Euler-Lagrange equations on a differentiable manifold

I am following the conventions of https://arxiv.org/abs/math-ph/9902027

Let $$M$$ be a differentiable manifold, $$E \to M$$ a vector bundle over $$M$$ with fibre $$F$$, $$J^1(E)$$ the rank-one jet bundle over $$M$$ and $$V(M)$$ the bundle of densities over $$M$$.
Furthermore, let $$L : J^1(E) \to V(M)$$ be a bundle map and define the action functional

$$\begin{equation} F : \Gamma(E) \to \mathbb{R}, \psi \mapsto \int\limits_M \left( L \circ j^1 \right) ( \psi ) \end{equation}$$

where $$\Gamma(E)$$ is the space of sections of $$E$$ and $$j^1 : \Gamma(E) \to J^1(E)$$ denotes the rank-one jet prolongation. How would I now derive the Euler-Lagrange equations? Assuming $$L$$ is nice enough, I obtain

\begin{equation} \begin{aligned} \left( D F \right)_\psi \left( \phi \right) &= \int\limits_M \left. \frac{\partial}{\partial t} \right \vert_{t = 0} \left[ \left( L \circ j^1 \right) \left( \psi + t \phi \right) \right] \\ &= \int\limits_M \left[ \left( D L \right)_{j^1 \left( \psi \right)} \circ \left( D j^1 \right)_\psi \right] \left( \phi \right) = \int\limits_M \left[ \left( D L \right)_{j^1 \left( \psi \right)} \circ j^1 \right] \left( \phi \right) \end{aligned} \end{equation}

where all $$D$$s can be read as Fréchet derivatives because they act on topological vector spaces. Now, I would need to perform partial integration and apply appropriate boundary conditions. One could probably introduce a positive definite inner product on $$F$$, translate it to the fibres and by introducing a positive definite 'reference' density $$\nu$$ and employing a Riesz-like representation theorem to arrive at something like

\begin{equation} \begin{aligned} \int\limits_M \left[ \left( D L \right)_{j^1 \left( \psi \right)} \circ j^1 \right] \left( \phi \right) = \int_M \langle \omega_x, \phi \rangle \, \nu(x) \end{aligned} \end{equation}

for all $$\phi \in \Gamma(E)$$. Here $$\omega_x \in \Gamma(E)$$ would be defined as

$$\begin{equation} \left[ \left( D L \right)_{j^1 \left( \psi \right)} \circ j^1 \right] \left( \phi \right) (x) = \langle \omega_x, \phi \rangle \, \mathrm{d} \nu(x) \end{equation}$$

This, however, seems rather ugly because of the arbitrariness of the reference density (I could probably live with the inner product).
The Euler-Lagrange equations should then read

$$\begin{equation} \omega_x = 0 \end{equation}$$

Of course the main issue is then still to obtain an expression for $$\omega_x$$ which seems difficult.

• Does the RHS in the bottom equation make sense ($\phi$ is not a scalar function)? Intuitively, you have to do some integration by parts to get rid of 'derivatives of $\phi$' and then whatever will multiply $\phi$ has to vanish, which will produce the Euler-Lagrange equations. The paper arxiv.org/abs/1406.3369 provides integration by parts formulae for expressions involving jet prolongations. – S.Surace May 16 at 14:21
• Yes, you are of course right! I fixed that and added my attempt at a solution. I have not found the time to read the paper yet, but thanks anyway! – iolo May 16 at 20:09