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Multi-Function Variation

Suppose that we wish to maximize or minimize the functional

$\displaystyle I = \int_a^b F(y_1,y_2,\cdots,y_{\cal F},y_1',y_2',\cdots,y_{\cal F}',x)\,dx.$ (E.33)

Here, the integrand $ F$ is now a functional of the $ {\cal F}$ independent functions $ y_i(x)$ , for $ i=1,{\cal F}$ . A fairly straightforward extension of the analysis in Section E.2 yields $ {\cal F}$ separate Euler-Lagrange equations,

$\displaystyle \frac{d}{dx}\!\left(\frac{\partial F}{\partial y_i'}\right)-\frac{\partial F}{\partial y_i} = 0,$ (E.34)

for $ i=1,{\cal F}$ , which determine the $ {\cal F}$ functions $ y_i(x)$ . If $ F$ does not explicitly depend on the function $ y_k$ then the $ k$ th Euler-Lagrange equation simplifies to

$\displaystyle \frac{\partial F}{\partial y_k'} = {\rm const}.$ (E.35)

Likewise, if $ F$ does not explicitly depend on $ x$ then all $ {\cal F}$ Euler-Lagrange equations simplify to

$\displaystyle y_i'\,\frac{\partial F}{\partial y_i'} - F = {\rm const},$ (E.36)

for $ i=1,{\cal F}$ .

Richard Fitzpatrick 2016-03-31