Legendre transformation and Hamiltonian mechanics

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, in the fixed inertial frame with adapted coordinates $(t,x)$ used there: $N\ge 1$ labels with masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U:I_0\times Q\to \mathbb{R}$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, the Euclidean inner product of ${\mathbb{R}}^{3N}$ is The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and gradients and Jacobians are those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$.

(a) Cartesian momentum. For a state $(t,x,v)\in I_0\times Q\times {\mathbb{R}}^{3N}$ define the momentum of the labels by

$p(t,x,v):=\mathsf{M}v=(m_1{v}_1,\dots ,m_N{v}_N),$

whose label components are $p_i=m_i{v}_i$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (d)); its dimension is $\mu \mathrm{\ell }{\tau }^{-1}$ (read in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$). The space ${\mathbb{R}}^{3N}$ is identified with its dual by the Euclidean inner product, so that $p$ is a covector acting on a velocity $w$ by the dot product $p\cdot w$. The pair $(x,p)$ is the cotangent datum of the state, and $I_0\times Q\times {\mathbb{R}}^{3N}$ with coordinates $(t,x,p)$ is the momentum phase chart of the presentation.

(b) Fibre derivative. The fibre derivative of the presentation is the velocity-slot gradient $p:=L_v(t,x,v)$ at each state. When $Q\mathrm{\ne }\mathrm{\varnothing }$, this is the momentum coordinate of the Legendre map of the fibre-full chart $(O,L)$ of clause (c) of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, in the sense of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity (the chart data are those of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Because $L=K-U$ with $K^{\mathrm{\prime }}(v)=\mathsf{M}v$ and $U$ independent of the velocity slot (clause (a) of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions), one has $L_v(t,x,v)=\mathsf{M}v$: the fibre derivative and the momentum of clause (a) are the same covector, and for $Q\mathrm{\ne }\mathrm{\varnothing }$ the Legendre map of the presentation is $(t,x,v)\mapsto (t,x,\mathsf{M}v)$. If $Q=\mathrm{\varnothing }$, the state and momentum domains are empty and the momentum and fibre-derivative assignments are the unique empty maps; there is no fibre-full chart instance of the cited mathematical definition.

(c) Generalized momenta. Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map of the model, with substituted Lagrangian $\tilde{L}$, substituted potential $\tilde{U}$ and kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clauses (a)-(b), Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, The transpose $A^{\mathsf{T}}$ of a matrix). The generalized momentum of the presentation in the coordinates $X$ is the covector

$p(t,q,v):={\tilde{L}}_v(t,q,v)=G(q)v,$

whose components $p_a=\mathrm{\partial }\tilde{L}\mathrm{/}\mathrm{\partial }{v}_a$ have dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}{\gamma }_a^{-1}$ (action divided by the dimension ${\gamma }_a$ of the coordinate $q_a$), while the pairing $p\cdot v={\sum }_a{p}_a{v}^a$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$. For the Cartesian chart when $Q\mathrm{\ne }\mathrm{\varnothing }$, $X=\mathrm{i}\mathrm{d}$, $\mathrm{\Omega }=Q$, $G=\mathsf{M}$, and clause (c) reduces to clause (a). If $Q=\mathrm{\varnothing }$, no generalized-coordinate map exists, since its domain $\mathrm{\Omega }$ is nonempty.

(d) Status and warnings. The item defines the momentum datum of a declared presentation as a model object. It introduces no force law, no equation of motion and no measured value; it depends on the declared chart and on the declared coordinate dimensions ${\gamma }_a$, and it is independent of the residual forces $d$ and of the normalization $U\mapsto U+c(t)$ of the potential representative (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c)). The momentum is not in general mass times coordinate velocity: in the coordinates $X$ it is $G(q)v$, and the identity $p_i=m_i{v}_i$ is specific to Cartesian data. The evaluation of the velocity Hessian, the invertibility of the fiberwise momentum map, the Hamiltonian of an inverse branch and the evolution equations are the subject of the other items of this page, not of this definition. All differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

• Why a covector. The physical content of the definition is the choice of the momentum as the fibre derivative ${\mathcal{L}}_v$ rather than the kinematic velocity: in an inertial Cartesian chart the two differ only by the positive definite mass operator, $p=\mathsf{M}v$, whereas in generalized coordinates the difference is the configuration-dependent kinetic metric $G(q)$, and $p_a$ has the dimension of action per unit of $q_a$. The covector $p$ pairs with velocities by the Euclidean product in coordinates; no metric on momentum space is introduced here.

