# Linearity of time evolution in classical mechanics

• Anupama

#### Anupama

I came to know that time evolution in classical mechanics is highly non linear. Is there any case that it become linear?

You get linear equations of motion for the important case of harmonic oscillators. The EoM reads
$$m \ddot{x}+2 m \gamma \dot{x}+m\omega^2 x=F,$$
where ##F=F(t)## is an external force, ##\gamma## the damping, and ##\omega## the eigenfrequency of the (undamped) oscillator.

It's among the most simple equations of state, and you should carefully study its solutions. It's often a good approximation for the bound motion around the minimum of a more complicated potential, if the deviation from this stable fix point doesn't become too large (small amplitudes of oscillations).

• Anupama
Well, linearity is ensured if you can define a potential ##V## such that ##L=T-V## and this potential is at most quadratic in ##x##.
As stated in the post above, the (possibly driven and damped) harmonic oscillator is the standard (and probably the most important) example, since every potential can be written locally around its minimum as a quadratic potential (Taylor series). Hence, the harmonic oscillator is a good approximation for any (conservative) system around its stable equilibrium.
If you later on study quantum mechanics, you will also come across the harmonic oscillator several times.

• Anupama