Optimization of multiple integrals

[Hiero]

The Euler Lagrange equation finds functions $x_i(t)$ which optimizes the definite integral $\int L(x_i(t),\dot x_i(t))dt$

Is there any extensions of this to multiple integrals? How do we optimize $\int \int \int L(x(t,u,v),\dot x(t,u,v))dtdudv$ ?

In particular I was curious to try to maximize the entropy $\int (p\ln p )dV$ over the phase space of a classical system of particles.

 

[Infrared]

I think this is the form of Euler-Lagrange that you want https://en.wikipedia.org/wiki/Euler...n_of_several_variables_with_single_derivative

 

[Hiero]

Yes thank you. It comes the same way except you integrate by parts on each dimension. I suppose keeping the volume of integration fixed is enough to ensure the N-1 dimensional integrals are zero because we’re evaluating the variation somewhere on the boundary where it must be zero.

We can also use Lagrange multipliers just the same to optimize, for example, the entropy $S=-\int_V p\ln p d^N x$ under constraints, for example, $\int_v p d^N x=1$ and $\int_v pE d^N x=<E>$ (<E> is fixed, p and E are functions over phase space) we would just use this condition with a lagrangian of $p\ln p + \lambda _1 +\lambda_2E$ from which we get the Boltzmann distribution.

I was just bothered because in some statistical mechanics lectures, the Boltzmann distribution was derived as a sum over discrete energy levels, but then it was just used as an integral over phase space. It’s a bit trivial since the Lagrangian doesn’t depend on any $\frac{\partial p }{\partial x_i}$ but I’m glad it works.