• Comparison with the sources. The identification of the generalized momentum with the velocity gradient of the Lagrangian, together with the warning that it equals the familiar momentum only in Cartesian coordinates, is the momentum (4.2) of Tong's Section 4.1; Likharev's Section 10.1 and Stewart's Section 1.2 use the same definition, and SICM's Section 3.1.1 defines the momentum state function as the partial derivative of the Lagrangian. None of these treatments is used here as a proof: the item only fixes the model objects and the notation, and the mathematical computations are those of the cited items of this page.

• Not a phase-space claim. Nothing here asserts that the momentum coordinates cover the states of a mechanical model globally in a coordinate-free sense, that the Legendre map is invertible, or that the model's evolution is Hamiltonian; those are conditional statements proved on this page under their own hypotheses.

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions in the fixed inertial frame with adapted coordinates $(t,x)$: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, and the Euclidean gradients and second partial derivatives in the velocity slot are those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$.

(a) Velocity Hessian. When $Q\mathrm{\ne }\mathrm{\varnothing }$, let $(O,L)$ be the fibre-full time-dependent Lagrangian chart of clause (c) of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions and let $\tilde{L}$ be the substituted Lagrangian of a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). The velocity Hessian of the presentation in Cartesian data is the matrix $L_{vv}(t,x,v)\in M_{3N}(\mathbb{R})$ of second partial derivatives of $L$ in its third slot at $(t,x,v)$; in the coordinates $X$ it is the matrix ${\tilde{L}}_{vv}(t,q,v)$ of second partial derivatives of $\tilde{L}$ in its third slot. Invertibility of the Hessian is invertibility of a linear map in the sense of Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, read by A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit through the determinant, and matrices act on vectors by Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes and Linear map between vector spaces over the same field; transposes are those of The transpose $A^{\mathsf{T}}$ of a matrix.

(b) Regular states and regular presentations. The presentation is velocity-regular at the state $(t,x,v)$ of the Cartesian chart when $\det L_{vv}(t,x,v)\mathrm{\ne }0$, and velocity-regular at the state $(t,q,v)$ of the chart of $X$ when $\det {\tilde{L}}_{vv}(t,q,v)\mathrm{\ne }0$. It is regular on a chart when it is velocity-regular at every state of that chart, and regular when this holds on the chart under discussion. This is exactly the regularity of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity instantiated, for $Q\mathrm{\ne }\mathrm{\varnothing }$, at the corresponding fibre-full chart: the Legendre map is regular at a state precisely when its velocity Hessian there is nonsingular. The evaluation of the Hessian for the model's mechanical presentations, and the conclusion that those presentations are regular at every state, are proved on this page (the local Legendre theorem) and are not part of this definition. If $Q=\mathrm{\varnothing }$, the Cartesian state domain is empty and regularity there is vacuous; there is no fibre-full chart instance of the cited mathematical definition and no generalized-coordinate map, whose domain $\mathrm{\Omega }$ must be nonempty.

(c) Local inverse branch data. Let the presentation be velocity-regular at a state $z$ of a chart with Legendre map $\mathbb{F}$ and momentum image $z^{\mathrm{\prime }}:=\mathbb{F}(z)$. A local inverse branch at $z$ is a pair of open sets $W\ni z$ and $W^{\mathrm{\prime }}\ni z^{\mathrm{\prime }}$ together with a $C^1$ map $v_W:W^{\mathrm{\prime }}\to W$ that is inverse to $\mathbb{F}{\mathrm{\mid }}_W:W\to W^{\mathrm{\prime }}$; equivalently, $\mathbb{F}{\mathrm{\mid }}_W$ is a $C^1$ diffeomorphism onto $W^{\mathrm{\prime }}$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. A local inverse branch assigns to every momentum covector in $W^{\mathrm{\prime }}$ a state in $W$ whose fibre derivative is that covector. Whether such a branch exists at a regular state, and when it extends to a global fibre inverse, are conditional statements of the local and global Legendre theorems of this page.

(d) Status. The item fixes the regularity vocabulary of the presentation and the data used by the local Legendre theorem; it asserts no existence of inverse branches, no hyperregularity, no equation of motion and no empirical value. The regularity condition is a condition on the declared chart data $(U,\mathsf{M})$ and, in the coordinates $X$, on the declared map $X$; no such condition is imposed on the residual forces $d$, which do not enter the velocity Hessian of $L=K-U$.

