Variational Principles For Scalars On Manifolds

We revisit the principle of stationary action for systems whose Lagrangian is intended to represent a scalar quantity (e.g.\ energy densities in mechanics and field theory). In deriving Euler-Lagrange equations, the integration-by-parts step treats the integrand ($L\,\mathrm{d}t$ or $L\,\mathrm{d}^n x$) as a top form, conflating the scalar of physical interest with the integration measure. We propose a \emph{scalar-proper} reformulation in which the Lagrangian $f\in C^\infty(\mathcal M)$ and the volume form $\omega\in\Omega^n(\mathcal M)$ are strictly independent geometric objects and the action is $S=\int_{\mathcal M} f\,\omega$. Under this decoupling, in variational bicomplex language, vertical variations of the configuration ($q\mapsto q+\varepsilon\eta$) are shown to be incompatible with measure independence. The next, most natural option is a simultaneous horizontal variation of both the configuration and the base coordinate, which yields a clean stationarity condition with no boundary fixing requirements. This scalar treatment often results in different field equations given the same Lagrangian density. We discuss some example scalar actions and when a given formalism is appropriate, as well as extensions to fields on $n$-manifolds and implications for the usual treatment of boundary terms.