Hamiltonian vector field
{{Short description|Vector field defined for any energy function}}
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.{{sfn|Lee|2003|loc=Chapter 18}}
Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions and on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of and .
Definition
Suppose that is a symplectic manifold. Since the symplectic form is nondegenerate, it sets up a fiberwise-linear isomorphism
between the tangent bundle and the cotangent bundle , with the inverse
Therefore, one-forms on a symplectic manifold may be identified with vector fields and every differentiable function determines a unique vector field , called the Hamiltonian vector field with the Hamiltonian , by defining for every vector field on ,
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 is a -dimensional symplectic manifold. Then locally, one may choose canonical coordinates on , in which the symplectic form is expressed as:{{sfn|Lee|2003|loc=Chapter 12}}
where denotes the exterior derivative and denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian takes the form:{{sfn|Lee|2003|loc=Chapter 18}}
- \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,
where is a square matrix
\begin{bmatrix}
0 & I_n \\
-I_n & 0 \\
\end{bmatrix},
and
\frac{\partial H}{\partial q^i} \\
\frac{\partial H}{\partial p_i}
\end{bmatrix}.
The matrix is frequently denoted with .
Suppose that is the -dimensional symplectic vector space 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 (see above). Then a curve is an integral curve of the Hamiltonian vector field if and only if it is a solution of Hamilton's equations:{{sfn|Lee|2003|loc=Chapter 18}}
\dot{q}^i & = \frac{\partial H}{\partial p_i} \\
\dot{p}_i & =-\frac{\partial H}{\partial q^i}.
\end{align}
- The Hamiltonian is constant along the integral curves, because . That is, is actually independent of . This property corresponds to the conservation of energy in Hamiltonian mechanics.
- More generally, if two functions and have a zero Poisson bracket (cf. below), then is constant along the integral curves of , and similarly, is constant along the integral curves of . This fact is the abstract mathematical principle behind Noether's theorem.See {{harvtxt|Lee|2003|loc=Chapter 18}} for a very concise statement and proof of Noether's theorem.
- The symplectic form is preserved by the Hamiltonian flow. Equivalently, the Lie derivative .
Poisson bracket
The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold , the Poisson bracket, defined by the formula
where denotes the Lie derivative along a vector field . Moreover, one can check that the following identity holds:{{sfn|Lee|2003|loc=Chapter 18}} ,
where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians and . As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity:{{sfn|Lee|2003|loc=Chapter 18}} ,
which means that the vector space of differentiable functions on , endowed with the Poisson bracket, has the structure of a Lie algebra over , and the assignment is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if is connected).
Remarks
{{Reflist|group=nb}}
Notes
{{Reflist|22em}}
Works cited
{{refbegin}}
- {{cite book| title = Foundations of Mechanics
| last1 = Abraham | first1 = Ralph
| last2 = Marsden | first2 = Jerrold E.
| author1-link = Ralph Abraham (mathematician)
| author2-link = Jerrold E. Marsden
| year = 1978
| publisher = Benjamin-Cummings | location = London
| isbn = 978-080530102-1
}}See section 3.2.
- {{cite book| title = Mathematical Methods of Classical Mechanics
| last = Arnol'd | first = V.I. | year = 1997
| author-link = Vladimir Arnold
| publisher = Springer | location = Berlin etc
| url = https://archive.org/details/mathematicalmeth0000arno | url-access = registration
| isbn = 0-387-96890-3
}}
- {{cite book| title = The Geometry of Physics
| last = Frankel | first = Theodore | year = 1997
| publisher = Cambridge University Press
| url = https://archive.org/details/geometryofphysic0000fran | url-access = registration
| isbn = 0-521-38753-1
}}
- {{citation| title = Introduction to Smooth manifolds
| last = Lee | first = J. M. | year = 2003
| volume = 218 | series = Springer Graduate Texts in Mathematics
| isbn = 0-387-95448-1
}}
- {{cite book| title = Introduction to Symplectic Topology
| last1 = McDuff | first1 = Dusa
| last2 = Salamon | first2 = D.
| author1-link = Dusa McDuff
| year = 1998
| series = Oxford Mathematical Monographs
| isbn = 0-19-850451-9
}}
{{refend}}