Information AboutSymplectomorphism |
| CATEGORIES ABOUT SYMPLECTOMORPHISM | |
| symplectic topology | |
| hamiltonian mechanics | |
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FORMAL DEFINITION Specifically, let (''M''1, ω1) and (''M''2, ω2) be symplectic manifolds. A map f is a symplectomorphism if it is a Diffeomorphism and the Pullback of ω2 under ''f'' is equal to ω1:
Examples of symplectomorphisms are the canonical transformations of Classical Mechanics and Theoretical Physics . A Hamiltonian symplectomorphism is a symplectomorphism that arises as the flow of a Hamiltonian vector field, and hence from some Hamiltonian function. FLOWS The Flow of a Symplectic Vector Field on a symplectic manifold is a symplectomorphism. This follows from the closedness of the symplectic form and Cartan's formula for the Lie Derivative in terms of the Exterior Derivative . Since Hamiltonian Vector Field s are symplectic, as a consequence we have Liouville's Theorem : the symplectic volume is invariant under a Hamiltonian flow. Since :{''H'',''H''} = ''X''''H''(''H'') = 0 the flow of a Hamiltonian vector field also preserves ''H''. In physics this is interpreted as the law of conservation of Energy . Liouville's theorem is interpreted as the conservation of phase volume in Hamiltonian systems, which is the basis for classical Statistical Mechanics . We have shown that there is a One-to-one Correspondence between Infinitesimal symplectomorphisms and closed one-forms on a symplectic manifold. If the first Betti Number of the manifold is zero, and it is connected, the latter set is the same as the space of Smooth Function s modulo addition of constants. COMPARISON WITH RIEMANNIAN GEOMETRY Unlike general for every point ''x'' in a symplectic manifold there is a local coordinate system with coordinates, called the Canonical Coordinates , p such that : QUANTIZATIONS Representations of finite-dimensional subgroups of the group of symplectomorphisms (after -deformations, in general) on Hilbert Space s are called ''quantizations''. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the Lie Algebra to the Lie algebra of continuous linear operators is also sometimes called the ''quantization''; this is a more common way of looking at it in physics. See Weyl Quantization , Geometric Quantization . Locally, symplectomorphisms can be generated by a generating function over a (local) Darboux coordinates. See Hamilton-Jacobi Equation . THE GROUP OF (HAMILTONIAN) SYMPLECTOMORPHISMS ARNOLD CONJECTURE A celebrated conjecture of V. I. Arnold relates the ''minimum'' number of Fixed Point s for a Hamiltonian symplectomorphism ''f'' on ''M'', in case ''M'' is a Closed Manifold , to Morse Theory . More precisely, the conjecture states that ''f'' has at least as many fixed points as the number of Critical Point s a smooth function on ''M'' must have (understood as for a ''generic'' case, Morse Function s, for which this is a definite finite number which is at least 2). It is known that this would follow from the Arnold-Givental Conjecture , which is a statement on Lagrangian Submanifold s. It is proven in many cases by the construction of symplectic Floer Homology . REFERENCES
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