- #1
jjustinn
- 164
- 3
When solving virtually any non-trivial system via Hamilton's equations, it seems that I'm ending up with equations of the form
## ∂H/∂p = dx/dt = p/m ##
## ∂H/∂x = -dp/dt = -F(x) ##
## p(t) = ∫(dp/dt)dt ##
## d^2x/dt^2 = F(x) ##
E.g., everything is given as a function of x, but we need to solve for t. I can change the variable of integration to x and get e.g. p(x), but that doesn't really help (especially since you can often directly solve for p(x)). One particular problem that's been irking me is where F(x) is either a delta function or a step function.
So...are there any tricks for solving Hamilton's equations (or Euler/Lagrange) for systems that are NOT like the ## d^2x/dt^2 = -Kx ## that invariably show up in textbooks?
## ∂H/∂p = dx/dt = p/m ##
## ∂H/∂x = -dp/dt = -F(x) ##
## p(t) = ∫(dp/dt)dt ##
## d^2x/dt^2 = F(x) ##
E.g., everything is given as a function of x, but we need to solve for t. I can change the variable of integration to x and get e.g. p(x), but that doesn't really help (especially since you can often directly solve for p(x)). One particular problem that's been irking me is where F(x) is either a delta function or a step function.
So...are there any tricks for solving Hamilton's equations (or Euler/Lagrange) for systems that are NOT like the ## d^2x/dt^2 = -Kx ## that invariably show up in textbooks?