\upsilon }}_1$ within $d^3{\overrightarrow{\upsilon }}_{\mathrm{i}}\left(\text{i}\text{.e}\text{.}\mathrm{Due\; north}f\left({\overrightarrow{\upsilon }}_{\mathrm{i}}\right)d{\overrightarrow{\upsilon }}_1\right)$ , and those with velocity ${\overrightarrow{\upsilon }}_2$ inside $d^{\mathrm{iii}}{\overrightarrow{\upsilon }}_2\left(\text{i}\text{.east}\text{.}\mathrm{North}f\left({\overrightarrow{\upsilon }}_2\right)d{\overrightarrow{\upsilon }}_{\mathrm{two}}\right)$ , and that spatial book chemical element. Thus:

(29) $N\left({\overrightarrow{v}}_1,{\overrightarrow{v}}_{\mathrm{ii}}\right)={\mathrm{Due\; north}}^{2}f\left({\overrightarrow{v}}_1\right)f\left({\overrightarrow{5}}_2\right)\Vert {\overrightarrow{\mathrm{five}}}_2-{\overrightarrow{v}}_{\mathrm{ane}}\Vert d^3{\overrightarrow{v}}_1{d}^3{\overrightarrow{\mathrm{five}}}_{2}bdbd\varphi dt.$

Due to the time reversal invariance of the collision laws, a similar consideration applies to the then-called inverse collisions, in which initial velocities ${\overrightarrow{\upsilon }}_1$ , ${\overrightarrow{\upsilon }}_2$ and last velocities ${\overrightarrow{\upsilon }}_1$ and ${\overrightarrow{\upsilon }}_2$ are interchanged. Their number is proportional to

(xxx) $\mathrm{Due\; north}({\overrightarrow{\upsilon }}_1^{\hspace{0.17em}\hspace{0.17em}\prime },{\overrightarrow{\upsilon }}_{\mathrm{two}}^{\hspace{0.17em}\hspace{0.17em}\prime })=N^{2}f({\overrightarrow{\upsilon }}_{\mathrm{i}}^{\hspace{0.17em}\hspace{0.17em}\prime })f({\overrightarrow{\upsilon }}_1^{\hspace{0.17em}\hspace{0.17em}\prime })\left|\right|{\overrightarrow{\upsilon }}_{\mathrm{two}}^{\hspace{0.17em}\hspace{0.17em}\prime }-{\overrightarrow{\upsilon }}_{\mathrm{ane}}^{\hspace{0.17em}\hspace{0.17em}\prime }\left|\right|d^{\mathrm{iii}}{\overrightarrow{\upsilon }}_1^{\hspace{0.17em}\hspace{0.17em}\prime }{d}^3{\overrightarrow{\upsilon }}_2^{\hspace{0.17em}\hspace{0.17em}\prime }bdbd\phi dt$

Maxwell argues that the distribution of velocities will remain stationary, i.e. unaltered in the course of fourth dimension, if the number of collisions of these two kinds are equal, i.eastward. if

(31) $N({\overrightarrow{\upsilon }}_{\mathrm{ane}}^{\hspace{0.17em}\hspace{0.17em}\prime },{\overrightarrow{\upsilon }}_2^{\hspace{0.17em}\hspace{0.17em}\prime })=\mathrm{Due\; north}({\overrightarrow{\upsilon }}_1^{},{\overrightarrow{\upsilon }}_2^{}).$

Moreover, the standoff laws entail that ${\Vert {\overrightarrow{\upsilon }}_{\mathrm{ii}}}^{\prime }-{\overrightarrow{\upsilon }}_1\Vert =\Vert {\overrightarrow{\upsilon }}_2-{\overrightarrow{\upsilon }}_{\mathrm{one}}\Vert$ and $d^3{\overrightarrow{\upsilon }}_{\mathrm{i}}{d}^3{\overrightarrow{\upsilon }}_2=d^3{\overrightarrow{\upsilon }}_1{d}^{\mathrm{three}}{\overrightarrow{\upsilon }}_{\mathrm{two}}$ . Hence, the condition (31) may be simplified to

(32) $f({\overrightarrow{\upsilon }}_1)f({\overrightarrow{\upsilon }}_{\mathrm{ii}})=f({\overrightarrow{\upsilon }}_1^{\hspace{0.17em}\hspace{0.17em}\prime })f({\overrightarrow{\upsilon }}_2^{\hspace{0.17em}\hspace{0.17em}\prime })\hspace{0.17em}\hspace{0.17em}\prime \text{for all}{\overrightarrow{\upsilon }}_1\hspace{0.17em}\hspace{0.17em}\prime {\overrightarrow{\upsilon }}_{\mathrm{ii}}.$

This is the case for the Maxwellian distribution (26). Therefore, Maxwell says, the distribution (26) is a "possible" form.

He goes on to claim that it is also the simply possible form for a stationary distribution.This claim, i.e. that stationarity of the distribution f tin only arise under (32) is present also called the principle of detailed balancing (cf. [Tolman, 1938, p. 165]). ¹⁴ Although his statement is rather brief, the idea seems to exist that for a distribution violating (32), there must (considering of the *Stoßzahlansatz) be two velocity pairs ¹⁵ ${\overrightarrow{\upsilon }}_1$ , ${\overrightarrow{\upsilon }}_{\mathrm{ii}}$ and ${\overrightarrow{u}}_1$ , ${\overrightarrow{u}}_2$ , satisfying ${\overrightarrow{\upsilon }}_1+{\overrightarrow{\upsilon }}_2={\overrightarrow{u}}_1+{\overrightarrow{u}}_2$ and ${\upsilon }_1^2+{\upsilon }_2^2=u_2^2$ , such that the collisions would predominantly transform $\left({\overrightarrow{\upsilon }}_1,{\overrightarrow{\upsilon }}_2\right)\to \left({\overrightarrow{u}}_{\mathrm{ane}},{\overrightarrow{u}}_2\right)$ rather than $\left({\overrightarrow{u}}_1,{\overrightarrow{u}}_2\right)\to \left({\overrightarrow{\upsilon }}_1,{\overrightarrow{\upsilon }}_{\mathrm{ii}}\right)$ . But then, since the distribution is stationary, there must be a third pair of velocities, $\left({\overrightarrow{w}}_1,{\overrightarrow{w}}_2\right)$ , satisfying like relations, for which the collisions predominantly produce transitions $\left({\overrightarrow{u}}_1,{\overrightarrow{u}}_2\right)\to \left({\overrightarrow{w}}_{\mathrm{i}},{\overrightarrow{w}}_2\right)$ , etc. Now, the distribution can only remain stationary if any such sequence closes into a bicycle. Hence in that location would be cycles of velocity pairs $\left({\overrightarrow{\upsilon }}_1,{\overrightarrow{\upsilon }}_{\mathrm{ii}}\right)\to \left({\overrightarrow{u}}_{\mathrm{i}},{\overrightarrow{u}}_2\right)\to \dots \to \left({\overrightarrow{\upsilon }}_{\mathrm{ane}},{\overrightarrow{\upsilon }}_2\right)$ which the colliding particles go through, somewhen returning to their original velocities.