• Regularity is not hyperregularity. Regularity is a pointwise nonsingularity condition, whereas the existence of a globally defined inverse Legendre branch on a whole momentum fibre is the separate condition of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity; the two are compared on this page, and the example $\mathcal{L}(v)=e^v$ of the companion page shows that a presentation can be regular at every state and still fail to admit a global fibre inverse. The mechanical presentations of this framework lie on the positive side for both conditions, as the local and global Legendre theorems record.

• Source caveat. Tong's Section 4.1.1 phrases the invertibility requirement through convexity of the Lagrangian and the existence of a maximum; the scaffold's source audit records that this phrasing is too restrictive for local mechanical Legendre inversion, where the relevant hypothesis is nonsingularity of the velocity Hessian and a global inverse is needed only for global equivalence. The definition above follows the nonsingularity criterion; convexity is nowhere assumed, and the cited mathematical items prove the criterion.

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance); dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework and differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

(a) Cartesian chart. The presentation is hyperregular on the Cartesian chart when for every $(t,x)\in I_0\times Q$ the fibre momentum map

${\mathbb{F}}_{t,x}:{\mathbb{R}}^{3N}⟶{\mathbb{R}}^{3N},{\mathbb{F}}_{t,x}(v):=L_v(t,x,v)=\mathsf{M}v,$

is a bijection of ${\mathbb{R}}^{3N}$ onto ${\mathbb{R}}^{3N}$ and the inverse assignment

$v:I_0\times Q\times {\mathbb{R}}^{3N}⟶{\mathbb{R}}^{3N},L_v(t,x,v(t,x,p))=p,$

is defined on the whole momentum phase chart $I_0\times Q\times {\mathbb{R}}^{3N}$ and is of class $C^1$. In that case $v$ is the inverse Legendre branch of the presentation on the Cartesian chart. For $Q\mathrm{\ne }\mathrm{\varnothing }$, hyperregularity of this chart is exactly hyperregularity of its fibre-full Lagrangian in the sense of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, read through the identification of the fibre derivative with the momentum of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta. If $Q=\mathrm{\varnothing }$, the fibre condition is vacuous and the inverse assignment is the unique empty $C^1$ map; this is an empty-domain convention, not an instance of that supplier's fibre-full chart, whose base must be nonempty.

(b) Generalized-coordinate chart. Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$ and kinetic metric $G$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). The presentation is hyperregular on the chart of $X$ when for every $(t,q)\in I_0\times \mathrm{\Omega }$ the fibre map $v\mapsto {\tilde{L}}_v(t,q,v)=G(q)v$ is a bijection of ${\mathbb{R}}^r$ onto ${\mathbb{R}}^r$ and the inverse assignment $v(t,q,p)$ on $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$ is of class $C^1$; $v$ is then the inverse Legendre branch in the coordinates $X$. Linear maps and bijectivity are read by Linear map between vector spaces over the same field, and the pairing of momentum covectors with velocities is the Euclidean pairing of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

(c) Status. The item defines the hyperregularity condition and the branch objects for the two chart types of the model. It asserts no branch existence, no relation between regularity Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts and hyperregularity, no equation of motion and no empirical value. The global Legendre theorem for the mechanical branch on this page establishes hyperregularity for every admitted Cartesian and generalized-coordinate presentation. In the broader class of fibre-full $C^2$ Lagrangian charts of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, regularity alone does not imply hyperregularity: the companion page's example $\mathcal{L}(v)=e^v$ has a nonsurjective fibre momentum map. That example is outside the model's mechanical class and witnesses the separation only in this broader class.

• Two degrees of globality. Hyperregularity has two quantifiers: the fibre map must be bijective for every $(t,q)$ (surjectivity and injectivity on each momentum fibre), and the inverse assignment must be defined on the whole phase domain and be $C^1$ there. Injectivity alone gives no description of which momenta occur. Fibrewise bijectivity determines a unique inverse assignment; its joint $C^1$ regularity on the whole phase domain is a further requirement of the definition.

