Scope of the Legendre transformation: singular velocity Hessians, regular versus hyperregular presentations, and the excluded constraint theory

Singular velocity Hessians are excluded. The Legendre map of a presentation is invertible through a state only when its velocity Hessian there is nonsingular (Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse, Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts); when $\det L_{vv}=0$ at a state, no local inverse branch through that state exists, the momentum covector of the state belongs to the image of a map that cannot be inverted there, and the definition The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions of the branch Hamiltonian has no input at that state. Nothing on this page assigns a Hamiltonian to such a state, and no theorem of this page is stated for a presentation that is not regular on the chart under discussion.

The mechanical branch is never singular. For the presentations of the framework the velocity Hessian is the mass operator $\mathsf{M}$ in Cartesian data and the kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ in a generalized-coordinate chart (Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian, Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). Both are symmetric positive definite because the masses are positive and $DX(q)$ is injective, so both are nonsingular at every state and the fibre momentum maps are linear isomorphisms; the excluded degenerate case simply does not arise for the model's own Lagrangians. This is a property of the adopted force branch and its kinetic energy, not a general fact about Lagrangians.

Regular does not imply hyperregular. Regularity is the pointwise nonsingularity of the velocity Hessian; hyperregularity additionally requires the fibre momentum map to be a bijection onto the whole momentum fibre with a globally defined inverse branch (Hyperregular potential presentations and their inverse Legendre branches in the model's charts). The two conditions are genuinely different: the Lagrangian $\mathcal{L}(v)=e^v$ is regular at every state, since ${\mathcal{L}}_{vv}=e^v>0$, yet its fibre Legendre map has image $(0,\mathrm{\infty })$ and no inverse at nonpositive momenta, so it admits no global branch Hamiltonian; Scope of the Legendre map: regular versus hyperregular, the meaning of the covector datum, and the limits of the branch description records this separation and further witnesses. The companion page of this pair computes the branch $H(p)=p\ln p-p$ of that example on $p>0$.

Constraints are covered only by elimination. The only constraint statement of this page is clause 4 of Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart, in which a scleronomic regular holonomic constraint is eliminated by a parametrization and the reaction never appears in the reduced equations. Constraint theory in phase space — multipliers adjoined to a Hamiltonian system, primary and secondary constraints, or any generalised constraint algorithm — is not defined, proved, or claimed on this page; the cited treatments likewise either restrict themselves to constraints imposed on Lagrangians or refer that subject elsewhere. In particular, the Lagrange-multiplier function of an ideal-reaction representation has a singular velocity Hessian in the ambient variables and is not itself a hyperregular presentation of this page, as the constraint items of the earlier pair record.

Empirical status. Every condition and conclusion above is a statement about the adopted model: the force branch, the declared potential presentation and the cited mathematics. No measurement is reported, no numerical value is asserted, and the vocabulary of regular, hyperregular and Hamiltonian presentations adds no empirical content to the model's equation of motion.