Equivalence of the construction of the Lagrangian in a book of Sternberg to the "usual" construction

Question

I have a question regarding the following cited text from [1]:

Let $F$ be a representation of the structure group $G$ of the principal bundle $P_G\to M$ (a (semi-)Riemannian manifold), and let $F(P)$ denote the quotient of $P_G\times F$ by the $G$ action. Let $P_G\to M$ be a principal bundle with structure group $G$. There are various geometrical objects that we have associated with $P_G$: ... the space of differential forms on M with values in $F(P)$. We denote the space of $k$-forms on $M$ with values in $F(P)$ by $A^k(P)$, and shall denote the sum of these spaces - and for various possible $F$s simply $A^{*}(P_G)$. We can also denote the space of all connections on $P$, which we shall denote by $\mathrm{Conn}(P_G)$.

He then goes on, later in the page:

... defines a map $L_1,L_2$, etc., from geometrical objects to functions on $M$ $$L_i:\mathrm{Conn}(P_G)\times A^{*}(P_G)\to C^\infty(M)$$ If we then integrate these expressions, or some (linear) combination, $\mathcal{L}$, of them over $M$ relative to the volume form, we obtain a functional on the space of geometrical objects (at least on the sets where this integral converges).

Here he's talking about the action, quite obviously, and he defines the "function" $$\mathcal{F}=\int_M\mathcal{L}d\mathrm{vol}$$

My question, then, is the following:

• How does this construction relate to the "usual" construction of the Lagrangian on a (semi-)Riemannian manifold, where "usual" construction is as follows: the Lagrangian is "a horizontal differential form of degree $n=\dim M$ on the jet bundle of $P_G$" [2]?

References

[1] Curvature in Mathematics and Physics (Dover Books on Mathematics), p. 352, Shlomo Sternberg

[2] $n$lab, Lagrangian

Answer 1

There is no essential contradiction between the definitions in the two references that you gave, except that the one given in the nLab entry is more general.

Sternberg defines the Lagrangian (function) as a scalar-valued differential operator. One can easily change this definition to be volume form valued by simply replacing $L$ with $L\,d\mathrm{vol}$ and obtain the Lagrangian (density). Now, any $k$-th (or lower) order differential operator $L[A]$ can always be written as the composition of a jet extension $j^kA$ with a function $\bar{L}$ defined on the $k$-th order jet bundle, $L[A] = \bar{L}(j^kA)$. That's the essential property of the jet bundle. For convenience, in order not to bound the possible allowed order of such a differential operator in advance, we take $k=\infty$. In the case of a volume form valued differential operator, like the Lagrangian density, the function $\bar{L}$ on the $\infty$-jet bundle can be equivalently represented as a bundle map from jets to volume forms or as a horizontal form on the $\infty$-jet bundle itself.

And just a small note on the terminology. While the definition of a Lagrangian density given in the nLab entry is fairly straight forward, while also very general and flexible, it is the Sternberg-like definition that you'll find in most physics or mathematical physics textbooks and is thus more "usual".