The mechanical Hamiltonian is the model's energy in the inertial chart, its balance is the energy balance, and the substituted-chart Hamiltonian is the energy of time-independent coordinates

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 and let, for a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with substituted Lagrangian $\tilde{L}$, kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ and substituted potential $\tilde{U}(t,q)=U(t,X(q))$, the symbol $\tilde{H}$ denote the Hamiltonian in the coordinates $X$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). Let $J\subseteq I_0$ be a nonempty open interval and let $x:J\to Q$ be a $C^2$ curve with $p(t):=\mathsf{M}{x}^{\mathrm{\prime }}(t)$; write $K(t)=\frac{1}{2}{x}^{\mathrm{\prime }}(t)\cdot \mathsf{M}{x}^{\mathrm{\prime }}(t)$ for the kinetic energy and $E(t):=K(t)+U(t,x(t))$ for the mechanical energy along $x$ (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Then:

1. (the Hamiltonian is the energy) For every $(t,x,p)\in I_0\times Q\times {\mathbb{R}}^{3N}$, $H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x)$; consequently along every $C^2$ curve $x:J\to Q$ one has $H(t,x(t),p(t))=\frac{1}{2}{x}^{\mathrm{\prime }}(t)\cdot \mathsf{M}{x}^{\mathrm{\prime }}(t)+U(t,x(t))=E(t)$ for every $t\in J$: on the Cartesian chart the Hamiltonian of the presentation is the model's mechanical energy.

2. (balance and conservation) If $x$ is a motion of the model (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model) with evaluated residual forces ${\mathfrak{d}}_i(t):=d_i(t,x(t),x^{\mathrm{\prime }}(t))$, then $t\mapsto H(t,x(t),p(t))$ is differentiable on $J$ and

$\frac{d}{dt}H(t,x(t),p(t))={\mathrm{\partial }}_{t}U(t,x(t))+{\sum }_{i=1}^N⟨{\mathfrak{d}}_i(t),x_i^{\mathrm{\prime }}(t)⟩(t\in J);$

in particular, if ${\mathrm{\partial }}_{t}U\equiv 0$ on $I_0\times Q$ and $d\equiv 0$, then $H(t,x(t),p(t))$ is constant on $J$.

3. (time-independent coordinates) Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map and let $q:J\to \mathrm{\Omega }$ be a $C^2$ curve with $x:=X\circ q$ and $p(t):=G(q(t))q^{\mathrm{\prime }}(t)$. Then $\tilde{H}(t,q,p)=\frac{1}{2}p\cdot G(q{)}^{-1}p+\tilde{U}(t,q)$ and

$\tilde{H}(t,q(t),p(t))=\frac{1}{2}{q}^{\mathrm{\prime }}(t)\cdot G(q(t))q^{\mathrm{\prime }}(t)+\tilde{U}(t,q(t))=K(t)+U(t,x(t))=E(t):$

the substituted-chart Hamiltonian of the induced configuration curve is the same mechanical energy of the model.

The statement is conditional on the adopted force branch and the declared presentation; it asserts no measured value, no conservation outside the stated hypotheses and no identity between $H$ and the mechanical energy for charts other than those of clauses 1 and 3.

• Energy is chart-relative. Clauses 1 and 3 identify the branch Hamiltonian with the mechanical energy for the inertial Cartesian chart and for time-independent generalized-coordinate charts of the model. For a time-dependent chart the substituted Lagrangian acquires a velocity-linear term, the branch Hamiltonian is $p\cdot v-L$ for the new chart data, and it need not equal $K+U$; the companion page computes a moving-chart example in which the chart Hamiltonian is conserved while the mechanical energy is not, so no general identity may be read off from the Legendre formula alone.

• Balance is inherited, not reproved. Clause 2 is the model's energy balance of Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form read through the identification of clause 1; the explicit time dependence of the potential and the power of the residual forces are the only sources, and a static potential with no residual forces conserves the Hamiltonian along motions. The representation-dependence recorded there is inherited: replacing $U$ by $U+c(t)$ shifts $H$ by $c(t)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions clause (d)) and shifts the balance by $c^{\mathrm{\prime }}(t)$.

• Relation to the equivalence theorem. Clause 1 also shows that along solutions of the unforced Hamilton equations, $H$ is the mechanical energy in the inertial chart. Clause 3 of The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity gives $\frac{d}{dt}H=-L_t={\mathrm{\partial }}_{t}U$ along those solutions; it gives conservation when ${\mathrm{\partial }}_{t}U$ vanishes along the trajectory. Their identification with model motions when $d\equiv 0$, and the corresponding forced equations when residual forces are present, are 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. Neither statement asserts that any apparatus conserves energy, and neither reports a measured value.