Euler-Lagrange equation: Difference between revisions
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The '''Euler-Lagrange equation''' of a functional <math>L(u)</math> is an equation which is necessarily satisfied (formally, at least) by critical points of that functional. It can be computed formally by starting with the equation | The '''Euler-Lagrange equation''' of a functional <math>L(u)</math> is an equation which is necessarily satisfied (formally, at least) by critical points of that functional. It can be computed formally by starting with the equation | ||
<center><math>\frac{d}{d\ | <center><math>\frac{d}{d\epsilon} L(u+\epsilon v)|_{\epsilon = 0} = 0 </math></center> | ||
for arbitrary test functions ''v'', and then using duality to eliminate ''v''. | for arbitrary test functions ''v'', and then using duality to eliminate ''v''. | ||
Latest revision as of 20:14, 11 June 2007
The Euler-Lagrange equation of a functional is an equation which is necessarily satisfied (formally, at least) by critical points of that functional. It can be computed formally by starting with the equation
for arbitrary test functions v, and then using duality to eliminate v.
Equations which are Hamiltonian can (in principle, at least) be expressed as the Euler-Lagrange equation of a functional, and conversely Euler-Lagrange equations can in principle be reformulated in a Hamiltonian manner.
