Physics / advanced
Mechanics through stationary action
Reformulate constrained mechanical systems using generalized coordinates, variational principles, cyclic variables, and coupled small oscillations.
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Module 01 advanced
Mechanics through stationary action
Reformulate constrained mechanical systems using generalized coordinates, variational principles, cyclic variables, and coupled small oscillations.
- 01
Generalized coordinates and configuration space
Choose independent coordinates, build a kinetic-energy metric, and separate coordinate singularities from physical constraints.
24 min - 02
Virtual work and d’Alembert’s principle
Project Newton's law onto allowed instantaneous displacements and keep generalized-force signs and units explicit.
25 min - 03
Euler–Lagrange equations from variations
Derive stationary action with fixed endpoints and test the distinction between a stationary path and a minimum.
27 min - 04
Holonomic and nonholonomic constraints
Test whether velocity restrictions integrate to configuration equations and distinguish physical rolling dynamics from constrained action variation.
28 min - 05
Cyclic coordinates and Routh reduction
Eliminate a cyclic velocity at fixed momentum while preserving the correct centrifugal sign in the reduced dynamics.
29 min - 06
Velocity-dependent potentials
Derive the electromagnetic Lorentz force from a velocity-dependent Lagrangian and distinguish canonical from mechanical momentum.
29 min - 07
Noether charges in particle mechanics
Connect continuous variational symmetries to conserved quantities, including transformations that change L by a boundary term.
29 min - 08
Coupled small-oscillation eigenmodes
Linearize a mechanical Lagrangian around equilibrium and solve a mass-weighted generalized eigenvalue problem.
30 min - 09
Normal coordinates and mode excitation
Project initial displacement and velocity onto mass-normalized modes and predict which collective motions are actually excited.
29 min - 10
Lagrangian stability and effective motion
Analyze local equilibria using effective potentials while distinguishing conservative stability, attraction, and a motor-driven constraint.
31 min