Hamiltonian vector field

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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[edit]

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

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

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

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[edit]

Suppose that M is a 2n-dimensional symplectic manifold. Then locally, one may choose canonical coordinates (q1, ..., qn, p1, ..., pn) on M, in which the symplectic form is expressed as[2]

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

where Ω is a 2n × 2n square matrix

and

The matrix Ω is frequently denoted with J.

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

  • If then
  • if then
  • if then
  • if then

Properties[edit]

  • The assignment fXf 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, ..., pn) 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 XH if and only if it is a solution of the Hamilton's equations:[4]
  • The Hamiltonian H is constant along the integral curves, because . 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

Poisson bracket[edit]

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

where denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds:[5]

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]

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

Remarks[edit]

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

Notes[edit]

  1. ^ Lee 2003, Chapter 18.
  2. ^ Lee 2003, Chapter 12.
  3. ^ Lee 2003, Chapter 18.
  4. ^ Lee 2003, Chapter 18.
  5. ^ Lee, Chaptter 18.
  6. ^ Lee, Chaptter 18.

References[edit]

  • 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.