Guest Question.

1. Show that the evolution equation of a harmonic oscillator of mass M, subjected to an elastic force $$F^{el} = −k r$$ and free of friction, is invariant under time reversal.
2. Demonstrate that the evolution equation of a damped harmonic oscillator of mass M, subjected to an elastic force $$F^{el} = −k r$$ and a viscous friction force $$F^{fr }= −b v$$, is not invariant under time reversal.

Solution

1. According to the definition (1.16) of momentum, the equation of motion (1.13) applied to the harmonic oscillator is written as,

P = Mν,

$$\dot{P} =F^{ext}$$.

−k r = Ma

According to the transformation laws (2.4)

r → r
ν → − ν
a → a

for the position r and acceleration a, the equation of motion is invariant under time reversal. Thus, this evolution is reversible.

2. The equation of motion of the damped harmonic oscillator is written,

−k r − bν = Ma

According to the transformation laws (2.4) for the position r, the velocity v and the acceleration a the equation of motion becomes,

−k r + b v = Ma

under time reversal. This equation is different from the previous one, because the sign of the second term on the left-hand side changed, which implies that the equation of motion is not invariant under time reversal. Thus, this evolution is not reversible. The irreversibility is due to the viscous friction force.