Maxwell then argues: "Now it is impossible to assign a reason why the successive velocities of a molecule should be bundled in this cycle rather than in the opposite gild" [Maxwell, 1867, p.45]. Therefore, he argues, these two cycles should exist equally likely, and, hence, a collision cycle of the type $\left({\overrightarrow{\upsilon }}_1,{\overrightarrow{\upsilon }}_2\right)\to ({\overrightarrow{\upsilon }}_{\mathrm{ane}}^{\hspace{0.17em}\hspace{0.17em}\prime },{\overrightarrow{\upsilon }}_2^{\hspace{0.17em}\hspace{0.17em}\prime })$ is already equally probable as a collision cycle of the type $({\overrightarrow{\upsilon }}_1^{\hspace{0.17em}\hspace{0.17em}\prime },{\overrightarrow{\upsilon }}_2^{\hspace{0.17em}\hspace{0.17em}\prime })\to \left({\overrightarrow{\upsilon }}_1,{\overrightarrow{\upsilon }}_2\right)$ , i.e. condition (32) holds.

Comments.

First, a clear advantage of Maxwell's 1867 derivation of the distribution function (26) is that the collisions play a crucial role. The argument would non utilize if at that place were no collisions between molecules. A second point to note is that the distribution (26) is singled out considering of its stationarity, instead of its spherical symmetry and factorization properties. This is also a major comeback upon his previous paper, since stationarity is essential to thermal equilibrium.

A crucial element in the statement is still an supposition about independence. But now, in the Stoßzahlansatz, the initial velocities of colliding particles are causeless independent, instead of the orthogonal velocity components of a unmarried particle. Maxwell does not expand on why we should assume this ansatz; he clearly regarded information technology as obvious. Yet it seems plausible to argue that he must have had in the back of his heed some version of the principle of insufficient reason, i.east., that nosotros are entitled to treat the initial velocities of two colliding particles as independent because we have no reason to assume otherwise. Although nonetheless an statement from insufficient reason, this is at least a much more plausible application than in the 1860 paper.

A main defect of the paper is his sketchy claim that the Maxwell distribution (26) would be the unique stationary distribution. This claim may be broken in 2 parts: (a) the cycle statement just discussed, leading Maxwell to debate for detailed balancing; and (b) the merits that the Maxwell distribution is uniquely uniform with this status.

A sit-in for part (b) was not provided past Maxwell at all; merely this gap was soon bridged by Boltzmann (1868) — and Maxwell gave Boltzmann due credit for this proof. Simply function (a) is more than interesting. We have seen that Maxwell here explicitly relied on reasoning from bereft reason. He was criticized on this point by [Boltzmann, 1872] and also past [Guthrie, 1874].

Boltzmann argued that Maxwell was guilty of begging the question. If we suppose that the 2 cycles did not occur equally oft, so this supposition by itself would provide a reason for assigning diff probabilities to the ii types of collisions. ¹⁶ This argument by Boltzmann indicates, at to the lowest degree in my stance that he was much less prepared than Maxwell to argue in terms of bereft reason. Indeed, as we shall encounter in Section 4, his view on probability seems much more thoroughly frequentist than Maxwell.

In fact Boltzmann later on repeatedly mentioned the counterexample of a gas in which all particles are lined up so that they only collide centrally, and movement perpendicularly between parallel walls [Boltzmann, 1872 (Boltzmann, 1909, I p. 358); Boltzmann, 1878 (Boltzmann, 1909, II p. 285)]. In this example, the velocity distribution

(33) $\frac{1}{\mathrm{ii}}(\delta \left(v-{\mathrm{five}}_0\right)+\delta \left(\mathrm{five}+5_0\right))$

is stationary likewise.

Some terminal remarks on Maxwell's work: As we have seen, it is not easy to pinpoint Maxwell's interpretation of probability. In his (1860), he identifies the probability of a particular molecular state with the relative number of particles that possess this state. ¹⁷ Yet, nosotros have also seen that he relates probability to a state of knowledge. Thus, his position may exist characterized as somewhere betwixt the classical and the frequentist view.

Note that Maxwell never fabricated whatever attempt to reproduce the second law. Rather he seems to accept been content with the statistical description of thermal equilibrium in gases. ¹⁸ All his writings after 1867 indicate that he was convinced that a derivation of the Second Law from mechanical principles was incommunicable. Indeed, his remarks on the 2d Police force by and large point to the view that the Second Law "has only statistical certainty" (letter to Tait, undated; [Garber et al., 1995, p. 180]), and that statistical considerations were foreign to the principles of mechanics. Indeed, Maxwell was quite amused to see Boltzmann and Clausius engage in a dispute almost who had been the first to reduce the 2d Constabulary of thermodynamics to mechanics:

It is rare sport to see those learned Germans contending the priority of the discovery that the 2d law of θΔcs is the 'Hamiltonsche Prinzip', […] The Hamiltonsche Prinzip, the while, soars along in a region unvexed by statistical considerations, while the German Icari flap their waxen wings in nephelococcygia ¹⁹ amid those cloudy forms which the ignorance and finitude of human science take invested with the incommunable attributes of the invisible Queen of Sky (letter to Tait, 1873; [Garber et al., 1995, p. 225])

Clearly, Maxwell saw a derivation of the 2d Law from pure mechanics, "unvexed past statistical considerations", as an illusion. This signal appears even more vividly in his thought experiment of the "Maxwell demon", by which he showed how the laws of mechanics could be exploited to produce a violation of the 2nd Constabulary. For an entry in the extensive literature on Maxwell's demon, I refer to [Earman and Norton, 1998; 1999; Leff and Rex, 2003; Bennett, 2003; Norton, 2005].

But neither did Maxwell make whatever try to reproduce the Second Law on a unified statistical/mechanical ground. Indeed, the scanty comments he made on the topic (e.k. in [Maxwell, 1873; Maxwell, 1878b]) rather seem to point in another direction. He distinguishes between what he calls the 'statistical method' and the 'historical' or 'dynamical' (or sometimes 'kinetic') method. These are ii modes of description for the same system. Only rather than unifying them, Maxwell suggests they are competing, or even incompatible — one is tempted to say "complementary" – methods, and that it depends on our ain cognition, abilities, and interests which of the two is appropriate. For example:

In dealing with masses of affair, while nosotros do non perceive the individual molecules, nosotros are compelled to adopt what I have described as the statistical method, and to abandon the strict dynamical method, in which we follow every move by the calculus [Maxwell, 1872, p. 309, emphasis added].

In this respect, his position stands in sharp contrast to that of Boltzmann, who made the project of finding this unified basis his lifework.

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Translational Movement

Paul Davidovits , in Physics in Biology and Medicine (Fifth Edition), 2019

3.4 Range of a Projectile

A trouble that is solved in most basic physics texts concerns a projectile launched at an angle $\theta$ and with initial velocity $v_0$ . A solution is required for the range $R$ , the distance at which the projectile hits the Earth (see Fig. 3.6). It is shown that the range is

Effigy 3.6. Projectile.

(3.17) $R=\frac{v_0^{\mathrm{two}}\sin 2\theta }{\mathrm{1000}}$

For a given initial velocity the range is maximum when $\sin \mathrm{two}\theta =\mathrm{one}$ or $\theta =45°$ . In other words a maximum range is obtained when the projectile is launched at a 45° angle. In that case the range is

(iii.xviii) $R_{\max }=\frac{5_0^2}{g}$

Using this result, we will gauge the distance attainable in broad jumping.

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Line-of-Sight Guidance

North.A. Shneydor B.Sc., Dipl. Ingénieur, M.Sc., Ph.D. , in Missile Guidance and Pursuit, 1998

Example three - Costless-falling T

In this case, an analytic solution of (ii.5) is not possible, and one has to resort to a numerical ane.

Suppose T starts a free autumn at t  =   0 from the point (0, h ) having the initial velocity ${\upsilon }_{T_o}$ in the x management. The y axis is vertical such that the acceleration of gravity g is in the – y direction. As in Example 1, K is LOS-guided from the origin (0,0) to intercept T. Its velocity υ_K is constant, the initial velocity ratio being $M_0={\upsilon }_K/{\upsilon }_{T_{\mathrm{o}}}$ .

