Question

A mathematical model for the position x(t) of a moving object is $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{=}\mathbf{0}$Use a numerical solver to graphically investigate the solutions of the equation subject to $\mathit{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{x}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{1}}\mathbf{⩾}\mathbf{0}$. Discuss the motion of the object for $\mathit{t}\mathbf{⩾}\mathbf{0}$and for various choices of x. Investigate the equation

$\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{=}\mathbf{0}$

in the same manner. Give a possible physical interpretation of the dx /dt term.

Short answer

The solution of the above question is isthat you can use WolframAlpha’s Runge Method.

Step-by-step solution

Motion in a force field:

We consider the motion of a particle in a random isotropic force field. Assuming that the force field arises from a Poisson field in , d ≥ 4, and the initial velocity of the particle is sufficiently large, we describe the asymptotic behaviour of the particle

Solving using graphs:

Graphs of motions are provided:

$\frac{d^{2}x}{d{t}^2}+\sin \left(x\right)=0$

The motion of the given equation appears to be periodic:

The graphs for x₁=x’(0)=0.5,1.0,1.5 are shown in the given figure;

$\frac{d^{2}x}{d{t}^2}+\frac{dx}{dt}+\sin \left(x\right)=0$

The graphs for x₁=x’(0)=0.5,1.0,1.5 are shown in the given figure;

Hence, the final answer isthat you can use WolframAlpha’s Runge Method.