Analysis of a simple harmonic oscillator

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We developed a program that calculates the actual velocity of a mass on a linear spring. [math] \def\MYdef{\mathrel{\stackrel{\rm def}=}} \def\({\left(} \def\){\right)} [/math] The Hooke's law has the form:

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

Dividing by [math]m[/math] and multiplying by [math]\dot u[/math] we arrive at:

[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]

By introducing notation

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

We obtain: [math] y^2 + x^2 = 0 [/math].

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