Question

What is the Hamiltonian for a system, and how does it relate to the Lagrangian?

Answer 1

In classical mechanics, the Hamiltonian is a function that summarizes the total energy of a system in terms of the generalized coordinates and momenta of the system. It is denoted by the symbol $H$ and is defined as:

$H=T+V$

where:

• $T$ represents the total kinetic energy of the system.
• $V$ represents the potential energy of the system.

The Hamiltonian is closely related to the Lagrangian, which is another important concept in classical mechanics. The Lagrangian, denoted by $L$, is defined as the difference between the kinetic and potential energy of a system:

$L=T-V$

The Lagrangian is a function of the generalized coordinates (e.g., position, velocity) and their time derivatives. It is used to derive the equations of motion for a system using the principle of least action, known as the Euler-Lagrange equations.

The relationship between the Hamiltonian and Lagrangian can be established through a transformation called the Legendre transformation. The Legendre transformation involves taking the partial derivative of the Lagrangian with respect to the velocities and expressing them in terms of the momenta. The resulting expression, when substituted into the Lagrangian, gives the Hamiltonian function.

Mathematically, the relationship between the Hamiltonian and Lagrangian is as follows:

$H={\sum }_i\frac{\partial L}{\partial \dot{q_i}}\dot{q_i}-L$

where:

• $q_i$