Ee [357,72]). (Make contact with) Hamiltonian Vector Fields. For any genuine valued function H on

Ee [357,72]). (Make contact with) Hamiltonian Vector Fields. For any genuine valued function H on a get in touch with c manifold (M,), there’s a corresponding contact vector field X H , defined as follows:c X H = – H, c X H d = dH – R( H),(108)where R could be the Reeb vector field. Here, H is named the (get in touch with) Hamiltonian function c and X H is known as the (contact) Hamiltonian vector field. We Nimbolide Apoptosis denote a make contact with Hamiltonian technique as a three-tuple (M, , H) exactly where (M,) is a make contact with manifold and H is really a smooth true function on M. A direct computation determines the conformal factor for a given Hamiltonian vector DMT-dC Phosphoramidite Autophagy fields asc c c L X H = d X H X H d = -R( H).(109)That’s, = R( H). Within this realization, the speak to Jacobi bracket of two smooth functions on M is defined by F, H c = [X c ,X c ] , (110)F Hwhere X F and X H are Hamiltonian vectors fields determined through (108). Here, [ could be the Lie bracket of vector fields. Then, the identityc c c – [ XK , X H ] = XK,H c(111)establishes the isomorphism(Xcon (M), -[) F (M), { c(112)between the Lie algebras of real smooth functions and contact vector fields. According to (109), the flow of a contact Hamiltonian system preserves the contact structure, but it does not preserve neither the contact one-form nor the Hamiltonian function. Instead, we obtain c L X H H = -R( H) H. (113) Being a non-vanishing top-form we can consider d n as a volume form on M. Hamiltonian motion does not preserve the volume form sincec L X H (d n ) = -(n 1)R( H)d n .(114)However, it is immediate to see that, for a nowhere vanishing Hamiltonian function H, the quantity H -(n1) (d)n is preserved along the motion (see [41]). Referring to the Darboux’s coordinates (qi , pi , z), for a Hamiltonian function H, the Hamiltonian vector field, determined in (108), is computed to bec XH =H H H H ( pi – H) , – p pi qi z i pi pi z qi(115)Mathematics 2021, 9,20 ofwhereas the contact Jacobi bracket (110) is F, H c =F H H F F H F H F – pi – H – pi . – i p pi qi pi z pi z q i(116)Thus, we receive that the Hamilton’s equations for H as qi = H , pi pi = – H H – pi , i z q z = pi H – H. pi (117)Evolution vector fields A further vector field could be defined from a Hamiltonian function H on a make contact with manifold ( M,): the evolution vector field of H [52], denoted as H , which is the one particular that satisfiesL H = dH – R( H),In regional coordinates, it is actually provided by H =( H) = 0.(118)H H H H – pi pi , i i pi q z pi pi z q(119)in order that the integral curves satisfy the evolution equations qi = H , pi pi = – H H – pi , i z q z = pi H . pi (120)The evolution and Hamiltonian vector fields are associated byc H = X H H R.(121)Quantomorphisms. By asking the conformal issue inside the definition (105) to be the unity, a single arrives the conservation with the contact types two = 1 . (122)We get in touch with such a mapping as a strict contact diffeomorphism (or quantomorphism). For any get in touch with manifold (M,) we denote the space of all strict get in touch with transformations as Diffst (M) = Diff(M) : = Diffcon (M). con (123)The Lie algebra of this group is consisting with the infinitesimal quantomorphisms Xst (M) = X Xcon (M) : L X H = 0 . con (124)If the speak to vector field is determined by way of a smooth function H as in (108), then X H falls into the subspace Xst (M) if and only if = -dH (R) = 0. This reads that, to con generate an infinitesimal quantomorphism, a function H will have to not rely on the fiber variable z. Now, take into account the canonical make contact with manifold (T Q, Q). For two functions, those which are not dependent on t.