Multi-Function Variation

(B.61) |

Here, the integrand is now a functional of the independent functions , for . A fairly straightforward extension of the analysis in Section B.5 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

(B.63) |

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

(B.64) |

for .

Richard Fitzpatrick 2016-01-25