Page - 68 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 TM. Toeachparameterizedcontinuous,piecewisesmoothcurve γ : [t0,t1]→M,definedonaclosed interval [t0,t1],withvalues inM, oneassociates thevalueatγof theaction integral: I(γ)= t1 t0 L ( dγ(t) dt ) dt (51) Thepartial differential of the function L :M×g→ with respect to its secondvariable d2L, which plays an important part in the Euler-Poincaré equation, can be expressed in terms of the momentandLegendremaps: d2L= pg∗ ◦ϕt◦L◦ϕ with J= pg∗ ◦ϕt(⇒ d2L= J◦L◦ϕ) themoment map, pg∗ :M×g∗→ g∗ thecanonicalprojectiononthesecondfactor, L :TM→T∗M theLegendre transform,with: ϕ :M×g→TM/ϕ(x,X)=XM(x)andϕt :T∗M→M×g∗/ϕt(ξ)= (πM(ξ), J(ξ)) (52) TheEuler-Poincaréequationcanthereforebewrittenunder the form:( d dt −ad∗V(t) ) (J◦L◦ϕ(γ(t),V(t)))= J◦d1L(γ(t),V(t))with dγ(t)dt =ϕ(γ(t),V(t)) (53) with H(ξ)= 〈 ξ,L−1(ξ) 〉 −L ( L−1(ξ) ) , ξ∈T∗M , L :TM→T∗M , H :T∗M→R . (54) Following the remarkmadebyPoincaréat theendofhisnote [105], themost interestingcase iswhen themap L :M×g→R only depends on its secondvariableX ∈ g. The Euler-Poincaré equationbecomes: ( d dt −ad∗V(t) )( dL(V(t)) ) =0 (55) WecanuseanalogyofstructurewhentheconvexGibbsensemble ishomogeneous [106].Wecan thenapplyEuler-PoincaréequationforLiegroupthermodynamics.ConsideringClairaut’sequation: s(Q)= 〈β,Q〉−Φ(β)= 〈 Θ−1(Q),Q 〉 −Φ ( Θ−1(Q) ) (56) withQ=Θ(β)= ∂Φ ∂β ∈ g∗,β=Θ−1(Q)∈ g, aSouriau-Euler-Poincaré equationcanbeelaboratedfor SouriauLiegroupthermodynamics: dQ dt = ad∗βQ (57) or d dt ( Ad∗gQ ) =0. (58) Thefirstequation, theEuler-Poincaréequation isareductionofEuler-Lagrangeequationsusing symmetriesandespecially the fact thatagroupisactinghomogeneouslyonthesymplecticmanifold: dQ dt = ad∗βQand ⎧⎨⎩ s(Q)= 〈β,Q〉−Φ(β)β= ∂s(Q)∂Q ∈ g , Q= ∂Φ(β)∂β ∈ g∗ (59) 68