PU-201 · Analytical Mechanics
From "write down the forces" to "choose a formalism" — the transition that defines the second year of a physics degree.
Before you start
Comfort with PU-101 · Newtonian Mechanics and PU-104 · Multivariable Calculus & ODEs. The variational machinery here rests on the stationary-action argument developed in D-018 — read it if functionals feel unfamiliar.
Lectures
| L06 |
The Euler–Lagrange equation
cites D-001
|
22 min |
| L11 |
Symmetry & conservation laws
cites D-002
|
28 min |
| L18 |
Small oscillations & normal modes
cites D-003
|
31 min |
| L24 |
The Hamiltonian formulation
cites D-004
|
26 min |
Derivations homed in this unit
The Euler–Lagrange equation
Stationary action → the equation of motion for any generalised coordinate.
Energy from time-translation symmetry
When the Lagrangian has no explicit time dependence, a conserved quantity appears.
Small oscillations as an eigenproblem
Linearise about equilibrium; normal modes fall out as generalised eigenvectors.
Legendre transform & Hamilton’s equations
Trade velocities for momenta; the second-order system becomes first-order in phase space.
Poisson brackets
The algebraic skeleton of classical dynamics — and the bridge to quantum commutators.
Central-force orbit equation
Reduce the two-body problem to a single radial equation for the orbit shape.