Page - 12 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Proof. Sincedω=0, the fact thatkerω is completely integrable isan immediateconsequenceof the Frobenius’ theorem([27],Chapter III,Theorem5.1,p. 132). Theexistenceandunicityofasymplectic formωronM/kerω suchthatω= p∗ωr results fromthefact thatM/kerω isbuiltbyquotientingM bythekernelofω. PresymplecticManifolds inMechanics Letusgobackto theassumptionsandnotationsofSection4.1.Wehaveseen inRemark6that the Poincaré-Cartan1-forminHamiltonianformalism ̂HL onR×T∗NandthePoincaré-Cartan1-form inLagrangianformalism ̂LonR×TNarerelatedby ̂L=(idR,LL)∗̂HL . Theirexteriordifferentialsd̂L andd̂HL botharepresymplectic2-formsontheodd-dimensional manifoldsR×TN andR×T∗N, respectively. At any point of these manifolds, the kernels of these closed 2-forms are one-dimensional. They therefore Proposition 1 determine foliations into smooth curvesof thesemanifolds. TheEuler-CartanTheorem1shows that eachof thesecurves is a possiblemotionof thesystem,describedeither in theLagrangian formalism,or in theHamiltonian formalism,respectively. Thesetofallpossiblemotionsof thesystem,calledbyJean-MarieSouriauthemanifoldofmotions of thesystem, isdescribedbythequotient (R×TN)/kerd̂L in theLagrangianformalism,andby thequotient (R×T∗N)/kerd̂HL in theHamiltonian formalism. Bothare (maybenon-Hausdorff) symplecticmanifolds, theprojectionsonthesequotientmanifoldsof thepresymplectic formsd̂L and d̂HL bothbeingsymplectic forms. Ofcourse thediffeomorphism (idR,LL) :R×TN→R×T∗N projects onto a symplectomorphismbetween theLagrangianandHamiltoniandescriptionsof the manifoldofmotionsof thesystem. 4.3. TheHamiltonEquation Proposition 2. Let N be the configuration manifold of a Lagrangian system whose Lagrangian L :R×TN→R, maybe time-dependent, is smooth and hyper-regular, and HL : R×T∗N → R be the associated Hamiltonian Definition 4. Let ϕ : [t0,t1] → N be a smooth curve parametrized by the time t∈ [t0,t1], and letψ : [t0,t1]→T∗Nbetheparametrizedcurve inT∗N ψ(t)=LL ( t, dγ(t) dt ) , t∈ [t0,t1] , whereLL :R×TN→T∗Nis theLegendremapDefinition1. Theparametrizedcurve t →γ(t) is amotionof the systemif andonly if theparametrizedcurve t →ψ(t) satisfies the equatin, called theHamiltonequation, i ( dψ(t) dt ) dθN=−dHLt , wheredHLt = dHL− ∂HL∂t dt is the differential of the function HLt : T ∗N →R inwhich the time t is consideredasaparameterwith respect towhich there isnodifferentiation. Whentheparametrizedcurveψ satisfies theHamiltonequationstatedabove, it satisfies too the equation, called the energyequation d dt ( HL ( t,ψ(t) )) = ∂HL ∂t ( t,ψ(t) ) . Proof. Theseresultsdirectly followfromtheEuler-CartanTheorem1. 12