# Identification of the variation on the boundary and why $\delta S_{\partial V}=0$

I recently asked this question about variational principles and how it all works. The essential answer I got was to go read a book on the calculus of variations, which I did, and this helped me make sense of what was going on. I have one lingering question.

For finite volume $$\mathcal{R}^3$$ and $$V=\mathcal{R}^3\otimes[t_i,t_f]$$, we write out the variation of the action of a classical field $$\varphi$$ as $$\delta S=\int_V\left(\frac{\partial\mathcal{L}}{\partial\varphi(t,x)}-\partial_\mu\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)\delta\varphi(t,x)+\int_V\left(\partial_\mu\left(\delta\varphi\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)\right)$$ and identify the terms on the right of the equality as $$\delta S_V$$ and $$\delta S_{\partial V}$$. I don't fully understand how the right-most term above is identified as the "variation on the boundary".

One way I tried to make sense of this is by applying Gauss's theorem, which yields $$\int_V\left(\partial_\mu\left(\delta\varphi\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)\right)=\int_{\partial V}d\Sigma_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}\delta\varphi.$$ Is this the mathematical underpinning of this identification? If so, why should we enforce this to be zero, which is what allows us to get the field equation?

On the other hand, say we have the field equation and consider a general symmetry transformation $$x\mapsto\widetilde{x}=\widetilde{x}(x)$$ and $$\varphi(x)\mapsto\widetilde{\varphi}(\widetilde{x})$$. We have $$\widetilde{x}^\mu=x^\mu+\delta x^\mu$$ and $$d^4\widetilde{x}=Jd^4x$$, where $$J=1+\partial_\mu(\delta x^\mu)$$. Expanding the action to lowest order in the coordinate yields $$S=\int d^4x(\mathcal{L}(\widetilde{\varphi}(x),\partial\widetilde{\varphi}(x))+\partial_\mu(\mathcal{L}\delta x^\mu)).$$ For an infinitesimal transformation, we have $$\delta S=\int_V\left(\frac{\partial\mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)\delta\varphi(t,x)+\int_V\left(\partial_\mu\left(\delta\varphi\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)+\partial_\mu(\mathcal{L}\delta x^\mu)\right).$$ Here, I can immediately justify the identification of $$\delta S_V$$. The equation of motion tells us that $$\int_V\left(\frac{\partial\mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)\delta\varphi(t,x)=0.$$ This equation governs the system between $$t_i$$ and $$t_f$$, so it seems fair to call this integral $$\delta S_V$$. Moreover, that we have a symmetry transformation tells us $$\delta S=0$$, so that $$\int_V\left(\partial_\mu\left(\delta\varphi\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi}\right)+\partial_\mu(\mathcal{L}\delta x^\mu)\right)=0,$$ to which I can also apply Gauss's theorem and justify the identification as $$\delta S_{\partial V}$$. To see that this is zero requires us to already have the field equation on hand, which required us to, believe it or not, assume that $$\delta S_{\partial V}$$ is zero.

Can someone explain to me the identification of the terms $$\delta S_V$$ and $$\delta S_{\partial V}$$ and why we should, in the derivation of the field equation, require $$\delta S_{\partial V}=0$$?

The normal thing done in the physics texts is to use Gauss' theorem to conclude that the far right term is a boundary term, i.e. an integral over the bounding surface of $$V$$. One of the most important reasons to accept that this integral goes to zero is that the dynamical system under consideration is thought to be finite in its spatial extension. This is probably not a bad idea, because such finite systems do not give us things like infinite energies and so forth; it is a lot easier to guarantee the existence of the action if the system does not have infinite extent. I think in terms of functional analysis, this allows us to think of the action as a distribution with test functions (the fields $$\phi$$) that have compact support. This generally guarantees that everything is mathematically rigorous.
• This is all very interesting to me. In college I took courses that required field theory, I even took GR with Wald himself, but I never really got it. I was a mathematician by intellectual vice thinking I could jump into physics and ego-math my way through all of it. I did research on $\gamma$-LQG and LCFT without ever understanding why or what was going on. I'm now in finance, and I've realized slight modifications to physical Hamiltonians coupled with field theory can be very powerful. Can you point me to any interesting/evocative ideas in physics so I can learn more? Commented Mar 22 at 3:44