JOVANA
Explore Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Least Action and the Euler-Lagrange Equation

Turn the principle of least action into a workable equation using the calculus of variations — and watch \mathbf{F}=m\mathbf{a} reappear as a special case.

What exactly is the action?

For a wide class of systems the Lagrangian is the difference between kinetic and potential energy, L=T-V. (Not their sum — that is the energy; the difference is what least action minimizes.) The action is L integrated over the time of the motion.

S[\mathbf{q}] = \int_{t_1}^{t_2} L(\mathbf{q},\dot{\mathbf{q}},t)\,dt, \qquad L = T - V

The action is a functional: feed it an entire path \mathbf{q}(t) and it returns a number.

Deform the trial path and watch the action change. The physical trajectory sits at the bottom of the well of S — nudge away from it and S goes up.

The calculus of variations

To find the stationary path, we use the calculus of variations. Take the true path q(t) and compare it with a nearby wiggled path q(t)+\delta q(t), where the wiggle \delta q vanishes at the two fixed endpoints. Demand that the resulting change in the action vanish to first order.

\delta S = \int_{t_1}^{t_2}\!\left(\frac{\partial L}{\partial q}\,\delta q + \frac{\partial L}{\partial \dot q}\,\delta \dot q\right)dt = 0

The first variation of the action, from perturbing both q and its velocity \dot q.

  1. Note that \delta\dot q = \tfrac{d}{dt}\delta q — varying the path also varies its slope.
  2. Integrate the second term by parts to move the time-derivative off \delta q.
  3. The boundary term \left[\tfrac{\partial L}{\partial \dot q}\delta q\right]_{t_1}^{t_2} vanishes because \delta q=0 at both endpoints.
  4. What remains is \int(\cdots)\,\delta q\,dt = 0 for every choice of \delta q.
  5. By the fundamental lemma of the calculus of variations, the bracket itself must be zero everywhere.

The Euler-Lagrange equation

The bracket that must vanish is the Euler-Lagrange equation — the beating heart of Lagrangian mechanics. There is one such equation for each generalized coordinate, and each is a second-order differential equation for the motion.

\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q_i}\right) - \frac{\partial L}{\partial q_i} = 0

One equation per coordinate q_i — the same form in Cartesian, polar or any coordinates you like.

Many trial paths connect the two events; the Euler-Lagrange equation is exactly the condition that singles out the highlighted, stationary-action one.

m\ddot x = -\frac{dV}{dx}

Newton's second law recovered as a special case of Euler-Lagrange.

What we assumed — and its limits

The great payoff is coordinate independence: the Euler-Lagrange equation has the identical form no matter which coordinates you chose, so you write L in whatever variables are convenient and turn the crank. That is why a hard Newtonian problem often becomes a short Lagrangian one.