A common exposition of Hamiltonian mechanics (e.g., Goldstein's Classical Mechanics) is to start with the action(adsbygoogle = window.adsbygoogle || []).push({});

[tex]S=\int{L dt }[/tex]

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?

[tex]S=\int{(\sum{p_j dq_j}-H(p_i,q_i)dt)}[/tex]

And do we have to rewrite it as

[tex]S=\int{(\sum{p_j \frac{dq_j}{dt}}-H(p_i,q_i))dt}[/tex]

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.

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# How Hamilton's equations extremize the action

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