T plus the extended cotangent bundles T R3 three T R3 generates the Lagrangian

T plus the extended cotangent bundles T R3 three T R3 generates the Lagrangian submanifold in (235). To be able to visualize the Lagrangian submanifold, we draw the following diagram by merging the Morse loved ones (231) as well as the left wing of the evolution speak to Tulczyjew’s triple (199)0 im( H ,R( H)) im( H ,R( H))RoLT R three R3 R T RT T R3 oH T R3 R(236)T RT R0 where T R3 could be the mapping provided in (196).( T RT 0R6. Inositol nicotinate In Vitro Discussion In this paper, we have applied the tangent speak to structure, around the extended tangent bundle T T Q, which was introduced in [70]. Referring to this, and by introducing the notion of unique contact structure, we’ve constructed a Tulcyzjew’s triple for get in touch with manifolds, see Diagram (176). This permits us to describe each the make contact with Lagrangian plus the make contact with Hamiltonian dynamics as Legendrian submanifolds of T T Q. Within this formulation, the Legendre transformation is defined as a passage in between two generators with the similar Legendrian submanifold. Note that, this strategy is totally free from the Hessian situation. That suggests, it really is applicable for degenerate theories also. We, further, present Tulcyzjew’s triple for evolutionary dynamics, see Diagram (199). Rather of speak to PD-168077 Technical Information structures, the evolution triple (199) consists of special symplectic structures. In this construction, the get in touch with manifold T T Q is substituted by the extended horizontal bundle H T Q R, which can be symplectic. We’ve got concluded the paper by applications in the theoretical outcomes to geometrical foundations of some thermodynamical models. Here are some further queries we want to pursue: In Section 4.2, we have established that the image space of a get in touch with Hamiltonian vector field is really a Legendrian submanifold from the tangent speak to manifold. Evidently, not all Legendrian submanifolds identify explicit dynamical equations. This observation motivates us to define the notion of an implicit Hamiltonian Contact Dynamics as a non-horizontal Legendrian submanifold on the tangent speak to manifold. We refer to [75] for any similar discussion carried out for the case of symplectic dynamics and integrability from the non-horizontal Lagrangian submanifolds. We find it intriguing to elaborate the integrability of implicit Hamiltonian speak to dynamics. Following, the initial question raised in this section, we plan to create a HamiltonJacobi theory for implicit Hamiltonian contact dynamics. Hamilton acobi theory for (explicit) Hamiltonian contact dynamics is lately examined in [72,76]. HamiltonJacobi theory for implicit symplectic dynamics is discussed in [77,78]. Within the literature, Tulczyjew’s triple for higher order classical dynamical systems is already readily available [14,15]. Larger order get in touch with dynamics is studied in [79]. As a futureMathematics 2021, 9,39 ofwork, we strategy to extend the geometry presented inside the present paper to higher order make contact with framework.Author Contributions: Writing–original draft, O.E., M.L.V., M.d.L. and J.C.M. All authors have contributed equally. All authors have read and agreed for the published version of the manuscript. Funding: M. de Le and M. Lainz acknowledge the partial finantial assistance from MICINN Grant PID2019-106715GB-C21 and the ICMAT Severo Ochoa project CEX2019-000904-S. M. Lainz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203. J.C. Marrero acknowledges the partial support from European Union (Feder) grant PGC2018-098265-B-C32. Institutional Assessment Board Statement: Not applicable. Informed Consent Stateme.