Hamiltonian vector field

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.^[1]

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of f and g.

Definition

Suppose that $(M, ω)$ is a symplectic manifold. Since the symplectic form $ω$ is nondegenerate, it sets up a fiberwise-linear isomorphism

\omega:TM\to T^*M,

between the tangent bundle $TM$ and the cotangent bundle $T*M$ , with the inverse

\Omega:T^*M\to TM, \quad \Omega=\omega^{-1}.

Therefore, one-forms on a symplectic manifold $M$ may be identified with vector fields and every differentiable function $H : M \to R$ determines a unique vector field $X H$ , called the Hamiltonian vector field with the Hamiltonian $H$ , by defining for every vector field $Y$ on $M$ ,

\mathrm {d} H(Y)=\omega (X_{H},Y).

Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.

Examples

Suppose that $M$ is a $2 n$ -dimensional symplectic manifold. Then locally, one may choose canonical coordinates $(q 1, ..., q n, p 1, ..., p n)$ on $M$ , in which the symplectic form is expressed as^[2]

\omega=\sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,

where $d$ denotes the exterior derivative and $\land$ denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian $H$ takes the form^[3]

\Chi_H=\left( \frac{\partial H}{\partial p_i}, - \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,

where $Ω$ is a $2 n \times 2 n$ square matrix

\Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix},

and

\mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.

The matrix $Ω$ is frequently denoted with $J$ .

Suppose that M = R²ⁿ is the 2n-dimensional symplectic vector space with (global) canonical coordinates.

If $H=p_{i}$ then $X_H=\partial/\partial q^i;$
if $H=q_{i}$ then $X_H=-\partial/\partial p^i;$
if $H=1/2\sum (p_i)^2$ then $X_H=\sum p_i\partial/\partial q^i;$
if $H=1/2\sum a_{ij} q^i q^j, a_{ij}=a_{ji}$ then $X_H=-\sum a_{ij} q_i\partial/\partial p^j.$

Properties

The assignment $f \mapsto X f$ is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
Suppose that $(q 1, ..., q n, p 1, ..., p n)$ are canonical coordinates on $M$ (see above). Then a curve $γ(t) = (q(t),p(t))$ is an integral curve of the Hamiltonian vector field $X H$ if and only if it is a solution of the Hamilton's equations:^[4]

\dot{q}^i = \frac {\partial H}{\partial p_i}

\dot{p}_i = - \frac {\partial H}{\partial q^i}.

The Hamiltonian $H$ is constant along the integral curves, because $\langle dH, \dot{\gamma}\rangle = \omega(X_H(\gamma),X_H(\gamma)) = 0$ . That is, $H (γ(t))$ is actually independent of $t$ . This property corresponds to the conservation of energy in Hamiltonian mechanics.
More generally, if two functions $F$ and $H$ have a zero Poisson bracket (cf. below), then $F$ is constant along the integral curves of $H$ , and similarly, $H$ is constant along the integral curves of $F$ . This fact is the abstract mathematical principle behind Noether's theorem.^{[nb 1]}
The symplectic form $ω$ is preserved by the Hamiltonian flow. Equivalently, the Lie derivative ${\mathcal {L}}_{X_{H}}\omega =0.$

Poisson bracket

The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold M, the Poisson bracket, defined by the formula

\{f,g\} = \omega(X_g, X_f)= dg(X_f) = \mathcal{L}_{X_f} g

where $\mathcal{L}_X$ denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds:^[5]

X_{\{f,g\}}= [X_f,X_g],

where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity^[6]

\{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,

which means that the vector space of differentiable functions on $M$ , endowed with the Poisson bracket, has the structure of a Lie algebra over $R$ , and the assignment $f \mapsto X f$ is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if $M$ is connected).

Remarks

↑ See Lee (2003, Chapter 18) for a very concise statement and proof of Noether's theorem.

Notes

↑ Lee 2003, Chapter 18.
↑ Lee 2003, Chapter 12.
↑ Lee 2003, Chapter 18.
↑ Lee 2003, Chapter 18.
↑ Lee, Chaptter 18.
↑ Lee, Chaptter 18.

References

Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN 9780805301021. See section 3.2.
Arnol'd, V.I. (1997). Mathematical Methods of Classical Mechanics. Berlin etc: Springer. ISBN 0-387-96890-3.
Frankel, Theodore (1997). The Geometry of Physics. Cambridge: Cambridge University Press. ISBN 0-521-38753-1.
Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, 218, ISBN 0-387-95448-1
McDuff, Dusa; Salamon, D. (1998). Introduction to Symplectic Topology. Oxford Mathematical Monographs. ISBN 0-19-850451-9.

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[5] See Lee (2003, Chapter 18) for a very concise statement and proof of Noether's theorem.

[1] Lee 2003, Chapter 18.

[2] Lee 2003, Chapter 12.

[3] Lee 2003, Chapter 18.

[4] Lee 2003, Chapter 18.

[6] Lee, Chaptter 18.

[7] Lee, Chaptter 18.