Page - 14 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Theorem4 (W.M.Tulczyjew). With thenotations specifiedaboveSection4.3.2, letXH :T∗N→TT∗Nbe theHamiltonianvectorfieldonthesymplecticmanifold(T∗N,dθN)associatedtotheHamiltonianH :T∗N→R, definedby i(XH)dθN=−dH.Then XH(T∗N)=β−1N ( dH(T∗N) ) . Moreover, the equality α−1N ( dL(TN) ) =β−1N ( dH(T∗N) ) holds if andonly if theLagrangianL ishyper-regularandsuch that dH=d ( EL◦L−1L ) , whereLL :TN→T∗Nis theLegendremapandEL :TN→R the energyassociated to theLagrangianL. The interestedreaderwillfindtheproofof that theoremintheworksofTulczyjew([34,35]). WhenL isnothyper-regular,α−1N ( dL(TN) ) still isaLagrangiansubmanifoldof thesymplectic manifold ( TT∗N,α∗N(dθTN) ) , but it isnomore thegraphofasmoothvectorfieldXH definedonT∗N. Tulczyjewproposes toconsider thisLagrangiansubmanifoldasan implicitHamiltonequationonT∗N. TheseresultscanbeextendedtoLagrangiansandHamiltonianswhichmaydependontime. 4.4. TheHamiltonianFormalismonSymplectic andPoissonManifolds 4.4.1. TheHamiltonFormalismonSymplecticManifolds In pure mathematics as well as in applications of mathematics to mechanics and physics, symplecticmanifoldsother thancotangentbundlesare encountered. A theoremdue to the french mathematicianGastonDarboux (1842–1917) asserts that any symplecticmanifold (M,ω) is of even dimension2nand is locally isomorphic to the cotangentbundle toan-dimensionalmanifold: ina neighbourhoodofeachof itspoint thereexist local coordinates (x1,. . . ,xn,p1,. . . ,pn), calledDarboux coordinateswithwhichthesymplectic formω is expressedexactlyas thecanonical symplectic formof acotangentbundle: ω= n ∑ i=1 dpi∧dxi . Let (M,ω) be a symplectic manifold and H : R×M → R a smooth function, said to be a time-dependentHamiltonian. Itdeterminesa time-dependentHamiltonianvectorfieldXH onM, suchthat i(XH)ω=−dHt , Ht :M→Rbeingthe functionH inwhichthevariable t is consideredasaparameterwithrespect to whichnodifferentiation ismade. TheHamiltonequationdeterminedbyH is thedifferentialequation dψ(t) dt =XH ( t,ψ(t) ) . TheHamiltonian formalismcan therefore be applied to any smooth,maybe timedependent HamiltonianonM, evenwhenthere isnoassociatedLagrangian. TheHamiltonianformalismisnot limitedtosymplecticmanifolds: it canbeapplied, forexample, toPoissonmanifolds [36], contactmanifoldsand Jacobimanifolds [37]. Forsimplicity Iwill consideronly Poissonmanifolds. Readers interested in Jacobimanifoldsandtheirgeneralizationsarereferredto the papersbyLichnerowiczquotedaboveandto thevery importantpaperbyKirillov [38]. 14