From simple mechanics it is known that, provided air-resistance effects are neglected, the trajectory of T is the parabola given by the parametric equations

${\mathrm{ten}}_T\left(t\right)={\upsilon }_{T_0}t,y_T\left(t\right)=h-m{t}^2/2\text{.}$

For the present case, ${\upsilon }_{T_0}$ is called to be $\sqrt{2\mathrm{thou}h}$ ; if, furthermore, h and $\sqrt{h/g}$ are called as the units for deportation and fourth dimension, respectively, one gets the nondi- mensional equations

$x_T=\sqrt{\mathrm{ii}t},y_T=\mathrm{one}-t^2/2\text{.}$

Fig. 2.10 shows the solution for several values of K ₀. There is a minimum value for Thousand ₀ required for intercept, which turns out to be virtually 1.29. However, accelerations a_M increase rapidly with K ₀, while lead angles δ decrease. (The one-time effect is due to the relation $a_{\mathrm{Thou}}={\upsilon }_{\mathrm{1000}}{\dot{\mathrm{\gamma }}}_G=K{\upsilon }_T{\dot{\mathrm{\gamma }}}_M$ ; the latter results from the fact that sin δ is inversely proportional to M, cf. (2.2)).

Effigy ii.10. Trajectories, accelerations a_Chiliad , and pb angles δ, Example iii

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Voltage Contrast Modes in a Scanning Electron Microscope and Their Application

V.Yard. Dyukov , ... M. Schönhense , in Advances in Imaging and Electron Physics, 2016

3 Method of Field Contrast (Trajectory Sensitive Dissimilarity Technique)

Allow u.s.a. consider the paradigm contrast formation of the potential relief in a SEM with a detector providing trajectory selectivity, ie, one that detects secondary electrons leaving the sample surface in a definite direction within some solid angle interval. The simplest detector of this type (Fig. iv) consists of a grid parallel to the object surface (at a distance fifty ₁ from information technology) and a collector with radius r. The filigree is at a positive voltage V ₀. Thus, at that place is an extraction field above the object $E_0=\frac{V_0}{l_{\mathrm{ane}}}$ . The coordinate origin is chosen in the eye of the voltage driblet region on the object with the effective extent 2a.

Fig. 4. Scheme of the secondary electron detector, which realizes the field contrast mode. The current density distribution in the detector aeroplane is presented. The signal change after application of voltage U to the object corresponds to the hatched area.

The center of the detector is shifted forth the Z-centrality by u. The secondary electron current received by the detector forms the detector output signal U _d at a load resistance R. When $U=0$ (the surface is equipotential) the distribution of the secondary emission current density in the detector airplane (dashed line) is symmetric with respect to the Z-axis. When local microfields appear ( $U\ne 0$ ), its tangential component influences the slow secondary electrons and their trajectories become curved and the density distribution becomes shifted (solid line). Therefore, the detected current changes by the value that corresponds to the hatched expanse between the curves, ie, in the situation shown in Fig. four, U _d increases. It is necessary to find the dependence $U_{\mathrm{d}}=F\left(U\right)$ , ie, transfer characteristic of the given detector.

The solution is divided into the following steps:

(one)

The move of an electron emitted with cipher initial velocity from the betoken with the maximal relief field magnitude. The dependence of the electron trajectory shift in the detector plane on the barrier pinnacle U is derived.

(two)

The spatial distribution of the secondary electron density in the detector plane is calculated by taking into business relationship their free energy distribution.

(3)

The current of secondary electrons received by the detector is calculated.

It is also causeless that the object surface potential depends merely on the coordinate x, the grid is characterized by accented transparency and simply deadening secondary electrons are detected.

The motion of secondary electrons in a higher place the object surface takes place nether the combined influence of the extraction field E ₀ and the potential relief microfields forming a barrier with height U. It is convenient to specify the one-dimensional surface potential distribution, which approximates the potential barrier in the form of (Nepijko, Sedov, & Schönhense, 2005)

(21) $\phi \left(x\right)=\frac{U}{\pi }arctan\frac{\mathrm{10}}{a}.$

Then, the potential in the one-half-space above the object will be determined past the expression

(22) $V\left(\mathrm{ten},y\right)=\frac{U}{\pi }arctan\frac{x}{z+a}+z{\mathrm{Eastward}}_0\text{.}$

The components of electric field are adamant by the equations

(23) $\frac{\partial }{\partial x}V\left(\mathrm{ten},z\right)=\frac{U}{\pi }\frac{z+a}{{\mathrm{10}}^{\mathrm{two}}+{\left(z+a\right)}^2}=E_{\mathrm{10}}\text{,}$

(24) $\frac{\partial }{\partial z}\mathrm{Five}\left(\mathrm{ten},z\right)=-\frac{U}{\pi }\frac{x}{{\mathrm{ten}}^{\mathrm{two}}+{\left(z+a\right)}^2}+E_0=E_z+E_0\text{.}$

The tangential component of E _x reaches the maximum of $\frac{U}{\pi a}$ when $\mathrm{ten}=z=0$ . To determine the sensitivity threshold, ie, minimal barrier height U at which can exist registered, the field E ₁₀ defined past the potential relief is assumed to be smaller than East ₀, ie, $E_0\gg {\mathrm{Due\; east}}_{x\text{max}}=\frac{U}{\pi a}\simeq \frac{U}{2a}$ . Then, component East _z with the maximum of $\frac{U}{\mathrm{two}\pi a}$ ( $\mathrm{10}=a$ , $z=0$ ) also meets the inequality $E_{z\text{max}}\ll E_0$ , as $2E_{z\text{max}}={\mathrm{Eastward}}_{x\text{max}}$ . It results in the fact that the electron move along the Z-axis occurs practically under the influence of the outer extraction field E ₀ only, and in further calculations, the commencement term of the correct side (24) will exist neglected. Simplified equations of motion in nonrelativistic approximation are

(25) $\ddot{\mathrm{ten}}=\frac{eU}{\pi \mathrm{yard}}·\frac{z+a}{x^2+{\left(z+a\right)}^2}\text{,}$

(26) $\ddot{z}=\frac{e}{\mathrm{thousand}}{\mathrm{Eastward}}_0\text{.}$

To simplify the solution, the electron displacement along the 10-centrality while the electron passes through the microfield region is estimated. The microfield region dimension forth the Z-axis is a (E ₁₀ is halved at this height). It follows from (26) that

(27) $z=\frac{\mathrm{east}{E}_0}{2m}{t}^2\text{.}$

Assuming that $z=a$ , we obtain that this time equals $\sqrt{\frac{2ma}{\mathrm{eastward}{\mathrm{Due\; east}}_0}}$ . Since from $\frac{m{v}_x^2}{\mathrm{two}}=eU$ the electron velocity Ten-component cannot exceed $\sqrt{\frac{2eU}{\mathrm{thousand}}}$ , the required displacement is $\Delta x_a=2\sqrt{\mathrm{two}\frac{E_{\mathrm{10}}}{{\mathrm{Eastward}}_0}}a\ll a$ . In other words, while the electron passes the microfield region with a feature dimension a, its displacement along the X-axis will be much smaller than the value of a. Thus, on the condition that $E_x\ll E_0$ , the field dependence on x in (25) can be neglected. Substituting $x=0$ in (25) nosotros obtain

