Functional approach vs jet approach to Lagrangian field theory

I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo Brunetti and Klaus Fredenhagen,

• R. Brunetti, K. Fredenhagen, P. L. Ribeiro, Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields, Commun. Math. Phys. 368 (2019) 519-584, arXiv:1209.2148 [math-ph].

There we discuss only scalar fields, but the discussion remains essentially unchanged for sections of fiber bundles or even fibered manifolds. In what follows, I will assume the former.

As a rule, the functional formalism is more general, for it is clear that given a smooth $d$-form $\omega$ on the total space of the $k$-th order jet bundle $J^k\pi$ of the fiber bundle $\pi:E\rightarrow M$ with $D$-dimensional typical fiber $Q$ over the smooth $d$-dimensional (space-time) manifold $M$ (we assume all finite-dimensional manifolds here to be smooth, Hausdorff, paracompact and connected) - think of $\omega$ as a "Lagrangian density" - we have that $$F_K(\varphi)=\int_K (j^k\varphi)^*\omega\ ,\quad\varphi\in\Gamma(\pi)$$ is a functional on the space $\Gamma(\pi)=\{\varphi\in\mathscr{C}^\infty(M,E)\ |\ \pi\circ\varphi=\mathrm{id}_M\}$ of smooth sections of $\pi$ for each compact region $K\subset M$ (i.e. $K$ has nonvoid interior) and each $k=0,1,...\infty$. The case $k=\infty$ can be handled just like for finite $k$ since the total space of the infinite-order jet bundle $J^\infty\pi$ is the countable projective limit of the finite-dimensional manifolds $J^k\pi$ and therefore is a metrizable Fréchet manifold, see e.g. the aforementioned paper or the book by Andreas Kriegl and Peter Michor, The Convenient Setting of Global Analysis (AMS, 1997) for a discussion of the manifold structure of $J^\infty\pi$. For a better handling of the boundary terms which appear when performing functional derivatives (see below), it is convenient to replace $K$ with the multiplication of $(j^k\varphi)^*\omega$ by some $f\in\mathscr{C}^\infty_c(M)$ and then integrate over the whole of $M$, thus yielding the functional $$F(\varphi)=\int_M f(j^k\varphi)^*\omega\ ,\quad\varphi\in\Gamma(\pi)\ .$$ Speaking of which, $\Gamma(\pi)$ has an infinite-dimensional manifold structure modelled on the locally convex vector spaces $\Gamma_c(\varphi^*V\pi)$ of smooth sections with compact support of the pullback of the vertical bundle $V\pi=\ker T\pi$ of $\pi$ by each $\varphi\in\Gamma(\pi)$ (these locally convex vector spaces are all canonically topologically isomorphic to each other) - the corresponding (topological) manifold structure is the so-called Whitney topology on $\Gamma(\pi)$. There is in addition a natural smooth structure on $\Gamma(\pi)$ modelled on that of $\Gamma_c(\varphi^*V\pi)$ - it can be proven that smooth curves $\gamma:\mathbb{R}\rightarrow\Gamma(\pi)$ are necessarily of the form $\gamma\in\mathscr{C}^\infty(\mathbb{R}\times M,E)$ with $\gamma(t,\cdot)=\gamma(t)=\gamma_t\in\Gamma(\pi)$ (i.e. $\pi\circ\gamma_t=\mathrm{id}_M$) for all $t\in\mathbb{R}$ and for all $a<b\in\mathbb{R}$ there is $K\subset M$ compact such that $\gamma(t,p)$ is constant in $t\in[a,b]$ for all $p\not\in K$. This entails that $\gamma'_t\in\Gamma_c(\gamma_t^*V\pi)$ for all $t\in\mathbb{R}$ since we are differentiating along a single fiber of $\pi$ when differentiating with respect to the curve parameter $t$. Given that notion of smooth curves, smooth maps are just the ones that map smooth curves to smooth curves (see A. Kriegl, P. Michor, loc. cit. for many more details). Given the specific notion of smooth curves and smooth maps that $\Gamma(\pi)$ has, it is easy to see why the "Poincaré-lemma-type" argument used by Anderson to solve the inverse problem of calculus of variations can be recast in functional form with essentially no change as you did, for the core of the argument still remains essentially finite-dimensional.

In order to see where the boundary terms come from, consider the second functional $F$ above. The variational derivative consists only in taking the derivative of $F(\gamma_t)$ with respect to $t$ at $t=0$, where $\gamma:\mathbb{R}\rightarrow\Gamma(\pi)$ is a smooth curve on $\Gamma(\pi)$ so that $\gamma_0=\varphi$, $\gamma'_0=\delta\varphi\in\Gamma_c(\varphi^*V\pi)$ and applying the chain rule (which, by the way, does hold in this setting). One then applies the standard variational formula $j^k\delta\varphi=\delta(j^k\varphi)$ (recall that $j^k\varphi$ only takes into account base = horizontal derivatives of sections, whereas $\delta\varphi$ is just a fiber = vertical derivative. The desired commutativity comes from local triviality of $\pi$ or, more generally, the implicit function theorem in the case of arbitrary fibered manifolds) together with integration by parts - the result is the Euler-Lagrange derivative of $\omega$ plus a sum of terms proportional to derivatives of positive order of the cutoff function $f$. This latter sum yields the boundary terms in the (distributional) limit when $f$ becomes the characteristic function of a compact region $K$ with smooth boundary $\partial K$ - in this limit, $F$ becomes $F_K$ defined above.

The functional formalism is genuinely more general than the jet bundle formalism also because it can handle a large class of nonlocal functionals. Allowing these is seen to yield better algebraic properties (closure under "pointwise" = "field-wise" products, etc.) than just considering local ones, which is convenient for field quantization at a later stage, among other things. Moreover, there is a very simple and elegant characterization of local functionals within the functional formalism which does not mention jet bundles anywhere - see e.g. Proposition 2.2, pp. 535-539 of the above paper.

This is meant as a long comment to the very good answer by Pedro Riberio.

There is a nice analog of the variational bicomplex in the functional framework. Namely, the space of differential forms on $M \times \Gamma(E)$ (where $E$ is a fiber bundle over $M$) comes with a natural bigrading $\Omega^{p, q}(M \times \Gamma(E))$ induced by the product structure, i.e. the dual of the decomposition $T_{m, \phi} (M \times \Gamma(E)) = T_m M \times T_\phi \Gamma(E)$ of the tangent space. Moreover, the jet map $$ j^k: M \times \Gamma(E) \to J^k E, \qquad (m, \phi) \mapsto j^k_m \phi $$ yields a morphism from the variational bicomplex $\Omega^{p, q}(J^k E)$ to the bicomplex $\Omega^{p, q}(M \times \Gamma(E))$ with the exterior differential. Personally, I find the functional bicomplex easier to understand than the variational one; and as remarked by Pedro the functional framework is more flexible as it also handles non-local Lagrangians. On the other hand, the jet bundle approach has advantages for simulation, because you stay in the finite-dimensional setting which makes it easier to discretize while preserving the (symplectic) geometry.