Scope of the stationary-action formulation: no minimisation, regular mechanical Lagrangians, no general velocity-constraint equivalence and empirical status

Stationarity is not minimisation. The defining condition of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions is the vanishing of the first variation for all fixed-endpoint variations, and The stationary-action criterion and the potential-part Euler-Lagrange equation identifies it with the potential-part equation, and with the full model equation when $d\equiv 0$. It is not claimed that a stationary curve minimises the action, and no second-variation or convexity hypothesis is imposed on this page: Scope of stationarity: stationary is not minimum, and no second-variation theory is developed here records the explicit quadratic model in which stationary curves are not local minima, and the verification page of this pair checks the analogous phenomenon for an oscillator of the model class.

The mechanical branch is regular. For the mechanical Lagrangians of this page the velocity Hessian in a chart is the kinetic matrix $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ (or $\mathsf{M}$ in Cartesian data), which is symmetric positive definite because $DX(q)$ is injective and $\mathsf{M}$ is; A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian proves that such a chart Lagrangian is hyperregular with the explicit Legendre inverse, so no failure of invertibility of the Legendre map The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity occurs on this branch and the Euler-Lagrange equations determine the coordinate accelerations. A singular velocity Hessian — for which the Legendre map has no $C^1$ local inverse and the second-order equations need not determine accelerations — is outside the regular mechanical branch described here; the admissible coordinate maps of the generalized-coordinate definition of this page exclude it by injectivity of $DX$ and positive definiteness of $\mathsf{M}$.

There is no general action equivalence for velocity constraints. The reduction theorem of this page applies to ideal holonomic constraints. For a genuine velocity constraint, the unrestricted stationary-action or augmented-action prescription can differ from the d'Alembert equation of the preceding page: The velocity constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$: virtual displacements, the d'Alembert law and the vakonomic law defines both prescriptions for the normalized relation $z_s=x{y}_s$ (corresponding physically to ${\dot{r}}_z=(r_x\mathrm{/}{L}_0){\dot{r}}_y$ with a declared length scale) and The d'Alembert and vakonomic motions of $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ differ from the same initial data proves that their evolutions differ for the specified constraint and initial data. Therefore that model branch is not obtained by "imposing a constraint inside the action", and the examples page quantifies this discrepancy. The claim is that equivalence does not hold in general; identifying the prescriptions in another model requires its own justification.

Empirical status. The equivalence of the stationary-action condition with the model's equation of motion is a conditional theorem of the adopted force branch, the declared potential presentation and the cited mathematics: it does not establish that nature extremises or stationarises an action, and it adds no empirical content beyond the force model and its potential presentation. When the presentation has residual forces, the equation in generalized coordinates retains the transformed residual term defined by Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, while the pure action reading gives only the potential-part equation; comparing predictions with observations tests the force law and the apparatus, not the variational reformulation. Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c) supplies the further reminder that the potential and the Lagrangian are defined only up to the declared normalization, so no numerical value of the action is a model prediction without fixing that convention.

Not asserted. No global existence of stationary curves, no theory of minimisers in function spaces, no further criteria for extrema, no free- or variable-endpoint problems, and no field-theoretic or infinite-dimensional actions are part of this page; the boundary-value non-uniqueness of the oscillator, for instance, is recorded in the verification page of the mathematics track, not here.