(28) $\ddot{x}=\frac{eU}{\pi m\left(z+a\right)}\text{.}$

While integrating (28), it is convenient to alter a variable. Taking into account (27), and $dt=\sqrt{\frac{m}{2e{E}_0}}\frac{dz}{\sqrt{z}}$ the following equation results

(29) $\dot{x}=\underset{0}{\overset{t}{\int }}\frac{\mathrm{due\; east}U}{\pi m\left(z+a\right)}dt=\frac{U}{\pi }\sqrt{\frac{\mathrm{east}}{\mathrm{ii}m{E}_0}}{\int }_0^z\frac{dz}{\left(z+a\right)\sqrt{z}}\text{.}$

As in reality $l_{\mathrm{one}}\gg 2a$ , the upper limit of integration tin can exist extended to ∞ and (29) will be

(30) $\dot{x}=v_{\mathrm{10}}=U\sqrt{\frac{e}{2\mathrm{grand}{E}_{0}a}}\text{.}$

The electron displacement from the Z-axis in the detector plane is the sum of displacements $\Delta x=\Delta x_1+\Delta {\mathrm{10}}_{\mathrm{two}}$ in the regions of uniformly accelerated move fifty ₁ and compatible motility l ₂. Assuming that the electron obtained a momentum corresponding to ${\dot{\mathrm{10}}}_{\infty }$ and was and so moving in a uniform field, we take $\Delta x_{\mathrm{i}}={\dot{x}}_{\infty }{t}_1$ , where t ₁ is the transit fourth dimension of l ₁, determined from (27). Taking into account that $E_0=\frac{5_0}{l_1}$ , $\Delta x_{\mathrm{ane}}=\frac{U}{E_0}\sqrt{\frac{{\mathrm{fifty}}_1}{a}}=\frac{U{l}_{\mathrm{ane}}}{V_0}\sqrt{\frac{l_1}{a}}$ . The time t ₂ of passing the region l _two is determined from the equation $t_{\mathrm{two}}=\frac{l_2}{v_{z\text{max}}}$ , where ${\mathrm{five}}_{z\text{max}}=\sqrt{\frac{\mathrm{two}e{V}_0}{g}}$ is the maximal speed component along the Z-centrality. Information technology results in $\Delta x_2={\dot{x}}_{\infty }{t}_2=\frac{U{\mathrm{50}}_2}{2V_0}\sqrt{\frac{l_1}{a}}$ . After summation we get

(31) $\Delta x=\frac{U}{\mathrm{two}{\mathrm{Five}}_0}\sqrt{\frac{l_1}{a}}\left(2l_{\mathrm{one}}+l_2\right)\text{.}$

Here, the beginning stage of the task solution is finished. It is worth mentioning that the numerical calculations achieved using the same equations of motion without whatsoever simplifying assumptions led to results that agree up to two decimal places with the approximate results for $\mathrm{twoscore}{\mathrm{Due\; east}}_{\mathrm{10}}\le {\mathrm{Eastward}}_0$ .

Calculation of the secondary electron current density distribution in the detector plane, which is determined past the initial velocity spread during the emission should first exist made for the example of an equipotential surface ( $U=0$ ). Any electron that leaves the surface with the component v _xy volition be deflected from the Z-centrality by the value r. This value depends on the time-of-flight through the region $l_1+l_2$ . If $V_0\gg U_{\mathrm{m}}=\frac{W_{\mathrm{k}}}{\mathrm{eastward}}$ , which is certainly fulfilled, the influence of velocity spread along the Z-centrality tin be neglected, and $t=t_1+t_2$ [used previously for the calculation of (31)] can be exploited. Taking into account this condition

(32) $r=v_{xy}t=\sqrt{\frac{m}{2e{\mathrm{Five}}_0}}\left(2l_1+l_2\right)v_{xy}\text{.}$

Electrons that have tangential components of speed in the interval from 5 _xy to $v_{xy}+{d\mathrm{five}}_{xy}$ , hit the ring with a radius r and a width dr in the detector aeroplane. Thus, the current density is

(33) $j\left(r\right)=\frac{\mathrm{Northward}\left(v_{xy}\right)d{v}_{\mathrm{ten}y}}{\mathrm{two}\mathit{\pi rdr}}\text{.}$

Substituting (18) and (32) into (33), nosotros get a Gauss distribution

(34) $j\left(r\right)=\frac{1}{\pi h^2}exp\left(-\frac{r^2}{h^2}\right)\text{,}$

where the effective width of the distribution h is

(35) $h=t\sqrt{\frac{\mathrm{ii}{\mathrm{West}}_{\mathrm{m}}}{\mathrm{one\; thousand}}}=\sqrt{\frac{U_{\mathrm{thou}}}{{\mathrm{Five}}_0}}\left(2l_1+l_2\right)\text{.}$

We assume a total current in the detector aeroplane $i_0=\mathrm{one}$ for normalization of (34). When $U_{\mathrm{m}}=2\mathrm{5}$ ( ${\mathrm{West}}_{\mathrm{m}}=2\mathrm{eV}$ ), $V_0=\mathrm{x}\mathrm{kV}$ , $2{\mathrm{50}}_{\mathrm{one}}+l_{\mathrm{two}}=\mathrm{xxx}\mathrm{mm}$ h is 0.42   mm.

Electrons with zero initial velocity striking the central point of the distribution. We now derive the shift of the beam along the X-centrality under the influence of the object microfield. Within the assumptions ${\mathrm{East}}_0\gg E_x$ and $5_0\gg U_{\mathrm{m}}$ , when the fourth dimension-of-flying through the microfield region is determined past the extraction field value, the whole Gauss distribution of electric current density is shifted past $\Delta x$ , see (31). All the electrons experience the same shift along the Ten-axis under the influence of microfields regardless of their initial speed.

As the detector center is shifted from the Z-centrality by u, the center of the secondary electron density distribution will be displaced by $\mathrm{due\; south}=\Delta x+u$ from the detector axis. The current of the electrons hit the detector (run across Fig. 5) volition be adamant as the integral of the current density over the detector expanse

Fig. 5. Calculation of the value of electron current that passes through the detector pigsty.

(36) $i\left(\mathrm{southward},r\right)=\frac{i_0}{\pi h^{\mathrm{ii}}}{\int }_0^r{\int }_0^{2\pi }exp\left(-\frac{r^{\mathrm{two}}+{\mathrm{southward}}^{\mathrm{two}}-2r\mathrm{due\; south}cos\phi }{h^2}\right)\mathit{rdrd\phi }\text{.}$

Assuming that $s<h$ , $r<h$ (the effective radius of the electric current density distribution exceeds the shift value and detector radius), which is easily accomplished, (36) can be written every bit

(37) $i\left(\mathrm{southward},r\right)=exp\left\{-\frac{s^2}{h^2}\right\}\left(\mathrm{i}-exp\left\{-\frac{r^2}{h^2}\right\}-\frac{s^2}{h^2}\left[\left(\frac{r^2}{h^2}+1\right)exp\left\{-\frac{r^2}{h^{\mathrm{two}}}\right\}-\mathrm{ane}\right]\right)\text{.}$

To determine the transfer characteristic, the shift $s=\Delta \mathrm{ten}+u$ has to be substituted into (37). Taking into business relationship (31) we obtain

