Hamiltonian Mechanics

Vrutang Shah29/4/2014

Overview of Classical Physics

• Three alternative approach for classical mechanics:

1. Newtonian mechanics

2. Lagrangian mechanics

3. Hamiltonian mechanics

Newtonian Mechanics

• In Newtonian mechanics, the dynamics of the system are defined by the force F, which in general is a function of position x, velocity ˙x and time t.

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(𝑚�̇� )=𝐹 (𝑥 , �̇� , 𝑡)

Lagrangian Mechanics

• Given a function L(q, q˙, t) (called the Lagrangian), the equations of motion for a dynamical system are given by:

Where, L=T-VT: Kinetic EnergyV: Potential Energyqi: Generalized Coordinates

Why use Lagrangian Mechanics?

• Advantage: Constraint forces can be easily taken care Can be generalized to relativistic mechanics Can be extended to continuous degrees of freedom(Maxwell equation of motion are from Lagrangian Equation of motion)

• Disadvantage: Not easy to quantized

Hamiltonian Mechanics

• Derivation from Lagrangian Mechanics:– Now to explicitly make 1st order equation, we can

take new variable in Lagrangian equation.– Hence, we want to make H(q, p, t) from L(q, q˙, t).

This can be done by using Legendre transform (change of variable)

Hamiltonian Mechanics

(1)Now, H (q, p, t)• (2)Equating (1) and (2)..

Hamiltonian Mechanics

• Given a function H(q, p, t) (called the Hamiltonian), the equations of motion for a dynamical system are given by Hamilton’s equations:

Where, H= T + V;T : Kinetic EnergyV: Potential Energy: Conjugate Momentum

Why use Hamiltonian Mechanics?

• Advantage: Explicitly give 1st order differential equationCan be used in Quantum mechanics

Unified Principle of Least Action

• Consider the path traced by a dynamical system on a plot of vs q. We can defined an Action S as,

Then system chose the path for which the action S is a minimum.

This lead to a formalization of Lagrangian equation of motion.