Multi-Function Variation

Suppose that we wish to maximize or minimize the functional

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

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

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

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

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

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