# Euler-Lagrange equation

## Meaning

### Noun

• A differential equation which describes a function $\mathbf{q}(t)$ which describes a stationary point of a functional, $S(\mathbf{q}) = \int L(t, \mathbf{q}(t), \mathbf{\dot q}(t))\,dt$, which represents the action of $\mathbf{q}(t)$, with $L$ representing the Lagrangian. The said equation (found through the calculus of variations) is $\left\{\partial L \over \partial \mathbf\left\{q\right\}\right\} = \left\{d \over dt\right\} \left\{\partial L \over \partial \mathbf\left\{\dot q\right\}\right\}$ and its solution for $\mathbf{q}(t)$ represents the trajectory of a particle or object, and such trajectory should satisfy the principle of least action.

## Origin

• Named after the Swiss mathematician and physicist Leonhard Euler (1707–1783), and the Italian-born French mathematician and astronomer Joseph Louis Lagrange (1736–1813).

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