Rrive at the following Lagrangian submanifold of TT QS-E = -1 (S) =qi , pi ,E E ,- i pi qTT Q :E =0 qiTT Q.(78)A direct computation proves that the Lagrangian submanifold S- E in (78) as well as the Lagrangian submanifold S L in (71) are the very same. So that the Legendre transformation is accomplished. In the event the Lagrangian function is non-degenerate, then from the equation L E (q, q, p) = pi – i (q, q) = 0 qi q (79)1 can explicitly determine the velocity qi with regards to the momenta (qi , pi). In other words, to get a non-degenerate Lagrangian function L = L(q, q) the fiber derivativeF L : T Q – T Q,(qi , q j) – qi ,L (q, q) qi(80)can be a neighborhood diffeomorphism. Within this case, the Morse family members E may be decreased to a well-defined Hamiltonian function H (qi , pi) = pi qi (q, p) – L q, q(q, p) (81) on T Q. Inverse Legendre Transformation. The inverse Legendre transformation is also attainable in a equivalent way. This time, a single begins having a Hamiltonian technique ( T Q, Q , H) exactly where H is a Hamiltonian function. See that, within this notation Hamiltonian vector field X H defined in (15) is determined by way of X H = -dH. (82)Notice that the Lagrangian submanifold determined by the equality (82) is written in coordinates as H H S- H = qi , pi , , – i TT Q TT Q. (83) pi q Evidently, this Lagrangian submanifold is precisely figuring out the Hamilton’s Equation (17). Inside the present image, the inverse Legendre transformation is Ikarugamycin supplier usually to create the Lagrangian submanifold (83) by referring to the suitable wing with the triple. When the Hamiltonian function is not common then one wants to employ a Morse loved ones F : T Q T Q – R,(u,) , u – H .(84)So, if we think about the Pontryagin bundle more than T Q and we proceed as within the prior subsection, we are going to get the inverse Legendre transformation. three. Make contact with Dynamics three.1. Speak to Manifolds A (2n 1)–dimensional manifold M is called make contact with manifold if it’s equipped with a contact one-form satisfying d n = 0 [4,34]. We denote a get in touch with manifold by a two-tuple (M,). The Reeb vector field R may be the exceptional vector field satisfying R = 1, R d = 0. (85)At each point from the manifold M, the kernel in the make contact with type determines the contact structure H M. The complement of this structure, denoted by V M, is determined by the kernel in the precise two-form d. These give the following decomposition of the tangent bundle T M = H M V M, H M = ker , V M = ker d. (86)Mathematics 2021, 9,16 ofHere, H M is actually a vector sub-bundle of rank 2n. The restriction of d to H M is nondegenerate to ensure that ( H M, d) is a symplectic vector bundle more than M. The rank of V M is 1, and it is actually generated by the Reeb field R. Contactization. It is actually probable to arrive at a contact manifold beginning from a symplectic manifold. To possess this, look at a symplectic manifold P admitting an integer symplectic two form . Introduce the principal circle (quantization) bundle S(M,) – (P ,).pr(87)The make contact with one-form on M would be the connection one-form linked using a principal connection around the principal S1 -bundle pr : M P with curvature . This process is called contactization. One more example of a contact manifold is often obtained from an precise symplectic manifold as follows. Think about a trivial line bundle more than a manifold Chelerythrine Biological Activity provided by Q R Q. The first jet bundle, denoted by T Q is diffeomorphic towards the solution space T Q R that is,T Q = T Q R.We get in touch with this space the extended cotangent bundle. There exist two projections1 Q : T Q = T Q R – T Q, 0 Q(88)(, z) (, z) Q,: T Q = T Q R – Q,(89)exactly where Q is the cotangent.

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