HOMEWORK8

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1.Abstract

This program computes general form for the equation of motion of harmonic motion with Eular-Cromer method.

2.equation and algorithm

For simple harmonic motion, the general form of the equation of motion is

$$\frac{d^2x}{dt^2}=-kx^\alpha$$
with $\alpha=1$. If we modify the parameter $\alpha$, we can get different result.
We use the Eular-Cromer method to find the numerical solution of the equation.
$$\frac{dx}{dt}=v$$
$$v_{i+1}=v_i-kx_i^{\alpha}dt$$
$$x_{i+1}=x_i+v_{i+1}dt$$
This is the complete code

3.result

I choose $k=10$, time interval $dt=0.1$, and $\alpha=1,3$, then I draw the two curves in the same coordinate system.

With the parameter $\alpha$ increasing, the curve becomes zigzag gradually.And $\alpha$ must be integer and odd, unless the method is invalid.