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What is the geometric interpretation for the Hamiltonian? Also, is there geometric interpretation of when and why it is not equal to the total energy of the system?
Lastly, what is the most general meaning of the Hamiltonian if it only corresponds to the total energy of the system at specific cases?

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  • $\begingroup$ 1. Related: physics.stackexchange.com/q/11905/2451 , physics.stackexchange.com/q/37725/2451 and links therein. 2. More on Hamiltonian vs. Energy. 3. For how to construct a Hamiltonian operator for a point-particle in Riemannian geometry, see e.g. my Phys.SE answer here. $\endgroup$
    – Qmechanic
    Sep 10 '16 at 16:40
  • $\begingroup$ symplectic geometry is one natural context - the momentum map there, for example, interprets the noether theorem; another is fibre bundles. $\endgroup$ Sep 10 '16 at 22:22
  • $\begingroup$ @MoziburUllah A geometric answer is what I am looking for here. If you could write an answer in a simple way(as I do not know sympletic geometry at the moment) it would be greatly appreciated! $\endgroup$ Sep 11 '16 at 7:02

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