Hamiltonian Formulation

The Hamiltonian formulation is a mathematical approach in classical mechanics and quantum mechanics that provides an alternative description of the dynamics of a physical system. It is named after Sir William Rowan Hamilton, who developed the formalism in the 19th century.

In classical mechanics, the Hamiltonian formulation is an equivalent but often more convenient way to describe the motion of particles or systems compared to the more familiar Lagrangian formulation. The key concept is the Hamiltonian function, denoted by H, which is defined as the Legendre transformation of the Lagrangian function. The Lagrangian, L, represents the difference between the kinetic and potential energies of a system and is expressed as the difference between these energies:

L=T−U

where T is the kinetic energy and U is the potential energy.

The Hamiltonian function is defined as:

∑H=∑i​pi​q˙​i​−L

where pi​ are the canonical momenta corresponding to the generalized coordinates qi​, and q˙​i​ is the time derivative of qi​.​

Substituting 2 in 1

Since g^𝛼𝛽 = g^𝛽𝛼 and interchanging dummy index 𝛽 by 𝛼, the 1st and 2nd term are same. Hence,

In the Hamiltonian's formulation we see that

1. It doesn't give total energy

2. H =0

In spite of these drawbacks we can derive Hamiltonian's equation. The Hamiltonian equation are

In quantum mechanics, the Hamiltonian operator plays a central role. The time-dependent Schrödinger equation, which describes the evolution of a quantum state.

The Hamiltonian formulation is a powerful tool in theoretical physics and is widely used in various branches, including classical mechanics, quantum mechanics, and field theory. It provides insights into the energy conservation and symmetries of physical systems.

This note is a part of the Physics Repository.