Physics / advanced
Phase space and canonical dynamics
Use Hamiltonian flow, canonical transformations, action variables, and perturbation theory to connect mechanics with modern theoretical physics.
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Module 01 advanced
Phase space and canonical dynamics
Use Hamiltonian flow, canonical transformations, action variables, and perturbation theory to connect mechanics with modern theoretical physics.
- 01
Legendre transforms and canonical momenta
Exchange velocities for momenta carefully, identifying when the local Hamiltonian construction exists and when it fails.
32 min - 02
Hamilton's equations and phase flow
Read a Hamiltonian as a vector field on phase space and distinguish a conserved contour from motion along it.
30 min - 03
Poisson brackets and generators
Compute canonical brackets, track sign conventions, and generate translations and reciprocal scalings from phase-space functions.
34 min - 04
Canonical transformations and generating functions
Build transformations with a type-2 generating function and verify both the bracket structure and time-dependent Hamiltonian shift.
36 min - 05
The Hamilton–Jacobi equation
Replace trajectory equations with a generating-function equation, recover motion from its constants, and recognize local and global limits.
38 min - 06
Action-angle coordinates
Normalize actions with 1/(2π), recover oscillator frequency from H(J), and state where torus coordinates exist.
35 min - 07
Adiabatic invariants in mechanics
Predict slow oscillator changes using approximate action conservation and identify why rapid changes and separatrices invalidate it.
34 min - 08
Liouville volume preservation
Derive incompressible canonical flow, follow density along trajectories, and separate volume preservation from mixing and recurrence.
33 min - 09
Hamiltonian perturbation and resonance
Average weak anharmonicity, recognize small denominators, and distinguish coherent addition from resonant energy transfer.
39 min - 10
Constrained Hamiltonian systems
Follow primary constraints through consistency, distinguish first- and second-class relations, and compute a simple Dirac bracket.
42 min