Analytical mechanics

The Euler–Lagrange equations

The Euler–Lagrange equations derive motion from a Lagrangian and apply to each generalized coordinate.

A system can be described using coordinates adapted to its constraints rather than all Cartesian coordinates. The Euler–Lagrange equations determine how these coordinates evolve when the action is stationary.

Example: the simple pendulum

Set $q=\theta$. The kinetic energy is $T=\tfrac12 ml^2\dot\theta^2$ and the potential, zeroed at the bottom, is $V=mgl(1-\cos\theta)$. Substituting $L=T-V$ gives nonlinear motion; only at small angles may we use $\sin\theta\simeq\theta$.

Quick check

If $L$ has no explicit dependence on a coordinate $q_j$, that coordinate is cyclic, and the Euler–Lagrange equation gives $d p_j/dt=0$, where $p_j=\partial L/\partial\dot q_j$. This extracts a conserved quantity before solving the equations of motion. For example, in a central potential the azimuthal angle is absent from the Lagrangian, so its conjugate angular momentum is conserved. A valid coordinate change leaves the physics intact, but both velocities and the Lagrangian must be transformed consistently.

In one dimension, $L=\tfrac12m\dot x^2-V(x)$ gives $m\ddot x=-dV/dx$, Newton’s second law. This equivalence is a useful check: the Lagrangian method does not change the prediction, but organizes it more effectively when many coordinates and constraints are present.