Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 851A Hamiltonian form H weakly associated with an almost regular LagrangianL exists iff the fibre bundle J 1 (R, V ) ∗ Q −→ N L admits a globalsection.The condition (iii) leads to the following property [Giachetta et. al.(1997); Mangiarotti and Sardanashvily (1998)]. The Poincaré–Cartan formH L for an almost regular Lagrangian L is constant on the connected pre–image ̂L −1 (z) of any point z ∈ N L .An immediate consequence of this fact is the following assertion [Giachettaet. al. (1997)]. All Hamiltonian forms weakly associated with analmost regular Lagrangian L coincide with each other on the Lagrangianconstraint space N L , and the Poincaré–Cartan form H L for L is the pull–backH L = ̂L ∗ H, π i ˙q i − L = H(t, q j , π j ),of any such a Hamiltonian form H.It follows that, given Hamiltonian forms H an H ′ weakly associatedwith an almost regular Lagrangian L, their difference is fdt, (f ∈ I N ).Above proposition enables us to connect Lagrangian and Cartan equationsfor an almost regular Lagrangian L with the Hamiltonian equations forHamiltonian forms weakly associated with L [Giachetta et. al. (1997)].Let a section r of V ∗ Q −→ R be a solution of the Hamiltonian equations(5.121) for a Hamiltonian form H weakly associated with an almost regularLagrangian L. If r lives in the constraint space N L , the section c = π Q ◦ rof Q −→ R satisfies the Lagrangian equations (5.111), while c = Ĥ ◦ r obeysthe Cartan equations (5.113).Given an almost regular Lagrangian L, let a section c of the jet bundleJ 1 (R, Q) −→ R be a solution of the Cartan equations (5.113). Let H be aHamiltonian form weakly associated with L, and let H satisfy the relationĤ ◦ ̂L ◦ c = j 1 (π 1 0 ◦ c). (5.141)Then, the section r = ̂L ◦ c of the Legendre bundle V ∗ Q −→ R is a solutionof the Hamiltonian equations (5.121) for H. Since Ĥ ◦ ̂L is a projectionoperator, the condition (5.141) implies that the solution s of the Cartanequations is actually an integrable section c = ċ where c is a solution of theLagrangian equations.Given a Hamiltonian form H weakly associated with an almost regularLagrangian L, let us consider the corresponding constrained Hamiltonianform H N (5.133). H N is the same for all Hamiltonian forms weakly associ-