FormulaProved

Euler–Lagrange equation

Statement

Let $J[q] = \int_{t_1}^{t_2} L(t, q(t), \dot q(t))\, dt$ be a functional on curves $q(t)$ with fixed endpoints. A curve $q$ is a stationary point of $J$ (in particular, if it minimizes or maximizes $J$ ) if and only if it satisfies, for each coordinate $q_i$ , $\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot q_i}\right) - \dfrac{\partial L}{\partial q_i} = 0$ .

Why is it true?

Among all the paths a system could conceivably follow between two fixed events, the one nature actually takes is the one that makes the 'action' insensitive to small wiggles — nudging the path slightly in any direction, to first order, changes the action not at all. The Euler–Lagrange equation is exactly the local, pointwise condition that guarantees no such nudge can lower or raise the action, turning the global search over all paths into a differential equation you can solve step by step.

Proof sketch

Consider a variation $q_\varepsilon(t) = q(t) + \varepsilon\, \eta(t)$ with $\eta(t_1) = \eta(t_2) = 0$ . Stationarity means $\frac{d}{d\varepsilon}\Big|_{\varepsilon=0} J[q_\varepsilon] = 0$ for every such $\eta$ . Differentiating under the integral gives $\int_{t_1}^{t_2} \left(\frac{\partial L}{\partial q}\eta + \frac{\partial L}{\partial \dot q}\dot\eta\right) dt = 0$ ; integrating the second term by parts (the boundary terms vanish since $\eta(t_1)=\eta(t_2)=0$ ) gives $\int_{t_1}^{t_2} \left(\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot q}\right)\eta\, dt = 0$ . Since $\eta$ is arbitrary, the fundamental lemma of the calculus of variations forces the bracketed expression to vanish identically, which is the Euler–Lagrange equation.

Stated by

stated 1755Leonhard Euler stated 1755Joseph-Louis Lagrange

Proved by

Leonhard Euler Joseph-Louis Lagrange

Topics that use this theorem

Lagrangian and Hamiltonian mechanics

Related theorems

Noether's theorem Liouville's theorem (Hamiltonian mechanics)

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

I. M. Gelfand, S. V. Fomin (1963). Calculus of Variations
Herbert Goldstein, Charles P. Poole, John L. Safko (2002). Classical Mechanics