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

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