- #1
Irid
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Homework Statement
I'm solving Goldstein's problems. I have proved by direct substitution that Lagrange equations of motion are not effected by gauge transformation of the Lagrangian:
[tex]L' = L + \frac{dF(q_i,t)}{dt}[/tex]
Now I'm trying to prove that Hamilton equations of motion are not affected by this type of transformation.
Homework Equations
Hamiltonian:
[tex]H = \dot{q}_i p_i - L[/tex]
Total time derivative:
[tex]\frac{dF(q_i,t)}{dt} = \frac{\partial F}{\partial q_i} \dot{q_i} + \frac{\partial F}{\partial t}[/tex]
Canonical momentum:
[tex]p_i = \frac{\partial L}{\partial \dot{q}_i}[/tex]
The Attempt at a Solution
Using the definition of canonical momentum we immediately see that the new canonical momentum is
[tex]p_i' = \frac{\partial L'}{\partial \dot{q}_i} = p_i + \frac{\partial F}{\partial q_i}[/tex]
But wait a moment! If the canonical momentum is altered, isn't the motion going to be effected? I'm missing something here... Anyway, we go on further to show that the new Hamiltonian is
[tex]H' = \dot{q}_i p_i' - L' = H - \frac{\partial F(q_i, t)}{\partial t}[/tex]
It satisfies one of Hamilton's equations of motion
[tex]\dot{q}_i' = \frac{\partial H}{\partial p_i} = \dot{q}_i[/tex]
but fails for the second one,
[tex]\dot{p}_i' = -\frac{\partial H}{\partial q_i} = \dot{p}_i + \frac{\partial^2 F}{\partial q_i \partial t}[/tex]
Now I'm a little lost... I don't know how to prove the invariance, and the most disturbing part is that the canonical momentum is clearly not invariant.