cotangent bundle वाक्य
उदाहरण वाक्य
मोबाइल
- If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metric induces a linear isomorphism between the tangent and cotangent bundles . ( See Musical isomorphism ).
- Where the ? " i " are the fiber coordinates on the cotangent bundle induced by the coordinate differentials d " x " " i ".
- Where ? is the holomorphic cotangent bundle and the notation ? [ 2 ] means the " tensor square " ( " not " the second exterior power ).
- Sikorav is known for his proof, joint with Fran�ois Laudenbach, of the Arnold conjecture for Lagrangian intersections in cotangent bundles, as well as for introducing generating families in symplectic topology.
- A volume form is a nowhere vanishing section & omega; of \ bigwedge ^ n T ^ * M, the top exterior power of the cotangent bundle of " M ".
- The explanation in geometric terms is that a general tensor will have contravariant indices as well as covariant indices, because it has parts that live in the tangent bundle as well as the cotangent bundle.
- Thus, every " frame field " is associated with a unique " coframe field ", and vice versa; a coframe fields is a set of four orthogonal sections of the cotangent bundle.
- The associated graded algebra is the commutative algebra of smooth functions on the cotangent bundle T ^ * M which are polynomial along the fibers of the projection \ pi \ colon T ^ * M \ rightarrow M.
- Just as we build differential forms out of exterior powers of the cotangent bundle, we can build exterior powers of the complexified cotangent bundle ( which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle ).
- Just as we build differential forms out of exterior powers of the cotangent bundle, we can build exterior powers of the complexified cotangent bundle ( which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle ).
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 ).
- 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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