Multi-Function Variation

Suppose that we wish to maximize or minimize the functional

$$I = \int_a^b F(y_1,y_2,\cdots,y_{\cal F},y_1',y_2',\cdots,y_{\cal F}',x)\,dx.$$ (707)

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 10.2 yields ${\cal F}$ separate Euler-Lagrange equations,

$$\frac{d}{dx}\!\left(\frac{\partial F}{\partial y_i'}\right)-\frac{\partial F}{\partial y_i} = 0,$$ (708)

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

$$\frac{\partial F}{\partial y_k'} = {\rm const}.$$ (709)

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

$$y_i'\,\frac{\partial F}{\partial y_i'} - F = {\rm const},$$ (710)

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