Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential

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, charts and Hamiltonian field of The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian: masses $m_1,\dots ,m_N>0$ and open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ with mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), time window $I_0$, $C^2$ 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 and SI unit conventions of the non-relativistic classical mechanics framework), the Cartesian Hamiltonian $H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x)$ and, for a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with kinetic metric $G=D{X}^{\mathsf{T}}\mathsf{M}DX$, the chart Hamiltonian $\tilde{H}(t,q,P)=\frac{1}{2}P\cdot G(q{)}^{-1}P+\tilde{U}(t,q)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, 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, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta). Let $J\subseteq I_0$ be a nonempty open interval. For clause 4, a time-dependent phase function is a map $f:I_0\times W\to \mathbb{R}$ of class $C^2$; write $f_t(z):=f(t,z)$ and use The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian to define the associated family of Hamiltonian fields $t\mapsto X_{f_t}$. The forced field is called Hamiltonian on this full chart when it equals $X_{f_t}$ for every $t\in I_0$. Then:

1. (potential branch, Cartesian chart) Assume $d\equiv 0$ on the state domain. For a $C^2$ curve $x:J\to Q$ with $p(t):=\mathsf{M}{x}^{\mathrm{\prime }}(t)$ the following are equivalent: (a) $x$ is a motion of the model (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model); (b) $\dot{x}(t)=H_p(t,x(t),p(t))$ and $\dot{p}(t)=-H_x(t,x(t),p(t))$ for every $t\in J$; (c) the curve $t\mapsto (x(t),p(t))$ is an integral curve of the time-dependent field $t\mapsto X_{H_t}$, i.e. satisfies $\dot{z}(t)=X_{H_t}(z(t))$ with $z=(x,p)$ (First-order systems, initial value problems, and solutions on intervals).
2. (generalized-coordinate chart) 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 the declared Lagrange equations with residual generalized force, $\mathrm{\Lambda }(\tilde{L})(q)(t)=Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))$, are equivalent to $q^{\mathrm{\prime }}(t)={\tilde{H}}_P(t,q(t),P(t))$ and $P^{\mathrm{\prime }}(t)=-{\tilde{H}}_q(t,q(t),P(t))+Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))$; in particular, when $d\equiv 0$, they are equivalent to $(q,P)$ being an integral curve of $X_{{\tilde{H}}_t}$.
3. (residual forces, Cartesian chart) For a $C^2$ curve $x:J\to Q$ with $p=\mathsf{M}{x}^{\mathrm{\prime }}$, the model's forced equation $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}=-{\mathrm{\nabla }}_{x}U(t,x)+d(t,x,x^{\mathrm{\prime }})$ holds on $J$ if and only if $\dot{x}=H_p$ and $\dot{p}=-H_x+d(t,x,{\mathsf{M}}^{-1}p)$ on $J$.
4. (when is the forced field Hamiltonian?) The forced field $(x,p)\mapsto ({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U(t,x)+d(t,x,{\mathsf{M}}^{-1}p))$ is Hamiltonian in the preceding time-dependent sense if and only if there is a $C^2$ function $V:I_0\times Q\to \mathbb{R}$ with $d(t,x,v)=-{\mathrm{\nabla }}_{x}V(t,x)$ for every $(t,x,v)$ of the state domain; in that case the phase function is $H+V$ up to an independently chosen additive $C^2$ function of $t$ on each connected component of $Q$, and the model's forced motions are the motions of the potential presentation $(U+V,0)$.

The statement is conditional on the adopted force branch, the declared presentation and the declared chart; it asserts no existence or uniqueness of solutions, no completeness, no conservation law and no empirical content.

• No solution theory is asserted. The theorem compares three descriptions of the same first-order equations; existence, uniqueness and maximal domains of solutions are provided separately by the model's initial-value theory and by the ordinary ODE theory used on the later items of this page.

• Why clause 4 matters. It delimits the model's Hamiltonian branch: a residual force contributes a Hamiltonian field only when it derives from a potential independent of the velocity slot, which is why the flow and volume-preservation items below state the potential branch explicitly.