Answer to Question #140654 in Mechanics | Relativity for Sneha

Question

What is shape of phase space trajectory of classsical harmonic oscillator.

Answer 1

The simplest Autonomous vibrating system is called "harmonic oscillator"; its dynamics is described by linear differential equation of the form:

${\displaystyle {\ddot {x}}+\omega _{0}^{2}x}=0$

Such a system performs periodic sinusoidal (harmonic) movements; oscillatory motion does not occur only in the case ${\displaystyle x_{0}=0}$ $x_{0}=0$ and ${\displaystyle {\dot {x}}_{0}=0}$ ${\displaystyle {\dot {x}}_{0}=0}$ , that is, when the oscillator is initially in a state of equilibrium — in this case, it continues to remain in it. The coordinate equation of the phase trajectory of such a system defines integral curves in the form of a family of similar (with a constant ratio of axes) ellipses, with one and only one ellipse passing through each point of the phase path. This state of equilibrium is a special point of this system, namely, the center. Sourse:https://en.wikipedia.org/wiki/Phase_space