I have 4 time evolution equations and the Hamiltonian $H(X_{1},X_{2},P_{1},P_{2})$ that generates the time evolution depends on 4 canonical coordinates but I would like to solve the differential equations(time evolution equations) but I cant be sure if the canonical coordinates depend on time or not. Here is the first evolution equation;

$$\frac{d X_{1}}{d t}=\frac{\partial H}{\partial P_{1}}=X_{2}=v$$

What I am missing is whether $X_{2}$ depends on time or not because if it does I can not solve the differential equation $\frac{d X_{1}}{d t}$. If opposite it's trivial to solve.

Thank you.

  • $\begingroup$ You should solve a system of four equations, not only the equation for $X_1$. There is equation $dX_2/dt = \dots$ and $X_2$ almost surely depends on $t$. $\endgroup$ – Gec Mar 15 at 15:47
  • $\begingroup$ Well if $\mathrm dX_2/\mathrm dt\neq0$, would it depend on time or no? $\endgroup$ – Kyle Kanos Mar 15 at 15:47
  • $\begingroup$ dX_2/dt is nonzero so it depends on time, actually they are Ostrogradsky's choices for the time evolution equations but then I do not know how to solve the differential equations because each one of them depends on other canonical coordinates and time. $\endgroup$ – TheDreamer Mar 15 at 15:54
  • $\begingroup$ Well you may have to do it numerically... $\endgroup$ – Kyle Kanos Mar 15 at 16:06
  • $\begingroup$ If your equations are linear with constant coefficients, then there are known standard methods how to solve systems of such equations. $\endgroup$ – Gec Mar 15 at 16:11

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