Hamiltonian

The Hamiltonian is equivalent to the below

$$E=H=T+U$$

It is the total energy within the system. See equation 7.130 in the theoretical mechanics textbook for a derivation.

The Hamiltonian is equivalent to the total energy $E$ if the following conditions are met:

• The equations connecting generalized to rectangular/whatever-system-you’re using have to be non-time dependent
• Potential energy can’t be velocity dependent