0

Consider an ordinary differential equation (ODE) of a 1D damped oscillator of the form

$$\ddot{x}+\gamma\dot{x}+\omega^2x=0.~~(\omega^2,\gamma>0)$$

I want to know if this ODE is reversible i.e., given the information at present i.e. $x(0)=x_0$ and $\dot{x}(0)=v_0$, I want to know whether we can reconstruct the past uniquely. If yes, I would like to see how. I think one needs to make the change $t\to -t$ in the original ODE and then solve for $x(t)$ to predict the past.

If not, I would like to understand why. The answer here by Vincent Fraticelli suggests to me that the time-reversed solution uniquely predicts the past. But I may be wrong.

Qmechanic
  • 201,751
Solidification
  • 11,483

2 Answers2

2

I )

$$ {\frac {d^{2}}{d{t}^{2}}}x \left( t \right) +{\gamma}\,{\frac {d}{dt}} x \left( t \right) +{\omega}^{2}x \left( t \right)=0\tag 1 $$

you can solve this differential equation and obtain the solution $x(t)$ and $x(t\mapsto -t)$

II )

for $~\tau=-t~$ from equation (1) you obtain:

$${\frac {d^{2}}{d{\tau}^{2}}}x \left( \tau \right) -{\gamma}\,{\frac {d }{d\tau}}x \left( \tau \right) +{\omega}^{2}x \left( \tau \right) $$

if the ODE is reversible the solution $~x(\tau)~$ must be equal to $x(-t)$ , this is the case if the initial velocity $\dot x(0)=0~$ equal zero, in this case is the ODE reversible!!

Eli
  • 11,878
  • @mithusengupta123 this ODE is reversible my mistake I forgot to substitute $\tau~$ back to t!! – Eli Sep 04 '21 at 17:14
1

One can solve this equation exactly and obtain the values of $x,v$ at any past moment... unless we are at the special point $x=v=0$. But this is not what we call reversibility of fundamental laws, since damping here is a phenomenological force.

Roger V.
  • 58,522
  • Is it fair to say that by time-reversing this equation though we are able to predict the past of the oscillator, we are unable to predict the past of the other degrees of freedom (which are outside this phenomenological description)? – Solidification Sep 04 '21 at 06:23
  • @mithusengupta123 yes, this is it! Note also that reversibility is not the same as reconstructing the past: if we take $t\rightarrow -t$ then instead of damped oscillator we will have one with growing amplitude. – Roger V. Sep 04 '21 at 06:37