Hamiltonian Formulation

Canonical conjugate momenta, Hamiltonian function, and Hamilton's canonical equations.

General idea of the topic

The Hamiltonian formulation replaces the generalized velocities by the conjugate momenta as independent variables. The Hamiltonian H(q, p) usually coincides with the total energy of the system and generates the evolution with two first-order equations for each coordinate.

The topic introduces the Legendre transformation, which allows passing from the Lagrangian to the Hamiltonian, and derives Hamilton's equations, q̇ = ∂H/∂p and ṗ = −∂H/∂q, which describe the motion as a flow in phase space. The formalism is completed with Poisson brackets, canonical transformations, Hamilton-Jacobi theory and the treatment of constrained systems through Dirac brackets.

What you will truly understand

By the end you will be able to build the Hamiltonian from the Lagrangian, write Hamilton's equations and recognize that H is conserved when it does not depend explicitly on time.

You will also be able to interpret the trajectory as a curve in the (q, p) plane.

Recommended learning route

1. Legendre Transformation: how to change variable (q̇ to p) without losing information, with the Legendre transformation.
2. Hamilton's Equations: canonical equations, conservation of H and representation in phase space.
3. Poisson Brackets: evolution of any quantity through the bracket with H and the condition for it to be conserved.
4. Canonical Transformations: changes of variables that preserve the form of Hamilton's equations, and their generating function.
5. Hamilton-Jacobi Theory: equation for the action as a function of the coordinates, leading to the integration of the motion.
6. Constrained Systems and Dirac Brackets: how to treat systems with constraints among the canonical variables using the Dirac bracket.

Physical criterion you must acquire

The essential criterion is to express H in terms of q and p, not of q̇. If the Lagrangian does not depend explicitly on time and the constraints do not either, H coincides with the energy and is conserved. For a harmonic oscillator, the trajectories in the (q, p) plane are ellipses.

Connection with real problems

• quantum mechanics.
• statistical physics.
• orbital dynamics.
• chaotic systems and phase space.

Final goal of the topic

The final goal is to master the canonical formulation of mechanics and use phase space to describe motion.