Analysis of a simple harmonic oscillator

[math] \def\MYdef{\mathrel{\stackrel{\rm def}=}} \def\({\left(} \def\){\right)} [/math]

Here is the program showing the dependence of the speed of movement of the sinker on a linear spring. For starters to get this dependency.

We write the Hooke's law:

[math] F = - Cu, [/math]

[math] m \ddot u + Cu = 0. [/math]

Divide by [math]m[/math] and multiply by [math]\dot u[/math]:

[math] \ddot u \dot u + \omega^2 \dot u u = 0, \quad \omega_0 \MYdef \sqrt{\frac{C}{m}}, [/math]

[math] \dot u^2 + \omega^2 u^2 = 0, \quad \(x^2(t)\)'_t = 2x(t)\dot u(t), [/math]

[math] \(\frac{\dot u}{\omega}\)^2 + u^2 = 0. [/math]

We introduce the notation:

[math]\frac{\dot u}{\omega} = y, \quad u = x[/math].

We get the formula of the circle: [math] y^2 + x^2 = 0 [/math].

We get the formula of the circle:

Разработчики: D.V. Tsvetkov, A.M. Krivtsov.