How Hamilton's equations extremize the action

[pellman]

A common exposition of Hamiltonian mechanics (e.g., Goldstein's Classical Mechanics) is to start with the action

$$S=\int{L dt }$$

then show that the Euler-Lagrange equations extremize the action, then define the hamiltonian in terms of the lagrangian, then translate the Euler-Lagrange equations into the equivalent Hamilton equations.

But what does it look like using the Hamiltonian from the beginning?

$$S=\int{(\sum{p_j dq_j}-H(p_i,q_i)dt)}$$

And do we have to rewrite it as

$$S=\int{(\sum{p_j \frac{dq_j}{dt}}-H(p_i,q_i))dt}$$

to accomplish it? I am eventually going to be relating it to some relativistic problems in which t becomes a coordinate.

Any info and/or link is appreciated.

 

[dx]

Just perform an independent variation of coordinates and momenta on the action as you have written it, with $\delta q = 0$ at the end points, and equate the first order change to zero. It will lead you to Hamilton's equations.

 

[pellman]

Thanks.