(38) $\begin{array}{ll}F\left(U,u\right)& =exp\left\{-{\left(\frac{U}{U_{\mathrm{Chiliad}}}+\frac{u}{h}\right)}^2\right\}\hfill \\ & \times \left(1-exp\left\{-\frac{r^2}{h^2}\right\}-{\left(\frac{U}{U_{\mathrm{G}}}+\frac{u}{h}\right)}^2\left[exp\left\{-\frac{r^{\mathrm{ii}}}{h^2}\right\}\left(\frac{r^2}{h^2}+1\right)-1\right]\right)\text{,}\hfill \end{array}$

where

(39) $U_{\mathrm{G}}=2\sqrt{U_{\mathrm{1000}}{5}_0\frac{a}{l_{\mathrm{ane}}}}=\mathrm{two}{U}_{\mathrm{m}}\sqrt{\frac{{\mathrm{Eastward}}_{0}a}{U_{\mathrm{m}}}}\text{.}$

An analysis of (38) shows that the transfer characteristic shape is close to Gauss distribution and that its width is affected by the value of U _K. It also defines the gradient of the transfer characteristic-inclined regions. The latter in turn defines sensitivity and its threshold. To increase the sensitivity (decrease its threshold), the value of U ₀ must be reduced. Information technology follows from (39) that the dimensionless parameter $\frac{E_{0}a}{U_{\mathrm{m}}}$ has to be decreased for this. Nevertheless, the obtained equations are valid when $\frac{E_{0}a}{U_{\mathrm{m}}}\gg 1$ . This means that for every value of microfield extent iia, the extraction field value Due east ₀ requires optimization. Such optimization can be carried out on the basis of the precise solution when the current density distribution in the detector airplane will not take Gaussian shape. For the estimations, it is reasonable to presume that the status $\frac{E_{0}a}{U_{\mathrm{m}}}=10$ corresponds to the optimal field value. This means that, for an object with the nearly probable energy of secondary electrons $W_{\mathrm{m}}=2\mathrm{eV}$ and extraction field extent $2a=20\mathrm{\mu m}$ , the optimal extraction field value is $E_0=20\mathrm{kV}/\mathrm{cm}$ .

The electron detector radius also needs optimization. Following uncomplicated considerations, it should not exceed one-half the effective radius of the density distribution, ie, $r\le h/2$ . Then the equation for F(U, u) can be simplified

(forty) $F\left(U,u\right)=\left(\mathrm{ane}-exp\left\{-\frac{r^2}{h^2}\right\}\right)exp\left\{-{\left(\frac{U}{U_{\mathrm{Thousand}}}+\frac{u}{h}\right)}^{\mathrm{ii}}\right\}\text{.}$

The pre-exponential factor in (twoscore) shows the fraction of secondary electrons, hitting the detector due to its finite size. The transfer characteristic shape according to (forty) with ${\mathrm{West}}_{\mathrm{grand}}=2\mathrm{eV}$ , $\mathrm{two}r=h=u$ and ${\mathrm{Eastward}}_0=20\mathrm{kV}/\mathrm{cm}$ is shown in Fig. 6. The complete collection of secondary electrons, ie, $U_{\mathrm{d}}=\mathrm{one}$ , will be at $u=0$ , $U=0$ and $r\to \infty$ . If $u=0$ , the feature is symmetric and the information about the polarity of voltage drop on the relief barrier is lost.

Fig. 6. Dependence of the secondary electron detector output bespeak U _d on the potential barrier pinnacle U (transfer feature). It is calculated for the field contrast mode (encounter Fig. iv) with $W_{\mathrm{grand}}=\mathrm{two}\mathrm{eV}$ , $2r=h=u$ , $E_0=20\mathrm{kV}/\mathrm{cm}$ , $a=10\mathrm{\mu m}$ .

Let us show that u has an optimal value. Since sensitivity to a potential relief depends on the shift of the electron detector from the Z-axis, then sensitivity (1) and (xiii), which connects it with the transfer characteristic shape, should be generalized

(41) $C\left(u\right)=\left|\frac{\frac{\partial F\left(U,u\right)}{\partial U}{⃒}_{U=0}}{F\left(0,u\right)}\right|\text{.}$

After substituting (40) into (41) and considering (39) for sensitivity of the field contrast method C _f(u) nosotros get

(42) $C_{\mathrm{f}}\left(u\right)=\frac{\mathrm{two}u}{U_{\mathrm{Grand}}h}=\frac{u}{h\sqrt{U_{\mathrm{m}}{E}_{0}a}}\text{.}$

It follows from (42) that when the detector is moved abroad from the Z-axis, sensitivity increases linearly but the measured current decreases. Sensitivity decreases proportionally to the square root of the extraction field and the most probable energy of secondary electrons.

The expression for the sensitivity threshold (14) can be written as

(43) $\Delta U=\mathcal{N}\sqrt{\frac{2e\Delta f}{\delta i_{\mathrm{p}}}}\frac{{F\left(0,u\right)}^{1/2}}{{\left|\frac{\partial F\left(U,u\right)}{\partial U}\right|}_{U=0}}\text{.}$

After substituting (twoscore) into (43) we become

(44) $\Delta U=\mathcal{N}\sqrt{\frac{\mathrm{two}\mathrm{east}\Delta f}{\delta i_{\mathrm{p}}}}\frac{U_{\mathrm{G}}}{2}\frac{h}{u}exp\left\{\frac{u^2}{\mathrm{two}{h}^{\mathrm{two}}}\right\}{\left(1-exp\left\{-\frac{r^{\mathrm{ii}}}{h^{\mathrm{ii}}}\right\}\right)}^{-\mathrm{i}/2}\text{.}$

The obtained expression has a minimum that is reached with the optimal position of the "operating point" on the transfer characteristic with $u=h$ . Thus,

(45) $\Delta U_{\text{min}}\simeq \mathcal{North}\sqrt{\frac{2e\Delta f}{\delta i_{\mathrm{p}}}}\frac{U_{\mathrm{1000}}}{\sqrt{\mathrm{one}-exp\left\{-\frac{r^2}{h^{\mathrm{ii}}}\right\}}}\text{.}$

And $F\left(u_{\mathrm{opt}}\right)=0.37F_{\text{max}}$ , which means that a displacement of the detector from the Z-axis by $u_{\mathrm{opt}}=h$ leads to a subtract of the "constant component" to 0.37 of the maximal level. For quantitative estimation, we should specify typical values of the parameters actualization in (45): $\mathcal{N}=\mathrm{ii}$ , $\delta =\mathrm{0.two}$ , $\Delta f=\mathrm{ten}\mathrm{Hz}$ , $U_{\mathrm{k}}=2\mathrm{Five}$ , $E_0=20\mathrm{kV}/\mathrm{cm}$ , $a=\mathrm{ten}\mathrm{\mu m}$ , $r=h/2=\mathrm{seventy}\mathrm{\mu m}$ . Within the range of currents 10^{−   12}–10^{−   6}  A, the threshold of sensitivity $\Delta U_{\text{min}}$ is correspondingly 100   mV to 100   μV. Considering relation (2), the probe diameters are 6   nm to 1   μm for the same values of the probe current. The sensitivity at u _opt is $C_{\mathrm{opt}}=\frac{\mathrm{two}}{U_0}\simeq \mathrm{0.sixteen}{\mathrm{5}}^{-1}$ .

The following conclusions event from the fulfilled analysis:

(1)

Qualitative correspondence of images to the microfield distribution on the surface takes place when the detector creates an extraction field exceeding local microfields. However, it has to detect secondary electrons merely in a confined solid angle.

(2)

Due to a bell-shaped transfer characteristic, information most the polarity of barrier potential is lost if the "operating indicate" is in the maximum, and its shift in the inclined region gives rise to the 1-dimensional detection characteristic while imaging a relief.

(3)

With the aforementioned values of tangential field components, more extended fields produce a larger contrast.

