Euler-Lagrange equation

Meaning

Noun

• A differential equation which describes a function $\backslash mathbf\{q\}(t)$ which describes a stationary point of a functional, $S(\backslash mathbf\{q\})\; =\; \backslash int\; L(t,\; \backslash mathbf\{q\}(t),\; \backslash mathbf\{\backslash dot\; q\}(t))\backslash ,dt$, which represents the action of $\backslash mathbf\{q\}(t)$, with $L$ representing the Lagrangian. The said equation (found through the calculus of variations) is $\{\backslash partial\; L\; \backslash over\; \backslash partial\; \backslash mathbf\{q\}\}\; =\; \{d\; \backslash over\; dt\}\; \{\backslash partial\; L\; \backslash over\; \backslash partial\; \backslash mathbf\{\backslash dot\; q\}\}$ and its solution for $\backslash mathbf\{q\}(t)$ 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).