View source for Analysis of a simple harmonic oscillator

[[ru:Исследование простейшей линейной колебательной системы]]
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]].

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