(4)

The blurring width of the potential barrier image (at half-height) exceeds the barrier characteristic width 2a. In a weak extraction field ( ${\mathrm{Eastward}}_{\mathrm{10}}\simeq {\mathrm{Eastward}}_0$ ), the dissimilarity maximum position on the paradigm is shifted with respect to the region of the maximal microfield value to a higher potential on the surface.

In Department 4, the methods considered practice non take the previously mentioned drawbacks of the field (trajectory) contrast method.

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Detached-Fourth dimension Control Arrangement Implementation Techniques

X. Rong Li , Yaakov Bar-Shalom , in Command and Dynamic Systems, 1995

C ATC TRACKING Example

This example deals with an application of the IMM algorithm to civilian aircraft tracking in Air Traffic Command (ATC) systems.

Consider the problem of tracking an aircraft flying in the (ten ¹, 10 ² ) airplane, starting with an initial velocity [0 120  m/s]′. The sensor sampling period T is 10 seconds. The aircraft executes a 90° coordinated turn (with a turn rate of 3°/south) to the right in 3 sampling periods, starting at yard  =   7, and then continues direct. This apparently unproblematic only by no means little exam scenario is challenging for ATC tracking in that the deviation from the (nonmaneuvering) uniform motion due to the onset of the maneuver is very large and that the duration of the maneuver is extremely short. For the same reason, this is as well a stringent examination scenario for a performance prediction technique. For such a scenario, the IMM algorithm is the just practical algorithm bachelor that tin provide much better estimates during the uniform motion while maintain estimates not worse than the unfiltered sensor information during the maneuver [46], as well as a rapid and reliable indication of the flying mode.

The second-gild kinematic system (with matrices F, G, and H for each coordinate) for this planar move is, with position but measurements,

(8) $\mathrm{10}\left(\mathrm{grand}+1\right)=\mathit{diag}\left(F,F\right)\mathrm{10}\left(k\right)+\mathit{diag}\left(\mathrm{Yard},G\right)v\left(\mathrm{grand}\right)$

(nine) $z\left(k\right)=\mathit{diag}\left(H,H\right)x\left(\mathrm{1000}\right)+w\left(k\right)$

where

$\mathrm{ten}\left(k\right)={\left[{\mathrm{10}}^{\mathrm{ane}}\left(k\right){\dot{x}}^1\left(k\right){\mathrm{10}}^{\mathrm{ii}}\left(\mathrm{thousand}\right){\dot{x}}^2\left(\mathrm{yard}\right)\right]}^{\prime }$ ` (10)

(11) $z\left(k\right)={\left[z^{\mathrm{ane}}\left(k\right)z^{\mathrm{two}}\left(k\right)\right]}^{\prime }$

(12) $F=\left[\begin{array}{cc}\hfill \mathrm{ane}\hfill & \hfill T\hfill \\ \hfill 0\hfill & \hfill \mathrm{one}\hfill \end{array}\right]G=\left[\begin{array}{c}\hfill \frac{1}{\mathrm{ii}}{T}^2\hfill \\ \hfill T\hfill \end{array}\right]H=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \end{array}\right]$

The covariances of the process and the zero-mean measurement noises are

(13) $Q=\left[\begin{array}{cc}\hfill q\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill q\hfill \end{array}\right]R=\left[\begin{array}{cc}\hfill {\left(100\mathrm{grand}\right)}^{\mathrm{two}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\left(100\mathrm{chiliad}\right)}^{\mathrm{two}}\hfill \end{array}\right]$

The mean of the process noise is

(14) $\overline{v}\left(\mathrm{thousand}\right)={\left[{\overline{v}}^1\left(g\right){\overline{v}}^2\left(k\right)\right]}^{\prime }$

For the true dynamics, qt  =   (0.4   grand/s²)^two and the maneuver (turn) is obtained via piecewise constant accelerations as the mean of the process noise:

(fifteen) $\begin{array}{l}{\overline{\mathrm{five}}}_t\left(7\right)={\left[\mathrm{six}\mathrm{k}/{\mathrm{s}}^2-\mathrm{ane.61}\mathrm{m}/{\mathrm{s}}^{\mathrm{two}}\right]}^{\prime }\\ {\overline{v}}_t\left(8\right)={\left[\mathrm{iv.39}\mathrm{grand}/{\mathrm{s}}^2-4.39\mathrm{thou}/{\mathrm{southward}}^2\right]}^{\prime }\\ {\overline{v}}_t\left(\mathrm{ix}\right)={\left[\mathrm{one.61}\mathrm{one\; thousand}/{\mathrm{south}}^{\mathrm{ii}}-6\mathrm{m}/{\mathrm{s}}^2\right]}^{\prime }\end{array}$

At other times, ${\overline{v}}_t$ is zero. Note that the first time that the effect of the maneuver shows up in the measurements is at k  =   8.

The IMM algorithm used in this example consists of two Kalman filters (respective, respectively, to the maneuvering style (style 1) and the nonmaneuvering way (mode 2)) with $\overline{v}\left(k\right)=\overline{\mathrm{west}}\left(k\right)=0$ and the Q and R values as defined in (13). The two filters differ only in the values of q: q ₁  =   (0.3   grand/due south^two)² (for mode i) and q _two  =   (3   m/s²)² (for mode 2). These q values are not the optimal designs. Yet, they are adopted here to illustrate the capability of the HYCA performance predictor in predicting not only the bodily interpretation errors but also the misjudgment of the IMM algorithm on these errors. The Markov chain transition matrix used in the IMM algorithm is

(16) $\left[{\pi }_{ij}\right]=\left[\begin{array}{cc}\hfill \mathrm{0.nine}\hfill & \hfill 0.1\hfill \\ \hfill 0.33\hfill & \hfill 0.67\hfill \end{array}\right]$

A comparison of the operation predictor with 100 Monte Carlo simulations is shown in Effigy one for the coordinate-combined position RMS estimation errors, in Figure ii for the coordinate-combined velocity RMS estimation errors, and in Figure 3 for the mode probabilities.

Figure 1. Coordinate-combined position RMS estimation errors.

Figure two. Coordinate-combined velocity RMS interpretation errors.

Figure iii. Mode probabilities.

From these figures, information technology is clear that the HYCA performance predictor does predict both the actual average errors and the average error standard deviations calculated by the IMM algorithm in real fourth dimension remarkably well. The predictor-calculated mode probabilities follow very closely their counterparts obtained in existent time in the IMM algorithm and both of them showroom a sudden jump right after the maneuver occurs. Such a expert accurateness, including the transient capture, is something that cannot be expected from whatever mistake leap or analytic model.

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Physically Based Animation

Rick Parent , in Reckoner Blitheness (Third Edition), 2012

A simple example

Using a standard two-dimensional physics case, consider a point with an initial position of (0, 0), with an initial velocity of (100, 100) feet per second, and under the strength of gravity resulting in a uniform acceleration of (0, 232) anxiety per second. Assume a delta time interval of one/30 of a second (corresponding roughly to the frame interval in the National Tv Standards Committee video). In this example, the acceleration is uniformly applied throughout the sequence, and the velocity is modified at each time step by the downward dispatch. For each fourth dimension interval, the average of the beginning and ending velocities is used to update the position of the signal. This process is repeated for each step in time (meet Eq. 7.21 and Figures 7.seven and 7.8).

Figure 7.7. Modeling of a betoken's position at discrete time intervals (vector lengths are for illustrative purposes and are not accurate).

Figure vii.8. Path of a particle in the simple example from the text.

