Lagrangian mechanics reformulates classical mechanics in terms of generalized coordinates and an energy-based function called the Lagrangian. Instead of writing forces component by component, one defines
L(q, q˙, t) = T - U
and derives equations of motion from a variational principle. The method is especially effective for constrained systems and for choosing coordinates adapted to the geometry.
Action and stationary action
The action is
S = ∫_{t1}^{t2} L dt
Physical motion makes S stationary (δS = 0) under small variations with fixed endpoints.
Euler-Lagrange equations
For each coordinate q_i:
d/dt(∂L/∂q˙_i) - ∂L/∂q_i = 0
These equations are equivalent to Newton's laws but often simpler when constraints are present.
Generalized coordinates and constraints
Generalized coordinates can be angles, lengths, or any variables that specify the configuration. Holonomic constraints can be handled by reducing coordinates, or with Lagrange multipliers that enforce constraints systematically.
Canonical momentum and cyclic coordinates
Define
p_i = ∂L/∂q˙_i
If q_k does not appear explicitly in L (∂L/∂q_k = 0), then p_k is conserved. This is a basic expression of the symmetry–conservation connection (Noether's theorem).
Example: simple pendulum
For a pendulum of length ℓ and mass m with angle θ:
T = 1/2 m ℓ^2 θ˙^2
U = m g ℓ (1 - cos θ)
L = T - U
The Euler-Lagrange equation yields
θ¨ + (g/ℓ) sin θ = 0
Why it is useful
- Clean handling of constraints and complex coordinates
- Symmetries and conserved quantities appear naturally
- Extends to Hamiltonian mechanics and field theory