• The inverse branch is attached to the declared chart. The next definition of this page defines the Hamiltonian of the presentation from the inverse Legendre branch introduced here; consequently each Hamiltonian statement names the chart on which hyperregularity and the branch are taken, and on a fixed hyperregular chart the inverse is unique by fibre bijectivity. Different charts express the inverse in their own coordinates. The later global Legendre theorem on this page proves its existence for every admitted mechanical presentation; that conclusion is not an existence assertion of the present definition.

• Source caveat. The cited treatments invert the velocity-momentum relation on the branch where it is invertible and do not distinguish the pointwise, local and global levels of the inverse; the scaffold's source audit notes that a global inverse is needed for global equivalence between the Lagrangian and Hamiltonian descriptions, which is why the condition above is stated globally in the fibre. The mathematical content of regularity, local inversion and global inversion is supplied by the cited items of this page, not by the physical treatments.

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance); dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework and the Euclidean pairing is The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

(a) Cartesian Hamiltonian. Assume the presentation is hyperregular on the Cartesian chart, with inverse Legendre branch $v:I_0\times Q\times {\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ (Hyperregular potential presentations and their inverse Legendre branches in the model's charts, The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity). The Hamiltonian of the presentation on the Cartesian chart is the function

$H:I_0\times Q\times {\mathbb{R}}^{3N}⟶\mathbb{R},H(t,x,p):=p\cdot v(t,x,p)-L(t,x,v(t,x,p)),$

where $p\cdot v(t,x,p)$ is the Euclidean pairing of the momentum covector with the inverse-branch velocity (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). The momentum phase chart $I_0\times Q\times {\mathbb{R}}^{3N}$ with coordinates $(t,x,p)$ is the Hamiltonian phase chart of the presentation on the Cartesian chart. If $Q=\mathrm{\varnothing }$, the formula denotes the unique empty function; the identification with the cited mathematical fibre-full chart applies only when $Q\mathrm{\ne }\mathrm{\varnothing }$.

(b) Generalized-coordinate Hamiltonian. Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$, kinetic metric $G$ and inverse Legendre branch $v$ on $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Hyperregular potential presentations and their inverse Legendre branches in the model's charts). The Hamiltonian of the presentation in the coordinates $X$ is

$\tilde{H}:I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r⟶\mathbb{R},\tilde{H}(t,q,p):=p\cdot v(t,q,p)-\tilde{L}(t,q,v(t,q,p))\mathrm{.}$

For the Cartesian chart when $Q\mathrm{\ne }\mathrm{\varnothing }$, $X=\mathrm{i}\mathrm{d}$ and $\tilde{H}=H$; the general definition is the same formula applied to the substituted fibre-full chart.

(c) Dimensions and conventions. Each term of the defining difference has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (energy): the pairing $p\cdot v$ multiplies $\mu \mathrm{\ell }{\tau }^{-1}$ by $\mathrm{\ell }{\tau }^{-1}$ in Cartesian data (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta), and $L$ has energy dimension (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). In the coordinates $X$ the pairing is dimensionally consistent for every declared coordinate dimension ${\gamma }_a$ and the dimension is unchanged. The Hamiltonian is $C^1$ on its domain, being a sum of products of $C^1$ maps ($C^k$ Euclidean maps are closed under componentwise algebra and composition, $C^k$ Euclidean maps and diffeomorphisms). The pairing is the coordinate Euclidean product; no metric on momentum space and no symplectic form is introduced by this definition.

(d) Dependence and status. The Hamiltonian depends on the declared chart, on its inverse Legendre branch, uniquely determined on that chart by fibre bijectivity (Hyperregular potential presentations and their inverse Legendre branches in the model's charts), and on the potential representative: replacing $U$ by $U+c(t)$ leaves the branch unchanged and replaces $L$ by $L-c(t)$, hence replaces $H$ by $H+c(t)$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c), The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions clause (b)). It asserts no equation of motion, no conservation property and no measured value; the explicit kinetic-plus-potential form of $H$ for the mechanical branch, the equivalence of its equations with the model's equation of motion and its relation to the mechanical energy are the theorems of this page. Matrix identities used by those theorems are read through Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes.

• Why the branch matters. The definition is the mathematical Legendre transform ${\mathcal{L}}_v\mapsto p$, $H=p\cdot v-\mathcal{L}$, evaluated on the branch $v(\cdot )$; the notation $H(t,q,p)$ presupposes that the branch is a function of the momentum, which is exactly the content of hyperregularity. At a merely regular state the formula defines a local Hamiltonian only, on the momentum image of the local branch of Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts; the global mechanical case is treated later on this page.

