E Reeb field on T R3 is /U, plus the directional derivative of your Hamiltonian function (225) is R( H) = -1. The image space with the coupling with the Hamiltonian vector field and R( H) determines a 7-Dehydrocholesterol Endogenous Metabolite https://www.medchemexpress.com/7-Dehydrocholesterol.html �Ż�7-Dehydrocholesterol 7-Dehydrocholesterol Protocol|7-Dehydrocholesterol In stock|7-Dehydrocholesterol supplier|7-Dehydrocholesterol Autophagy} Legendrian submanifold, of your extended tangent bundle T T R3 , given byMathematics 2021, 9,37 ofc N- H = im( X H , R( H))= (S, V, N, T, – P, U; S – NR, 0, N, 0, – P, RT, U; -1) T T R3 .(229)Evidently, the Legendrian submanifolds in (226) and (229) are connected together with the contactomorphism c : T T R3 T T R3 by satisfying the relation c N- H = im(-T H) as we have established in (147). Now, we want to create the Legendrian Ikarugamycin Technical Information submanifold in (229) referring to the left wing with the get in touch with Tulczyjew’s triple (176), which is to generate it by means of a Lagrangian function (most likely as a Morse family) defined around the extended tangent bundle T R3 . In the light of Section 4.five, we now apply the inverse Legendre transformation. The initial step will be to apply c in (162) for the Legendrian submanifold N- H in (229). The image space is c (N- H) = (S, V, N, S – NR, 0, N, U; – T, 0, RT – T, – P, 1; U) T T R3 (230)which is a Legendrian submanifold of T T R3 . We assume the coordinates (S, V, N, S, V, N, U) on the extended tangent bundle T R3 and define the Whitney sum on the extended tangent and also the extended cotangent bundles T R3 three T R3 with coordinates (S, V, N, S, V, N, T, – P, U). Note that, in the Whitney sum, we repair the base coordinates (S, V, N) in R3 plus the extension U in R. The subscript R3 R in the notation of Whitney sum manifests these possibilities. A calculation offers that the Legendrian submanifold c (N- H) in (230) is generated by a Morse household on T R3 3 T R3 . In other words, the Lagrangian function L(S, V, N, S, V, N, U; T, – P, = T (S – S NR) N – N) PV U. (231)can be understood as defined around the extended tangent bundle T R3 , but depending on the auxiliary variables ( T, – P, . By merging the Lagrangian function using the left wing on the get in touch with Tulzyjew’s triple (176), we get the following diagram:c (N- H)N- HcRoLT R3 R three R T RT T R3 oT T R(232)T R0 T RT RT R(According to the local realization in the producing loved ones in (104), it is a direct calculation to show that the Legendrian submanifold c (N- H) is generated by the Morse household L in (231). We remark that in this formalism of thermodynamics, Hamiltonians are often singular, as in the case above, so there isn’t any Lagrangian formulation inside the classical sense. Therefore, we consider that Tulcyzjew triples may be a useful tool in this situation. 5.4. Evolutionary Flow and Its Lagrangian Realization We after more take into account the Hamiltonian function H exhibited in (225) and defined on the extended cotangent bundle T R3 . The evolutionary vector field H is defined in (119). A direct calculation determines the evolutionary vector field for the Hamiltonian function (225) as H = (S – NR) N P RT ( TS – NRT ) S N P U (233)on T R3 . By referring to Corollary 1, we establish that the image spaceMathematics 2021, 9,38 ofim( H , R( H)) = (S, V, N, T, – P, U; S – NR, 0, N, 0, – P, RT, TS – NRT ; -1) T T R(234)turns out to become a Lagrangian submanifold of H T R3 R defined in (179). Via 0 in (193), we map the Lagrangian submanifold (234) to a Lagrangian submanifold 0 im( H , R( H)) = (S, V, N, S – NR, 0, N, U; – T, 0, RT – T, – P, 1) T T R3 (235) in the cotangent bundle T T R3 . Based on (30), it is instant to see that the Lagrangian function L provided in (231) defined on the Whitney sum in the extended tangen.