Page - 13 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Remark8. The2-formdθN isa symplectic formonthe cotangentbundleT∗N,called its canonical symplectic form.WehaveshownthatwhentheLagrangianL ishyper-regular, the equationsofmotioncanbewritten in three equivalentmanners: 1. as theEuler–Lagrange equationsonR×TM, 2. as the equations given by the kernels of the presymplectic formsd̂L ord̂HL which determine the foliations intocurvesof the evolutionspacesR×TMintheLagrangian formalism,orR×T∗Minthe Hamiltonian formalism, 3. as theHamilton equation associated to theHamiltonian HL on the symplecticmanifold (T∗N,dθN), oftencalled thephase spaceof the system. 4.3.1. TheTulczyjewIsomorphisms Around1974, Tulczyjew [34,35] discovered (βN wasprobably known longbefore 1974, but I believe thatαN,muchmorehidden,wasnoticedbyTulczyjewfor thefirst time) tworemarkablevector bundles isomorphismsαN :TT∗N→T∗TNandβN :TT∗N→T∗T∗N. The first one αN is an isomorphism of the bundle (TT∗N,TπN,TN) onto the bundle (T∗TN,πTN,TN), while the second βN is an isomorphismof the bundle (TT∗N,τT∗N,T∗N) onto thebundle (T∗T∗N,πT∗N,T∗N). Thediagrambelowiscommutative. T∗T∗N πT∗N TT∗N βN τT∗N TπN αN T∗TN πTN T∗N πN TN τN N Sincetheyarethetotalspacesofcotangentbundles, themanifoldsT∗TNandT∗T∗Nareendowed with theLiouville1-forms θTN and θT∗N, andwith thecanonical symplectic formsdθTN anddθT∗N, respectively.UsingtheisomorphismsαN andβN,wecanthereforedefineonTT∗N two1-formsα∗NθTN andβ∗NθT∗N, andtwosymplectic2-formsα∗N(dθTN)andβ∗N(dθT∗N). Theveryremarkablepropertyof the isomorphismsαN andβN is that the twosymplectic formssoobtainedonTT∗Nareequal: α∗N(dθTN)= β∗N(dθT∗N) . The 1-forms α∗NθTN and β∗NθT∗N are not equal, their difference is the differential of a smoothfunction. 4.3.2. LagrangianSubmanifolds In viewof applications to implicitHamiltonian systems, let us recall here that a Lagrangian submanifold of a symplectic manifold (M,ω) is a submanifold N whose dimension is half the dimensionofM, onwhichthe forminducedbythesymplectic formω is0. Let L : TN → R and H : T∗N → R be two smooth real valued functions, defined on TN andonT∗N, respectively. ThegraphsdL(TN) anddH(T∗N)of theirdifferentials areLagrangian submanifolds of the symplectic manifolds (T∗TN,dθTN) and (T∗T∗N,dθT∗N). Their pull-backs α−1N ( dL(TN) ) and β−1N ( dH(T∗N) ) by the symplectomorphisms αN and βN are therefore two Lagrangian submanifolds of the manifold TT∗N endowed with the symplectic form α∗N(dθTN), which isequal to thesymplectic formβ∗N(dθT∗N). Thefollowingtheoremenlightenssomeaspectsof therelationshipsbetweentheHamiltonianand theLagrangianformalisms. 13