Multi-Function Variation

Suppose that we wish to maximize or minimize the functional

$$I = \int_a^b F(y_1,y_2,\cdots,y_N,y_1',y_2',\cdots,y_N',x)\,dx.$$ (777)

Here, the integrand $F$ is now a functional of the $N$ independent functions $y_i(x)$, for $i=1,N$. A fairly straightforward extension of the analysis in Sect. 11.2 yields $N$ separate Euler-Lagrange equations,

$$\frac{d}{dx}\!\left(\frac{\partial F}{\partial y_i'}\right)-\frac{\partial F}{\partial y_i} = 0,$$ (778)

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

$$\frac{\partial F}{\partial y_k'} = {\rm constant}.$$ (779)

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

$$y_i'\,\frac{\partial F}{\partial y_i'} - F = {\rm constant},$$ (780)

for $i=1,N$.