Adopt stationary Maxwell action as alternative physical formulation

Adopt as a physical formulation principle of this framework, as an alternative to postulating the field equations and the force law directly, the following assumption about the functionals, variation classes and stationarity predicate of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional.

(i) [Field sector] A smooth field $A:N\to {\mathbb{R}}^4$ on the region $U$ of clause (a) of that definition is dynamically admissible exactly when it is stationary for the field action $S[A]$ under all compactly supported smooth variations $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$ in the sense of clause (b) of that definition: $\frac{d}{ds}{\mid }_{s=0}S[A+sh]=0$ for every such $h$.

(ii) [Particle sector] For a prescribed external four-potential $A_{\mu }$ on a neighbourhood of the image of a $C^2$ future timelike massive worldline $X$, the motion is dynamically admissible exactly when $X$ is stationary for the particle functional $S_{\mathrm{p}\mathrm{t}}$ of clause (c) of that definition under all fixed-endpoint $C^2$ variations.

This is an adopted assumption of the model, not a theorem, not a derivation from the remaining postulates and not an empirical result. It asserts stationarity only: it does not assert that a stationary configuration is a local or global minimum, that stationary configurations exist or are unique, that the action functional is uniquely determined or forced by symmetry, that no other action formulation is possible, or that the two sectors can be coupled into a single well-posed action for singular sources. No Noether theorem, no symmetry-to-conservation machinery and no matter dynamics beyond the prescribed potential are assumed.

The conditional deductions of this pair are: on the declared smooth classes, (i) is equivalent to the inhomogeneous tensor field equation ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ and, for potential-represented fields, to the smooth-branch Maxwell equations, and (ii) is equivalent to the framework's adopted four-force equation $mdU\mathrm{/}d\tau =q\mathrm{\Phi }(U)$; both deductions are recorded separately on this page and in Action stationarity implies the inhomogeneous Maxwell equations and Relativistic prescribed-field particle action yields the Lorentz force, and this postulate asserts neither.

The cited treatments present the action principle of classical electrodynamics as adopted, not derived: Tong §5.5 introduces the action and obtains the field equations from its variation, and Landau–Lifshitz §16 states in the footnote on p. 44 that the form of the interaction term cannot be fixed on general considerations alone. The Gaussian-unit expressions of the monograph are read only for the structure of the principle; the functionals adopted here are exactly the SI functionals of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional. No measurement, no empirical claim and no uniqueness of the formulation is contained in this item.