Page - 5 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 diffeomorphisms (TΦX−t)T is thereducedflowofavectorfield X̂onT∗M,which is the canonical liftof X to thecotangentbundleT∗M. The canonical lifts of a vector field to the tangent and cotangent bundles are used in Sections3and5. Theycanbedefinedtoofor time-dependentvectorfields. 2.TheLagrangianFormalism 2.1. TheConfigurationSpaceand theSpaceofKinematicStates The principles of mechanics were stated by the great English mathematician Isaac Newton (1642–1727) inhisbookPhilosophiaNaturalisPrincipiaMathematicapublished in1687 [19].Onthisbasis, a littlemore thanacentury later, JosephLouisLagrange (1736–1813) inhisbookMécaniqueanalytique [20] derivedtheequations (todayknownas theEuler–Lagrange equations)whichgovern themotionofa mechanicalsystemmadeofanynumberofmaterialpointsorrigidmaterialbodies interactingbetween thembyverygeneral forces,andeventuallysubmittedtoexternal forces. Inmodernmathematical language, these equations arewrittenon the configuration space and on the space of kinematic states of the considered mechanical system. The configuration space is a smooth n-dimensional manifold N whose elements are all the possible configurations of the system (a configuration being the position in space of all parts of the system). The space of kinematic states is the tangent bundle TN to the configuration space, which is 2n-dimensional. Eachelementof thespaceofkinematic states isavector tangent to theconfigurationspaceatoneof its elements, i.e., ataconfigurationof themechanical system,whichdescribes thevelocityatwhichthis configurationchangeswithtime. Inlocalcoordinatesaconfigurationofthesystemisdeterminedbythe ncoordinatesx1, . . . ,xnofapoint inN, andakinematicstatebythe2ncoordinatesx1, . . . ,xn,v1, . . .vn ofavector tangent toNatsomeelement inN. 2.2. TheEuler–LagrangeEquations Whenthemechanical systemis conservative, theEuler–Lagrangeequations involveasinglereal valuedfunctionLcalledtheLagrangianofthesystem,definedontheproductofthereal lineR (spanned bythevariable t representing the time)with themanifoldTNofkinematic statesof thesystem. In local coordinates, theLagrangianL is expressedasa functionof the2n+1variables, t,x1, . . . ,xn,v1, . . . ,vn andtheEuler–Lagrangeequationshavetheremarkablysimple form d dt ( ∂L ∂vi ( t,x(t),v(t) ))− ∂L ∂xi ( t,x(t),v(t) ) =0, 1≤ i≤n , wherex(t) stands forx1(t), . . . ,xn(t)andv(t) forv1(t), . . . ,vn(t)with,ofcourse, vi(t)= dxi(t) dt , 1≤ i≤n . 2.3.Hamilton’sPrinciple ofStationaryAction The great IrishmathematicianWilliamRowanHamilton (1805–1865) observed [21,22] that the Euler–LagrangeequationscanbeobtainedbyapplyingthestandardtechniquesofCalculusofVariations, due toLeonhardEuler (1707–1783)and JosephLouisLagrange, to theaction integral (Lagrangeobserved that factbeforeHamilton,but in the last editionofhisbookhechose toderive theEuler–Lagrange equationsbyapplicationof theprinciple ofvirtualworks,usingaverycleverevaluationof thevirtual workof inertial forces forasmooth infinitesimalvariationof themotion). IL(γ)= ∫ t1 t0 L ( t,x(t),v(t) ) dt , withv(t)= dx(t) dt , 5