Question

In classical mechanics, the Hamiltonian is well defined by the Lagrangian. Whereas, energy is a very ambiguous term. We just say E=T+U

, and usually it equals to Hamiltonian. Does there exist a way that, by just looking at the Lagrangian mathematically, we immediately know the relationship between the Hamiltonian and the energy of the system?

And if we have a system, the Hamiltonian of which does not equal to energy, what is the physical meaning of that difference?

Answer 1

The Hamiltonian is related to the energy of the system and can be calculated using the equation H = T + U. However, in some cases, the Hamiltonian may not be equal to the energy of the system.

Step-by-step explanation:

In classical mechanics, the Hamiltonian is related to the energy of the system. The Hamiltonian is defined as the sum of the kinetic energy and potential energy, which is represented by the equation H = T + U. This equation allows us to calculate the total energy of a system. However, it is important to note that in certain cases, the Hamiltonian may not be equal to the energy of the system. When this happens, the difference between the Hamiltonian and the energy can have physical implications specific to the system being studied.