Page - 11 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 4.1.Hyper-RegularLagrangians AssumptionsMadein thisSection Weconsider in this sectionasmooth,maybetime-dependentLagrangianL :R×TN→R,which is such that theLegendremapDefinition 1LL :R×TN → T∗N satisfies the followingproperty: for eachfixedvalue of the time t ∈R, themap v → LL(t,v) is a smoothdiffeomorphismof the tangent bundle TN onto the cotangent bundle T∗N. An equivalent assumption is the following: themap (idR,LL) : (t,v) → ( t,LL(t,v) ) is a smooth diffeomorphismofR×TN ontoR×T∗N. TheLagrangianL is thensaid tobehyper-regular. TheequationsofmotioncanbewrittenonR×T∗N insteadofR×TN. Definition4. Under theassumptionSection4.1, the functionHL :R×T∗N→Rgivenby HL(t,p)=EL◦(idR,LL)−1(t,p) , t∈R , p∈T∗N , (EL :R×TN→Rbeing the energy functiondefined inDefinition1) is called theHamiltonianassociated to thehyper-regularLagrangianL. The1-formdefinedonR×T∗N ̂HL = θN−HLdt , whereθN is theLiouville1-formonT∗N, is called thePoincaré-Cartan1-formin theHamiltonian formalism. Remark 6. The Poincaré-Cartan 1-form ̂L onR×TN, defined inDefinition 1, is the pull-back, by the diffeomorphism (idR,LL) :R×TN→R×T∗N, of thePoincaré-Cartan1-form ̂HL in theHamiltonian formalismonR×T∗Ndefinedabove. 4.2. PresymplecticManifolds Definition5. Apresymplectic formonasmoothmanifoldMisa2-formω onMwhich isclosed, i.e., such that dω=0. AmanifoldMequippedwithapresymplectic formω is calledapresymplecticmanifoldanddenoted by (M,ω). The kernelkerω of apresymplectic formωdefinedona smoothmanifoldMis the set of vectors v∈TMsuch that i(v)ω=0. Remark7. Asymplectic formω onamanifoldMisapresymplectic formwhich,moreover, isnon-degenerate, i.e., such that for eachx∈Mandeachnon-zerovectorv∈TxM, there exists anothervectorw∈TxMsuch thatω(x)(v,w) =0. Or inotherwords, apresymplectic formωwhosekernel is the set ofnullvectors. Thekernel of apresymplectic formω onasmoothmanifoldMisavector sub-bundleofTMif andonly if for eachx∈M,thevector subspaceTxMofvectorsv∈TxMwhichsatisfy i(v)ω=0 is of afixeddimension, the same forall points x∈M.Apresymplectic formwhichsatisfies that condition is said tobeof constant rank. Proposition1. Letω beapresymplectic formof constant rankRemark7onasmoothmanifoldM.Thekernel kerω ofω is acompletely integrablevector sub-bundleofTM,whichdefinesa foliationFω ofMintoconnected immersedsubmanifoldswhich, at eachpointofM,have thefibreofkerω at thatpointas tangentvector space. Wenowassume inaddition that this foliation is simple, i.e., such that the set of leavesofFω, denotedby M/kerω, has a smooth manifold structure for which the canonical projection p : M → M/kerω (which associates to each point x ∈ M the leaf which contains x) is a smooth submersion. There exists onM/kerω aunique symplectic formωr such that ω= p∗ωr . 11