Setup: let me first begin with point particles. Noether first theorem states that if the action functional is invariant, equivalently the lagrangian invariant up to one total time derivate, then there is a conservation law. Varying the action, or the lagrangian, you can get a form for the Noether charge...Essentially:
$$\delta S=0\rightarrow E(L)\delta q+\dfrac{d}{dt}B(q,\dot{q}).$$
For a first order lagrangian, we could add in principle boundary terms dependent up to the generalized derivatives but no further. For $r$-order lagrangians, the boundary terms can also go up to $r$-order derivative. First order lagrangians have the conserved charge ($\dot{Q}=0$):
$$Q=\dfrac{\partial L}{\partial \dot{q}}\delta q-\Delta,$$
where $\Delta$ is an arbitrary function having $\dot{\Delta}(t_2)-\dot{\Delta}(t_1)=0$. Am I correct? My question is simple...Is there any general formula for the boundary term giving you conservation laws in the Noether first theorem or is that lagrangian/theory dependent? I have myself deduced, after lengthy calculations that for third order lagrangian:
$$B(q,\dot{q},\ddot{q},\dddot{q},\delta q,\delta \dot{q},\delta \ddot{q})=_2E(\delta q)+_1E(\delta \dot{q})+_0E(\delta \ddot{q})$$
where
$$_2E(\delta q)=\left(\dfrac{\partial L}{\partial \dot{q}}-\dfrac{d}{dt}\dfrac{\partial L}{\partial \ddot{q}}+\dfrac{d^2}{dt^2}\dfrac{\partial L}{\partial \dddot{q}}\right)\delta q$$
$$_1E(\delta\dot{q})=\left(\dfrac{\partial L}{\partial \ddot{q}}-\dfrac{d}{dt}\dfrac{\partial L}{\partial \dddot{q}}\right)\delta \dot{q}$$
$$_0E(\delta \ddot{q})=\dfrac{\partial L}{\partial \dddot{q}}\delta \ddot{q}.$$
From the above recipe, you can guess by induction the general, I guess up to an arbitrary $\Delta$ vanishing in the endpoints, formula for the boundary term and the conserved Noether Charge. Also, you can generalize the above algorithm to the field formalism, making the changes $d/dt\rightarrow \partial_\mu$, $q(t)\rightarrow \phi(x^\mu)$, $L(q,\dot{q},\ddot{q},\ldots)\rightarrow \mathcal{L}(\phi,\partial \phi,\partial^2\phi,\ldots)$. My question is simple:
Is the $\Delta$ term "arbitrary" or does it depend on the theory, and thus, lagrangian? How is it related to the symmetry group of $L$ or the global invariant action of the variational problem?
Related: is $\Delta$ related to higher derivatives or is it independent of them? After all, you could, in principle, have $B$ depending on higher derivatives than those of the prime lagrangian, but then, it would turn into a higher order lagrangian! Are equivalent lagrangians the same theory or not at the end?
Extra: From my guessed formula above, it seems the conserved charge depends not only on a general transformation of the generalized coordinates but also on the generalized velocities and so on! How can it be related to the general prescription of Noether first theorem in terms of a Lie derivative?
In summary, what is the origin of the boundary terms in the Noether first theorem, giving us the associated conservation law? In fact, as a simple exercise, I can see that if $\delta q=\dot{q}$ and $\Delta =L(q,\dot{q})$ we have the hamiltonian or total energy as conserved constant.
BONUS(complicated): For certain dimensions and lagrangian types (and their corresponding actions), we can even get more complicated boundary terms such as topological Chern-Simons forms (they are intrinsically gauge invariant!), or worst, topological currents such as
$$J^\mu_N=\varepsilon^{\mu\nu}\partial_\nu F$$
$$J^\mu_N=\varepsilon^{\mu \alpha_1\alpha_2}\partial_{\alpha_1\alpha_2}F$$
and more generally
$$J^\mu_N=\varepsilon^{\mu \alpha_1\alpha_2\cdots \alpha_n}\partial_{\alpha_1\alpha_2\cdots \alpha_n}F$$
that are trivially conserved due to the antisymmetry of the epsilon symbol and the symmetry of the partial derivatives, so there is a big ambiguity in the boundary term of the Noether theorem! But is $F$ an arbitrary function?
Remark: I would be glad if someone provided and answer both with indices and differential forms language, and if the boundary term is also relevant with symmetries mixing fermionic and bosonic variables, that is, SUSY.
