Physical Significance of Hamiltonian

Like Lagrangian, H also posses the dimensions of energy but in all circumstances it is not equal to the total energy E. Restriction for this equality, i.e., E= H are:

1. The system be conservative one, i.e. P.E is coordinate dependent and not velocity dependent and

2. Coordinate transformation equations be independent of time, so that ∑pjqj = 2T

There conditions are necessary for Hamiltonian H to represent the total energy E. It may happen that though coordinate transformation equation involve time explicitly yet L be still independent of time, then

This note is taken from Classical Mechanics, Msc physics, Nepal.