Ivancevic_Applied-Diff-Geom

940 Applied Differential Geometry: A Modern IntroductionNow we consider relations between Lagrangian and Hamiltonian formalisms.A Hamiltonian form H is defined to be associated with a Lagrangiandensity L if it satisfies the relationŝL ◦ Ĥ| Q = Id Q , Q = ̂L(J 1 (X, Y )),H = H bH + L ◦ Ĥ,which take the coordinate form∂ µ i L(xα , y j , ∂ j αH) = p µ i , L(xα , y j , ∂ j αH) = p µ i ∂i µH − H.Note that there are different Hamiltonian forms associated with the samesingular Lagrangian density.Bearing in mind physical application, we restrict our consideration toso–called semiregular Lagrangian densities L when the preimage ̂L −1 (q) ofeach point of q ∈ Q is the connected submanifold of J 1 (X, Y ). In this case,all Hamiltonian forms associated with a semiregular Lagrangian density Lcoincide on the Lagrangian constraint space Q, and the Poincaré–Cartanform Ξ L is the pull–backΞ L = H ◦ ̂L, π α i y i α − L = H(x µ , y i , π α i ),of any associated multimomentum Hamiltonian form H by the Legendremorphism ̂L [Zakharov (1992)]. Also the generating form (5.401) is thepull–back ofΛ L = E H ◦ J 1 ̂Lof the Hamiltonian operator (5.413) of any Hamiltonian form H associatedwith a semiregular Lagrangian density L. As a consequence, we getthe following correspondence between solutions of the Euler–Lagrangianequations and the Hamiltonian equations [Sardanashvily (1994); Zakharov(1992)].Let a section r of Π −→ X be a solution of the Hamiltonian equations(5.397) for a Hamiltonian form H associated with a semiregular Lagrangiandensity L. If r lives on the Lagrangian constraint space Q, the sections = Ĥ ◦ r of J 1 (X, Y ) −→ X satisfies the first–order Euler–Lagrangianequations (5.386). Conversely, given a semiregular Lagrangian density L,let s be a solution of the first–order Euler–Lagrangian equations (5.386).Let H be a Hamiltonian form associated with L so thatĤ ◦ ̂L ◦ s = s. (5.415)