Multi-Function Variation

(E.33) |

Here, the integrand is now a functional of the independent functions , for . A fairly straightforward extension of the analysis in Section E.2 yields separate Euler-Lagrange equations,

for , which determine the functions . If does not explicitly depend on the function then the th Euler-Lagrange equation simplifies to

(E.35) |

Likewise, if does not explicitly depend on then all Euler-Lagrange equations simplify to

(E.36) |

for .

Richard Fitzpatrick 2016-03-31