(vii.21) $\begin{array}{l}v\left(0\right)=\left(100,100\right)\\ x\left(0\right)=\left(0,0\right)\\ v\left(\frac{1}{\mathrm{xxx}}\right)=\left(100,100\right)+\left(0,-32\right)\frac{\mathrm{i}}{\mathrm{xxx}}\\ x\left(\frac{\mathrm{i}}{30}\right)=\left(0,0\right)+\frac{1}{\mathrm{two}}\left(v\left(0\right)+v\left(\frac{1}{30}\right)\right)\\ 5\left(\frac{2}{\mathrm{xxx}}\right)=v\left(\frac{1}{30}\right)+\left(0,-32\right)\frac{1}{30}\\ \mathrm{eastward}tc\text{.}\end{array}$

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Stochastic Hydrodynamics

B. Ferrario , in Encyclopedia of Mathematical Physics, 2006

Statistical Solutions

Let u(t, x) be the fluid velocity at time t and signal $x\in D\subseteq {\mathbb{R}}^d;$ since the initial velocity is e'er affected by experimental errors, it is reasonable to assign a measure out ν determining the probability that the initial velocity belongs to a Borel ready Γ of the space $\mathbb{H}$ of all admissible velocity fields u = u(x).

A spatial statistical solution is a family of probability measures μ(t, ·),t ≥ 0, each supported on the prepare $\mathbb{H}$ such that, given any Borel set up Γ in $\mathbb{H}$ , we take

[i] $\text{Prob}\left\{u\left(t,x\right)\in \text{Γ}\right\}=\mu \left(t,\text{Γ}\right),\forall t>0$

with the initial condition $\mu \left(0,\text{Γ}\right)=\nu \left(\text{Γ}\right)$ . The structure and analysis of statistical solutions μ(t,·) is i of the crucial mathematical problems in stochastic hydrodynamics (see, east.g., Vishik and Fursikov (1988)).

Hopf gave the first mathematical formulation of the problem of describing turbulent flows by statistical solutions. The first result on the existence of statistical solutions is by Foias in 1973. Hopf (1952) presented an equation in variational derivatives satisfied by the characteristic functional χ(t,ϕ) of the family of measures μ(t,·) associated with the Navier–Stokes equations. The characteristic functional χ(t,ϕ) is the Fourier transform of the measure μ(t,·):

[2] $\chi \left(t,\varphi \right)={\displaystyle {\int }_{\mathbb{H}}{\text{e}}^{\text{i}〈\varphi ,u〉}\mu \left(t,\text{d}u\right)}$

defined for any smooth test role ϕ.

We now derive the development equation for χ(t,ϕ), by bold that the dynamics takes place in the phase space $\mathbb{H}$ and follows the nonlinear equation

[3] $\frac{\text{d}u}{\text{d}t}=F\left(u\right)$

If u^five (t) is the solution started from v at time t = 0, and so its probability distribution is represented by the time-evolved mensurate μ(t,·). Therefore, nosotros have that

[4] ${\displaystyle {\int }_{\mathbb{H}}{\text{due east}}^{\text{i}〈\varphi ,u〉}\mu \left(t,\text{d}u\right)={\displaystyle {\int }_{\mathbb{H}}{\text{e}}^{\text{i}〈\varphi ,u^v\left(t\right)〉}\mu \left(0,\text{d}v\right)}}$

Differentiating in time, we obtain

[5] $\begin{array}{l}\frac{\text{d}}{\text{d}t}\chi \left(t,\varphi \right)={\displaystyle {\int }_{\mathbb{H}}{\text{e}}^{\text{i}〈\varphi ,u^v\left(t\right)〉}\text{i}〈\varphi ,F\left(u^{\mathrm{five}}\left(t\right)\right)〉\mu \left(0,\text{d}v\right)}\hfill \\ =\text{i}{\displaystyle {\int }_{\mathbb{H}}{\text{e}}^{\text{i}〈\varphi ,v〉}〈\varphi ,F\left(v\right)〉\mu \left(t,\text{d}5\right)}\hfill \end{array}$

The terminal integral is uniquely determined by χ, since the measure out μ(t,·) is uniquely determined by χ(t,ϕ). We denote past Φχ(t,ϕ) the last integral in [v]. The development equation thus obtained for the characteristic functional χ is

[vi] $\frac{\text{d}}{\text{d}t}\chi \left(t,\varphi \right)=\text{i}\text{Φ}\chi \left(t,\varphi \right),\forall \varphi$

This is chosen the Hopf equation associated with the dynamical system [3].

Some other way to analyze the evolution of measures is through the moments; instead of the measure μ(t,·) describing the spatial statistical solution, we deal with the moments of μ(t,·) of any guild. For a nonlinear dynamics [3], the moments equations are an infinite chain of coupled equations, the so-called Friedman–Keller equations.

A prominent role amongst statistical solutions is played by stationary solutions. They comprise all the statistical information in the case of equilibrium in time. We take that the characteristic functional of an invariant measure is constant in time. Therefore,

$\frac{\text{d}}{\text{d}t}\chi \left(t,\varphi \right)=0$

Bearing in heed equation [v], this is equivalent to say that the signed measure $〈\varphi ,F\left(v\right)〉\mu \left(t,\text{d}v\right)$ vanishes, for any test function ϕ and time t. Setting t = 0, we obtain that an invariant mensurate ν in the space $\mathbb{H}$ satisfies the Liouville equation

[seven] ${\displaystyle {\int }_{\mathbb{H}}〈\varphi \left(v\right),F\left(v\right)〉\text{d}\nu \left(5\right)=0}$

for advisable examination functions ϕ. This equation is besides called the relation of infinitesimal invariance and the measure ν is said to be infinitesimally invariant.

The stationary measures are natural candidates to describe the statistical asymptotic behavior of the system when t → ∞. Notice that, in a chaotic system two motions that are arbitrarily shut to one another at t = 0 tin can evolve in completely different ways. So, to depict satisfactorily the dynamics nosotros take average over a large number of experiments. This is the and so-called ensemble boilerplate. These averages are assumed to exist with respect to an invariant mensurate μ. The invariant measures must exist and either they are unique or at nearly ane has physical meaning and enters in the functional integral defining the ensemble average. According to the ergodic principle (an assumption not nonetheless proved in hydrodynamics), ensemble averages supersede long-time averages: for every initial velocity field v, except for a prepare of initial values negligible in some sense, the time boilerplate of an observable ψ tends, as time goes to infinity, to the ensemble average

[8] $\underset{T\to \infty }{\lim }\frac{1}{T}{\displaystyle {\int }_0^T\psi \left(u^v\left(t\right)\right)\text{d}t={\displaystyle {\int }_{\mathbb{H}}\psi \text{d}\mu }}$

However, it is extremely difficult to testify the existence of stationary probability measures for the Navier–Stokes equations solving straight equation [7]. The state of affairs is formally the same equally in equilibrium statistical mechanics, where the Liouville equation is in fact solved, leading to the Boltzmann–Gibbs distribution. Nevertheless, the results in statistical hydrodynamics are far from being satisfactory.

Recent studies to prove the existence of invariant measures for the Navier–Stokes equations are based on stochastic models (see the section "The stochastic Navier–Stokes equations"). On the other paw, for the Euler equations it is possible to construct formally invariant measures, by means of invariant quantities of the classical motion (see the next section).

Finally, we point out that there are techniques using invariant measures to testify some results for the time evolution (e.chiliad., the motion exists for almost all initial values with respect to an invariant mensurate).

