The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions

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.