The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta

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.