• Energy is not defined by the formula alone. The value $p\cdot v-L$ is an energy-dimension scalar built from the declared chart; whether it is the mechanical energy $K+U$ of the model, and under which chart choices, is the conditional statement of the energy item of this page. In particular a time-dependent chart can produce a Hamiltonian that differs from the mechanical energy, and the companion page records such a computation.

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U:I_0\times Q\to \mathbb{R}$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$ and kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). Then:

1. (velocity Hessian) For every $(t,x,v)\in I_0\times Q\times {\mathbb{R}}^{3N}$ one has $L_v(t,x,v)=\mathsf{M}v$ and $L_{vv}(t,x,v)=\mathsf{M}$, and for every $(t,q,v)\in I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$ one has ${\tilde{L}}_v(t,q,v)=G(q)v$ and ${\tilde{L}}_{vv}(t,q,v)=G(q)$. Both matrices are symmetric and positive definite; consequently the Cartesian presentation and the presentation in the coordinates $X$ are velocity-regular at every state in the sense of Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts.

2. (local inverse branch) At every state $z$ of either chart there is a local inverse branch: open sets $W\ni z$ and $W^{\mathrm{\prime }}\ni \mathbb{F}(z)$ with $\mathbb{F}{\mathrm{\mid }}_W:W\to W^{\mathrm{\prime }}$ a $C^1$ diffeomorphism, where $\mathbb{F}$ is the Legendre map of the corresponding fibre-full chart (Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts clause (c), The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity).

The statement is conditional on the adopted force branch and on the declared presentation; it asserts no global inverse Legendre branch on a momentum fibre and no empirical content.

• What is used and what is not. The proof uses only the positivity of the mass operator and of the kinetic metric, the identification of the model's Lagrangian with a mechanical chart Lagrangian, and the local inverse-function criterion of Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse; it does not use hyperregularity, and it does not claim that the local branches globalise. The global statement for the mechanical branch is proved later on this page, and the companion example $\mathcal{L}(v)=e^v$ shows that regularity at every state can fail to give a global fibre inverse.

• Source reading. The worked case $p=m\dot{r}$, $H=p^2\mathrm{/}(2m)+V(r)$ of Tong's Section 4.1.3 is the Cartesian instance of clauses 1 and 2; the general quadratic form with positive definite kinetic matrix appears in Stewart's Section 1.2, and SICM's Section 3.1.1 states explicitly that the Legendre transform is available in closed form for velocity-quadratic Lagrangians because the velocity derivative is then invertible. The source audit's caveat that convexity is not the relevant hypothesis is respected: the hypothesis used here is the nonsingularity recorded in clause 1.

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U:I_0\times Q\to \mathbb{R}$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$, kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ and substituted potential $\tilde{U}(t,q)=U(t,X(q))$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces), and let $H$ and $\tilde{H}$ be the Hamiltonians of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions. Then:

1. (Cartesian chart) The fibre momentum map ${\mathbb{F}}_{t,x}(v)=\mathsf{M}v$ is a bijection of ${\mathbb{R}}^{3N}$ onto ${\mathbb{R}}^{3N}$ for every $(t,x)\in I_0\times Q$, and the inverse Legendre branch is the $C^2$ map $v(t,x,p)={\mathsf{M}}^{-1}p$ on the whole phase chart $I_0\times Q\times {\mathbb{R}}^{3N}$. Hence the presentation is hyperregular on the Cartesian chart in the sense of Hyperregular potential presentations and their inverse Legendre branches in the model's charts, and the Hamiltonian of the branch is

$H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x),$

with the dot product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$,

a $C^2$ function of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$.

2. (generalized-coordinate chart) The fibre map $v\mapsto G(q)v$ is a bijection of ${\mathbb{R}}^r$ onto ${\mathbb{R}}^r$ for every $(t,q)\in I_0\times \mathrm{\Omega }$, and the inverse Legendre branch is the $C^2$ map $v(t,q,p)=G(q{)}^{-1}p$ on $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$. Hence the presentation is hyperregular on the chart of $X$, and the Hamiltonian in the coordinates $X$ is

$\tilde{H}(t,q,p)=\frac{1}{2}p\cdot G(q{)}^{-1}p+\tilde{U}(t,q),$

a $C^2$ function of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$.

The statement is conditional on the adopted force branch and the declared presentation; it asserts no equation of motion, no conservation law and no empirical content.

