In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a Hamiltonian vector field on a symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

 is a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton

William Rowan Hamilton

Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

. The integral curve

Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...

s of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphism

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

s of a symplectic manifold arising from the flow

Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...

 of a Hamiltonian vector field are known as canonical transformation

Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates  →  that preserves the form of Hamilton's equations , although it...

s in physics and (Hamiltonian) symplectomorphism

Symplectomorphism

In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...

s in mathematics.

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold

Poisson manifold

In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra C^\infty\, of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra...

. The Lie bracket

Lie bracket

Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...

 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

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...

 of f and g.

Definition

Suppose that (M,ω) is a symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

. Since the symplectic form ω is nondegenerate, it sets up a fiberwise-linear isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

between the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

  and the cotangent bundle

Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...

 , with the inverse

Therefore, one-form

One-form

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...

s on a symplectic manifold M may be identified with vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s and every differentiable function

Differentiable function

In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...

  determines a unique vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 X_H, called the Hamiltonian vector field with the Hamiltonian H, by requiring that for every vector field Y on M, the identity


must hold.

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

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics...

  on M, in which the symplectic form is expressed as


Then the Hamiltonian vector field with Hamiltonian H takes the form


where Ω is a 2n by 2n square matrix


Suppose that M = R²ⁿ is the 2n-dimensional symplectic vector space

Symplectic vector space

In mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...

 with (global) canonical coordinates.

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

Properties

• The assignment is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.

• Suppose that are canonical coordinates on M (see above). Then a curve is an integral curve
  Integral curve
  In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
  
   of the Hamiltonian vector field X_H if and only if it is a solution of the Hamilton's equations:

• The Hamiltonian H is constant along the integral curves, that is, is actually independent of t. This property corresponds to the conservation of energy
  Conservation of energy
  The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
  
   in Hamiltonian mechanics
  Hamiltonian mechanics
  Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
  
  .

• More generally, if two functions F and H have a zero Poisson bracket
  Poisson bracket
  In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
  
   (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
  Noether's theorem
  Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
  
  .

• Symplectic form  is preserved by Hamiltonian flow; or equivalently, Lie derivative
  Lie derivative
  In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
  
   

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

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...

, defined by the formula


where denotes the Lie derivative

Lie derivative

In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

 along a vector field X. Moreover, one can check that the following identity holds:

where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence, the Poisson bracket satisfies the Jacobi identity

Jacobi identity

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...

which means that the vector space of differentiable functions on M, endowed with the Poisson bracket, has the structure of a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

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