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

${\displaystyle \omega :TM\to T^{*}M,}$

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

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

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

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

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 2n-dimensional symplectic manifold. Then locally, one may choose canonical coordinates (q¹, ..., qⁿ, p₁, ..., p_n) on M, in which the symplectic form is expressed as: ^[2] ${\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}$

where d denotes the exterior derivative and ∧ denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian H takes the form: ^[1] ${\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}$

where Ω is a 2n × 2n square matrix

${\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}$

and

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

The matrix Ω is frequently denoted with J.

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

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

Properties

• The assignment f ↦ X_f is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that (q¹, ..., qⁿ, p₁, ..., 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 Hamilton's equations: ^[1] ${\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}}$

${\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}.}$

• The Hamiltonian H is constant along the integral curves, because ${\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}$. 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 ${\displaystyle {\mathcal {L}}_{X_{H}}\omega =0.}$

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

${\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}$

where ${\displaystyle {\mathcal {L}}_{X}}$ denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds: ^[1] ${\displaystyle X_{\{f,g\}}=[X_{f},X_{g}],}$

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: ^[1] ${\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,}$

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 ↦ X_f is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if M is connected).

Remarks

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

Notes

1. Lee 2003, Chapter 18.
2. Lee 2003, Chapter 12.

Works cited

• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN   978-080530102-1.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 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.