Classical Mechanics

Generalized Coordinates & Equations of Motion

Pretty Much Physics · Classical Mechanics

Degrees of freedom and generalised coordinates — count these first.

Derivation of Euler-Lagrange Equations

Pretty Much Physics · Classical Mechanics

Properties of the Lagrangian

Pretty Much Physics · Classical Mechanics

Cyclic coordinates and the conserved momenta they give.

Cyclic coordinates and the conservation laws they hand youinteractive

Central force in polar coordinates — angular momentum falls out of the Lagrangian in one line.

Unlock interactive visuals

Lagrangian mechanics

The recipe

Choose generalised coordinates $q_{1}, \dots, q_{n} (n =$ degrees of freedom = 3N − constraints).
Write T (kinetic) and V (potential) in terms of q, q̇.
L = T − V.
Euler–Lagrange: d/dt $(\partial L / \partial q$ ̇ $_{i}) = \partial L / \partial q_{i}$ .

Conservation laws (Noether)

Symmetry	Conserved
L independent of $q_{i} ($ cyclic)	$p_{i} = \partial L / \partial q$ ̇ $_{i}$
L independent of t	energy $h = \sum q$ ̇ $_{i} p_{i} - L$
translatio

Unlock notes

Part B2023 Decembershow ▾

Previous-year question — unlock to view

Part Cshow ▾

A particle of mass m moves in a plane under a central potential V(r). In polar coordinates

(r, θ)

, which of the following are true?

A.The Lagrangian is L = ½m(ṙ $^{2} + r^{2} θ$ ̇ $^{2}) - V (r)$ .✓
B. $θ$ is cyclic, so $p_{θ} =$ mr $^{2} θ$ ̇ is conserved.✓
C.The conservation of $p_{θ}$ expresses conservation of angular momentum.✓
D.Energy is not conserved, because the motion depends on $θ$ .

Solution

Kinetic energy in polar coordinates gives (1). L contains $θ$ only through $θ$ ̇, so $\partial L / \partial θ = 0$ and the conjugate momentum mr $^{2} θ$ ̇ — the angular momentum — is a first integral. (4) is backwards: L has no explicit time dependence, so ½m(ṙ $^{2} + r^{2} θ$ ̇ $^{2}) + V (r)$ is conserved as well.

Save for spaced revision (free account)

1. Lagrangian formalism and generalised coordinates

Cyclic coordinates and the conservation laws they hand youinteractive

Lagrangian mechanics

The recipe

Conservation laws (Noether)

2. Hamiltonian formalism and conservation laws

From Lagrangian to Hamiltonian, and what the flow preservesinteractive

Hamiltonian mechanics, Poisson brackets and canonical transformations

From L to H

Poisson brackets