Proving Hamiltonian Invariance with Goldstein Problems

[Irid]

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:

$$L' = L + \frac{dF(q_i,t)}{dt}$$

Now I'm trying to prove that Hamilton equations of motion are not affected by this type of transformation.

Homework Equations

Hamiltonian:

$$H = \dot{q}_i p_i - L$$


Total time derivative:

$$\frac{dF(q_i,t)}{dt} = \frac{\partial F}{\partial q_i} \dot{q_i} + \frac{\partial F}{\partial t}$$

Canonical momentum:

$$p_i = \frac{\partial L}{\partial \dot{q}_i}$$

The Attempt at a Solution

Using the definition of canonical momentum we immediately see that the new canonical momentum is

$$p_i' = \frac{\partial L'}{\partial \dot{q}_i} = p_i + \frac{\partial F}{\partial q_i}$$

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

$$H' = \dot{q}_i p_i' - L' = H - \frac{\partial F(q_i, t)}{\partial t}$$

It satisfies one of Hamilton's equations of motion

$$\dot{q}_i' = \frac{\partial H}{\partial p_i} = \dot{q}_i$$

but fails for the second one,

$$\dot{p}_i' = -\frac{\partial H}{\partial q_i} = \dot{p}_i + \frac{\partial^2 F}{\partial q_i \partial t}$$

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.

 

[siddharth]

But wait a moment! If the canonical momentum is altered, isn't the motion going to be effected? I'm missing something here...

Well, the Euler-Lagrange equation is what gives you the equation of motion, and that is invariant on adding the total derivative. So, the motion shouldn't be affected, right?

Also, for the last couple of steps, shouldn't you find

$$\frac{\partial H}{\partial {p_i'}}$$

as the canonical momentum is defined by the new lagrangian?