Where T ^ * Y and T ^ * X are the cotangent bundles of Y, respectively, and V ^ * Y \ to Y is the dual bundle to VY \ to Y, called the vertical cotangent bundle.
32.
Where T ^ * Y and T ^ * X are the cotangent bundles of Y, respectively, and V ^ * Y \ to Y is the dual bundle to VY \ to Y, called the vertical cotangent bundle.
33.
The Weinstein conjecture was first proved for contact hypersurfaces in \ mathbb R ^ { 2n } in 1986 by Viterbo, then extended to cotangent bundles by Hofer-Viterbo and to wider classes of aspherical manifolds by Floer-Hofer-Viterbo.
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The Legendre transform gives the Hamiltonian H ( p, q ) as a function of the coordinates of the cotangent bundle T ^ * \ mathcal M; the inner product used to define the Legendre transform is inherited from the pertinent canonical symplectic structure.
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In this more general situation, the wave front set is a closed conical subset of the cotangent bundle " T " * ( " X " ), since the ? variable naturally localizes to a covector rather than a vector.
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As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold.
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One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle TQ of Q coordinated by ( t, q ^ i, p, p _ i ) and provided with the canonical Hamiltonian is p-H.
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Using this set-up we can locally think of " M " as being the cotangent bundle T * "'R " "'n ", and the Lagrangian fibration as the trivial fibration This is the canonical picture.
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The strongest results are obtained for over-determined systems ( holonomic systems ), and on the characteristic variety cut out by the symbols, in the good case for which it is a Lagrangian submanifold of the cotangent bundle of maximal dimension ( involutive systems ).
40.
Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a Lie Group on a coadjoint orbit.
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