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Handbook of Chemometrics and Qualimetrics: Part B

B.G.Thou. Vandeginste , ... J. Smeyers-Verbeke , in Data Handling in Scientific discipline and Technology, 1998

39.4 Linearization of not-linear models

A classical non-linear model of chemical kinetics is defined past the Michaelis–Menten equation for rate-limited reactions, which has already been mentioned in Section 39.1.1:

(39.114) $V=-\frac{\mathrm{d}X}{\mathrm{d}t}=\frac{V_{\max }X}{K_{\mathrm{chiliad}}+X}$

where V represents the rate (or velocity) of the process and 10 represents the amount (or concentration) of the substrate.

The parameters of this model are the maximal charge per unit of the procedure V _max and the saturation constant K _k. The latter can also be defined as the corporeality of substrate which produces a rate V which is half the maximal rate V _max, as tin be verified past substituting K _m by X in the higher up relation.

Usually, one plots the initial charge per unit V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The charge per unit and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis–Menten equation can exist linearized past taking reciprocals on both sides of eq. (39.114) (Section 8.ii.xiii), which leads to the and then-called Lineweaver–Burk class:

Fig. 39.17. Schematic illustration of Michaelis–Menten kinetics in the absence of an inhibitor (solid line) and in the presence of a competitive inhibitor (dashed line). (a) Plot of initial rate (or velocity) Five against amount (or concentration) of substrate 10. Note that the two curves tend to the same horizontal asymptote for large values of 10. (b) Lineweaver–Burk linearized plot of 1/V against i/10. Note that the two lines intersect at a common intercept on the vertical axis.

(39.115) $\begin{array}{l}\frac{\mathrm{one}}{V}=\frac{K_{\mathrm{g}}}{V_{\max }}\frac{\mathrm{i}}{X}+\frac{1}{V_{\max }}\hfill \\ \mathrm{slope}={\mathrm{Yard}}_{\mathrm{m}}/V_{\max }\hfill \\ intercept=\mathrm{one}/V_{\max }\hfill \end{array}$

The parameters Five _max and Chiliad _yard can exist obtained from the intercept and gradient of the linear relationship between ane/Five and 1/X as shown in Fig. 39.17b.

To the lowest degree-squares linear regression (Affiliate 8) of 1/5 upon ane/10, although a unproblematic solution to the trouble, suffers from a lack of robustness which is induced by the reciprocal transformation of V and X. Serious heteroscedasticity (or lack of homogeneity of variance) occurs in the differences betwixt observed and computed values of 1/V. Indeed, the variances of these residuals tend to increase with large values of 1/X. (It should exist noted that the variance of one/5 is approximately equal to the variance of V divided by the 4th power of V, where 5 becomes vanishingly small when 10 tends to naught.) This could be remedied by means of weighted linear regression, in which each indicate is assigned a weight which is inversely proportional to the corresponding variance. (Heteroscedasticity and weighted regression have been explained in Department 8.2.3.) Unfortunately, the distribution of 1/V at the diverse levels of 1/Ten is commonly unknown. It has been recommended, therefore, to use distribution-complimentary methods of linear regression[xiv]. A robust procedure is to compute the intercept and slopes of the linear relationship by ways of the single median regression method (Section 12.1.5.1).

Another linearized class of the Michaelis–Menten equation is defined past:

(39.116) $\begin{array}{l}\frac{\mathrm{10}}{\mathrm{Five}}=\frac{\mathrm{i}}{5_{\max }}\mathrm{10}+\frac{{\mathrm{Chiliad}}_{\mathrm{m}}}{V_{\max }}\hfill \\ \mathrm{slope}=1/V_{\max }\hfill \\ intercept=G_{\mathrm{m}}/V_{\max }\hfill \end{array}$

This variant can be derived from the Lineweaver-Burk course in eq. (39.115) by multiplying both sides past X. From a statistical signal of view, it does not seem to have an reward over the Lineweaver-Burk form [fourteen]. The latter variant, still, can be more than hands extended to more circuitous systems of substrate-enzyme reactions, every bit will be shown below.

In the case of competitive inhibition, the substrate is displaced past a substance which has greater affinity for the enzyme (or receptor protein) than its natural substrate. For example, a competitive inhibitor (or antagonist) will endeavour to occupy the bounden sites such that the enzyme is prevented from exerting its normal action on the substrate. Information technology is assumed here that the binding between inhibitor and enzyme is reversible. The inhibiting activeness depends on the corporeality (or concentration) Y of the competing substance and on the inhibition constant M _i of the inhibitor–enzyme complex. The potency of a competitive inhibitor is inversely proportional to its M _i value. The human relationship between the rate of the enzymatic reaction V and the amount of substrate X can be expressed in a linearized form:

(39.117) $\begin{array}{l}\frac{1}{\mathrm{Five}}=\left(1+\frac{Y}{K_{\mathrm{i}}}\right)\frac{{\mathrm{Chiliad}}_{\mathrm{g}}}{V_{\max }}\frac{\mathrm{i}}{X}+\frac{1}{V_{\max }}\hfill \\ \mathrm{slope}=\left(1+Y/{\mathrm{Thou}}_i\right)K_{\mathrm{m}}/{\mathrm{Five}}_{\max }\hfill \\ intercept=\mathrm{i}/V_{\max }\hfill \end{array}$

Note the shut analogy with the Lineweaver–Burk class of the uncomplicated Michaelis–Menten equation. In a diagram representing ane/5 against 1/Thousand one obtains a line which has the same intercept as in the elementary example. The gradient, however, is larger by a cistron (1   + Y/Thou _i) as shown in Fig. 39.17b. Commonly, one first determines V _max and K _thousand in the absence of a competitive inhibitor (Y  =   0), as described above. Subsequently, one obtains Thou _i from a new set up of experiments in which the initial charge per unit V is determined for various levels of X in the presence of a stock-still amount of inhibitor Y. The slope of the new line tin be obtained by ways of robust regression.

The cases of not-competitive inhibition and even more circuitous non-linear reaction kinetics will non exist discussed farther here.

Example

We consider the initial velocities V, observed with different substrate concentrations 10 in a rate-limited enzymatic reaction[xv]:

10 (mM)	1.25	ane.67	2.50	5.00	10.0	20.0
5 (mg/min)	0.101	0.130	0.156	0.250	0.303	0.345

Ordinary to the lowest degree squares regression of 1/5 upon 1/10 produces a slope of 9.32 and an intercept of two.36. From these we derive the parameters of the simple Michaelis–Menten reaction (eq. (39.116)):

$\begin{array}{ll}{V}_{\max }\hfill & =0.424\mathrm{mg}/\min \hfill \\ G_{\mathrm{k}}\hfill & =3.95\mathrm{m}\mathrm{M}\hfill \end{array}$

Nosotros at present consider the case of a competitive inhibitor which has been added to the higher up reaction at the fixed concentration of 40   mM [15]. The post-obit initial velocities of the competitively inhibited Michaelis–Menten process are observed at the same substrate concentrations equally above:

X (mM)	1.25	one.67	2.50	5.00	10.0	xx.0
V (mg/min)	0.061	0.074	0.106	0.169	0.227	0.313

Ordinary to the lowest degree squares regression of 1/V upon 1/X yields a slope of 17.seven and an intercept of 2.46. Using the previously derived values for V _max and K _m, and setting Y equal to forty, we tin derive the inhibition constant of the competitively inhibited Michaelis–Menten reaction (eq. (39.117))

$K_{\mathrm{i}}=44.4\mathrm{m}\mathrm{One\; thousand}\text{.}$

Non-linear models, such equally described by the Michaelis–Menten equation, tin can sometimes exist linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the utilise of non-linear regression tin be obviated. As we take pointed out, the price for this convenience may accept to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5).

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