Symplectic manifold
Symplectic manifolds are the central objects of symplectic geometry , a sub-area of differential geometry . The symplectic manifolds have a very strong relation to theoretical physics .
definition
A symplectic manifold is a smooth manifold together with a symplectic form , that is, a global, smooth and closed 2-form that has not degenerated point by point (see also symplectic space ). "Closed" means that the exterior derivative of the differential form disappears .
Symplectic manifolds always have an even dimension, since antisymmetric matrices in odd dimensions cannot be inverted and therefore antisymmetric bilinear forms have degenerated in odd dimensions.
Poisson bracket
Since the shape has not degenerated, its inverse defines a bilinear mapping of one-shapes and at every point
and the Poisson bracket of the functions and ,
Lagrangian submanifold
A Lagrangian submanifold of a 2n-dimensional symplectic manifold is an n-dimensional submanifold with
- ,
d. H. the restriction of the symplectic form to the tangent space of L vanishes.
Hamilton River
In an Euclidean space , the gradient of a function is that vector field whose scalar product for any given vector field corresponds to the application of on ,
In a symplectic manifold, the vector field belongs to a given f and a given arbitrary function
that derives functions along an integral curve of the Hamiltonian equations belonging to (interpreted as the so-called Hamilton function of the system) . The role of w is assumed here by h , and the symplectic geometry or the Hamiltonian dynamics are used for h .
The vector field is therefore the symplectic gradient of or the infinitesimal Hamiltonian flow of .
Darboux's theorem
Darboux's theorem, named after the mathematician Jean Gaston Darboux, says:
In the vicinity of every point of a symplectic manifold there are local coordinate pairs with
The coordinate pairs defined in this way are called canonically conjugate .
Relationship to Hamiltonian mechanics
In Hamiltonian mechanics , the phase space is a symplectic manifold with the closed, symplectic form
This is not a special case, because according to Darboux's theorem, local coordinates can always be written as. Symplectic manifolds are the phase spaces of Hamiltonian mechanics.
The mathematical statement regarding is equivalent to the so-called canonical equations of theoretical physics, especially in analytical mechanics .
In this context, the Liouville theorem , which plays a role in statistical physics, is also important . It essentially states that with Hamiltonian fluxes the phase space volume remains constant, which is important for determining the probability measures of this theory.
See also
- Canonical transformation , especially the paragraph "Symplectic structure"
literature
- VI Arnold : Mathematical Methods of Classical Mechanics (= Graduate Texts in Mathematics 60). 2nd edition, Springer, New York NY et al. 1989, ISBN 0-387-96890-3 .
- Rolf Berndt: Introduction to Symplectic Geometry. Vieweg, Braunschweig et al. 1998, ISBN 3-528-03102-6 .
Web links
- Article Symplectic Structure in Springer Online Reference
- Article in Weisstein, Encyclopedia of Mathematics, at Math World
- Dusa McDuff Symplectic structures - a new approach to geometry , Notices AMS, November 1998, PDF file
Individual evidence
- ^ Definition of symplectic manifolds according to Vladimir I. Arnold Mathematical Methods of Classical Mechanics. 2nd edition, Springer, 1989, ISBN 0-387-96890-3 , p. 201 (Chapter 8 - Symplectic Manifolds). Likewise in Ana Cannas da Silva: Lectures on Symplectic Geometry . Springer, Berlin 2001, ISBN 3-540-42195-5 .
- ^ A proof can be found in VI Arnold : Mathematical Methods of Classical Mechanics. 2nd Edition. Springer, 1989, ISBN 0-387-96890-3 , chapter 8.