Английская Википедия:Hamiltonian vector field

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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.Шаблон:Sfn

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 Шаблон:Math is a symplectic manifold. Since the symplectic form Шаблон:Math is nondegenerate, it sets up a fiberwise-linear isomorphism

<math>\omega:TM\to T^*M, </math>

between the tangent bundle Шаблон:Math and the cotangent bundle Шаблон:Math, with the inverse

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

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

<math>\mathrm{d}H(Y) = \omega(X_H,Y).</math>

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 Шаблон:Math is a Шаблон:Math-dimensional symplectic manifold. Then locally, one may choose canonical coordinates Шаблон:Math on Шаблон:Math, in which the symplectic form is expressed as:Шаблон:Sfn <math>\omega=\sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,</math>

where Шаблон:Math denotes the exterior derivative and Шаблон:Math denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian Шаблон:Math takes the form:Шаблон:Sfn <math>\Chi_H=\left( \frac{\partial H}{\partial p_i}, - \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,</math>

where Шаблон:Math is a Шаблон:Math square matrix

<math>\Omega =

\begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix},</math>

and

<math> \mathrm{d}H=\begin{bmatrix} \frac{\partial H}{\partial q^i} \\

\frac{\partial H}{\partial p_i} \end{bmatrix}.</math>

The matrix Шаблон:Math is frequently denoted with Шаблон:Math.

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

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

Properties

• The assignment Шаблон:Math is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that Шаблон:Math are canonical coordinates on Шаблон:Math (see above). Then a curve Шаблон:Math is an integral curve of the Hamiltonian vector field Шаблон:Math if and only if it is a solution of Hamilton's equations:Шаблон:Sfn <math>\dot{q}^i = \frac {\partial H}{\partial p_i}</math>

<math>\dot{p}_i = - \frac {\partial H}{\partial q^i}.</math>

• The Hamiltonian Шаблон:Math is constant along the integral curves, because <math>\langle dH, \dot{\gamma}\rangle = \omega(X_H(\gamma),X_H(\gamma)) = 0</math>. That is, Шаблон:Math is actually independent of Шаблон:Math. This property corresponds to the conservation of energy in Hamiltonian mechanics.
• More generally, if two functions Шаблон:Math and Шаблон:Math have a zero Poisson bracket (cf. below), then Шаблон:Math is constant along the integral curves of Шаблон:Math, and similarly, Шаблон:Math is constant along the integral curves of Шаблон:Math. This fact is the abstract mathematical principle behind Noether's theorem.
• The symplectic form Шаблон:Mvar is preserved by the Hamiltonian flow. Equivalently, the Lie derivative <math>\mathcal{L}_{X_H} \omega= 0.</math>

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

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

where <math>\mathcal{L}_X</math> denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds:Шаблон:Sfn <math> X_{\{f,g\}}= [X_f,X_g], </math>

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:Шаблон:Sfn <math> \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0, </math>

which means that the vector space of differentiable functions on Шаблон:Math, endowed with the Poisson bracket, has the structure of a Lie algebra over Шаблон:Math, and the assignment Шаблон:Math is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if Шаблон:Math is connected).

Works cited

• Шаблон:Cite bookSee section 3.2.
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Citation
• Шаблон:Cite book

External links

• Hamiltonian vector field on nLab