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

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 momentum charts and two-form datum 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 and the canonical coordinates of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $\omega ={\sum }_{a=1}^{m}d{\theta }^a\wedge d{\theta }^{m+a}$ with $m$ the chart's half-dimension, and the model's own momentum charts: the Cartesian phase domain $W=Q\times {\mathbb{R}}^{3N}$, its two-form $\omega$, a generalized-coordinate chart map ${\mathrm{\Phi }}_X$ and the form ${\omega }_X$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces); the presentation is hyperregular on every chart with the Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions and 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,

$H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x),\tilde{H}(t,q,P)=\frac{1}{2}P\cdot G(q{)}^{-1}P+\tilde{U}(t,q)\mathrm{.}$

Differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms, and gradients those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case.

(a) The field of a phase function. Let $W_0\subseteq W$ be nonempty open and let $f:W_0\to \mathbb{R}$ be of class $C^2$. By clause 1 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ the form $\omega$ is nondegenerate on $W_0$, so there is exactly one $C^1$ vector field $X_f$ on $W_0$ with

${\iota }_{X_f}\omega =df$

(the sign convention of Hamiltonian vector field and Hamiltonian function and The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$); it is the Hamiltonian field of $f$ for the presentation. In the Cartesian coordinates $(x,p)$, writing $\mathrm{\nabla }f=(f_{x^1},\dots ,f_{x^{3N}},f_{p_1},\dots ,f_{p_{3N}})$,

$X_f=(f_p,-f_x)=J\mathrm{\nabla }f,J=\left(\begin{array}{cc}{0}_{3N}& I_{3N}\\ -I_{3N}& 0_{3N}\end{array}\right);$

in any canonical chart $({\theta }^1,\dots ,{\theta }^{2m})$ of the presentation the same field reads $X_f=(f_{{\theta }^{m+1}},\dots ,f_{{\theta }^{2m}},-f_{{\theta }^1},\dots ,-f_{{\theta }^m})$. A vector field on $W_0$ is called Hamiltonian for the presentation when it is $X_f$ for some $C^2$ function $f$; in particular $X_f=0$ exactly when $f$ is locally constant on $W_0$. Vector fields are read as tangent-bundle sections of A smooth vector field is a smooth section of the tangent bundle.

(b) The field of the branch Hamiltonian. For each $t\in I_0$ the function $H_t:=H(t,\cdot )$ is $C^2$ on $W$, and the Hamiltonian field of the presentation at time $t$ is $X_{H_t}$:

$X_{H_t}(x,p)=({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U(t,x)),$

whose components are a velocity and a force. In the coordinates $X$ the corresponding field is $X_{{\tilde{H}}_t}(q,P)=(G(q{)}^{-1}P,-{\mathrm{\partial }}_q\tilde{H}(t,q,P))$, where the second slot retains the kinetic contribution $-\frac{1}{2}P\cdot {\mathrm{\partial }}_q(G(q{)}^{-1})P$ when the metric varies. If $d\equiv ̸0$ the model's forced evolution carries the additional momentum source $d$; the corresponding vector field $(x,p)\mapsto ({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U(t,x)+d(t,x,{\mathsf{M}}^{-1}p))$ is generally not a Hamiltonian field for the presentation, and it is one exactly when the residual force has a potential independent of the velocity slot (the criterion is proved in clause 4 of the corresponding theorem of this page).

(c) Time dependence and status. For a time-dependent $H$ the assignment $t\mapsto X_{H_t}$ is a time-dependent field on the phase domain; each $X_{H_t}$ is $C^1$, hence locally Lipschitz in the state variable, and no completeness of its integral curves is asserted here. The item defines the field datum and its chart expressions; the identification of the model's motions with the integral curves, and all conservation or invariance properties, are the subject of the other items of this page. The field is defined by the intrinsic equation ${\iota }_X\omega =dH$ and therefore does not depend on the canonical chart used to compute it.