• The model is never singular. Clauses 1 and 2 show that the mechanical presentations of the framework are hyperregular on every chart of the model: the positive definiteness of the mass operator and of the kinetic metric makes each fibre momentum map a linear isomorphism, so the inverse branch is global in the fibre and given in closed form. This is the positive case anticipated by Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts and Hyperregular potential presentations and their inverse Legendre branches in the model's charts; no such conclusion holds for general Lagrangians, as the companion page's examples record.

• A linear specialisation of the cited proposition. The mathematical content is the proposition A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian instantiated at the model's data; the physical content is the identification of those data with the declared presentation, which fixes the meanings $p=\mathsf{M}v$, $v={\mathsf{M}}^{-1}p$ and $H=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U$ in terms of the labels, masses and potential. The residual forces $d$ do not enter the branch or the Hamiltonian; they reappear as momentum sources in the equivalence theorem of this page.

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U:I_0\times Q\to \mathbb{R}$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Let $H$ be the Cartesian Hamiltonian of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$, kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$, substituted potential $\tilde{U}(t,q)=U(t,X(q))$ and residual generalized force $Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q,v)=DX(q{)}^{\mathsf{T}}d(t,X(q),DX(q)v)$, and let $\tilde{H}$ be the Hamiltonian in the coordinates $X$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions). Let $J\subseteq I_0$ be a nonempty open interval and let $x:J\to Q$ be a $C^2$ curve, with $p(t):=\mathsf{M}{x}^{\mathrm{\prime }}(t)$ (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta). Then:

1. (potential branch) Assume $d\equiv 0$ on the state domain. The following are equivalent: (a) $x$ is a motion of the model on $J$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model); (b) $(x,p)$ satisfies $x^{\mathrm{\prime }}(t)=H_p(t,x(t),p(t))$ and $p^{\mathrm{\prime }}(t)=-H_x(t,x(t),p(t))$ for every $t\in J$; (c) for all reals $a<b$ with $[a,b]\subseteq J$ the restriction $x{\mathrm{\mid }}_{[a,b]}$ is stationary for the action $S$ of $L$ on $[a,b]$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, The stationary-action criterion and the potential-part Euler-Lagrange equation).

2. (residual forces) For a $C^2$ curve $x:J\to Q$ with $p(t)=\mathsf{M}{x}^{\mathrm{\prime }}(t)$ the following are equivalent: (a) $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=-{\mathrm{\nabla }}_{x}U(t,x(t))+d(t,x(t),x^{\mathrm{\prime }}(t))$ for every $t\in J$; (b) $x^{\mathrm{\prime }}(t)=H_p(t,x(t),p(t))$ and $p^{\mathrm{\prime }}(t)=-H_x(t,x(t),p(t))+d(t,x(t),{\mathsf{M}}^{-1}p(t))$ for every $t\in J$.

3. (coordinate form) Let $q:J\to \mathrm{\Omega }$ be a $C^2$ curve and put $p(t):=G(q(t))q^{\mathrm{\prime }}(t)$. Then the declared Lagrange equations of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clause (e) are equivalent to the Hamilton system of $\tilde{H}$: for every $t\in J$,

$\mathrm{\Lambda }(\tilde{L})(q)(t)=Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))⟺q^{\mathrm{\prime }}(t)={\tilde{H}}_p(t,q(t),p(t)),p^{\mathrm{\prime }}(t)=-{\tilde{H}}_q(t,q(t),p(t))+Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t));$

in particular, when $d\equiv 0$ the vanishing of the Lagrange operator is equivalent to $q^{\mathrm{\prime }}={\tilde{H}}_p$ and $p^{\mathrm{\prime }}=-{\tilde{H}}_q$.

4. (constraint elimination) Assume in addition that $d\equiv 0$ and that $X$ is a parametrization of a scleronomic regular holonomic-constraint locus in the sense of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clause (d), with the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Then the equivalent conditions of clause 3 with $Q^{\mathrm{r}\mathrm{e}\mathrm{s}}\equiv 0$ hold for $q$ if and only if $x=X\circ q$ is a constrained motion of the model with an ideal reaction (Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation).

The statement is conditional on the adopted force branch, the declared presentation, the declared chart and, in clause 4, the constraint datum and ideal-reaction law; it asserts no existence or uniqueness of solutions, no conservation law and no empirical content.