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In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?

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OP's question(v1) asks precisely the opposite of this Phys.SE question. –  Qmechanic Oct 31 '12 at 17:54

3 Answers 3

The Hamiltonian in a conservative system describes the total internal energy of the system.

The formula $H=T+V$ with the traditional form of the kinetic energy is valid for a frictionless nonrelativistic system in Cartesian coordinates, possibly with external forces.

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The Hamiltonian is a constant of the motion as long as it is independent of time. More deeply that means that the Lagrangian it comes from must be independent of time.

A constant Hamiltonian is the total energy if the potential is velocity independent.

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But $H=T+V$ also holds for a forced harmonic oscillator, where $H$ is not a constant of the motion. –  Arnold Neumaier Nov 1 '12 at 14:09

By definition, the Hamiltonian is always related to the total energy of the system via $\langle E \rangle = \mathrm{Tr}\{ H \rho \}$. But depending on what do you mean by $T$ and $V$ you equation may or may not be general.

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