Question

What is Lagrangian and Hamiltonian?

Answer 1

Hint: Lagrangian mechanics in physics is a formulation of classical mechanics which is founded on the stationary action principle which is also called the principle of least action. According to the stationary action principle, the action functional of the system derived from the Lagrangian should remain at a stationary point which is a maximum, saddle, or minimum throughout the time evolution of that system. This constraint allows us to calculate the equations of the motion of the given system with the help of Lagrange’s equation.

Complete answer:
Hamiltonian mechanics concept contributes to the formulation of statistical mechanics and quantum mechanics and it can be defined as the direct representation of the evolution of time. It is the sum of the potential and kinetic energies. One of the best known Hamiltonian equations can be represented as $H=T+V$ Here $T\,\And V$ represents the kinetic energy and potential energy respectively.

Hamiltonian mechanics basically emerged as a reformulation of Lagrangian mechanics. Generally Lagrangian can be defined as the input to an external principle which is used to solve the time evolution problem in physics. It is actually the difference between kinetic and potential energies.

Therefore Lagrange ‘s equation can be represented as $L=T-V$ ; Here $T\And V$ denotes the Kinetic energy and potential energy respectively.

Hamiltonian mechanics replaces the velocities that are used in Lagrangian mechanics with momenta. Hamiltonian mechanics serves as a link between quantum mechanics and classical mechanics and it also has a close relationship with geometry, mainly Poisson structures and symplectic geometry.

Note: The main reason behind preferring Lagrangian and Hamiltonian approach more than Newtonian mechanics is that the Lagrangian and Hamiltonian approch are cast in terms of potential energy and kinetic energies which involve only scalar functions but the Newtonian mechanics is vectorial in nature.