Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 843Since E L| J 2 (R,Q)= E L , the Cartan equations (5.113) are equivalent to theLagrangian equations (5.111) on integrable sections c = ċ of J 1 (R, Q) → R.These equations are equivalent in the case of regular Lagrangians.On sections c : R −→ J 1 (R, Q), the Cartan equations (5.113) are equivalentto the relationc ∗ (u⌋dH L ) = 0, (5.114)which is assumed to hold for all vertical vector–fields u on J 1 (R, Q) −→ R.With the Poincaré–Cartan form H L (5.110), we have the Legendre mapĤ L : J 1 (R, Q) −→ T ∗ Q, (p i , p) ◦ ĤL = (π i , L − π i ˙q i ).Let Z L = ĤL(J 1 (R, Q)) be an imbedded subbundle i L : Z L ↩→ T ∗ Q ofT ∗ Q → Q. It admits the pull–back de Donder form i ∗ LΞ. We haveH L = Ĥ∗ LΞ L = Ĥ∗ L(i ∗ LΞ).By analogy with the Cartan equations (5.114), the Hamilton–de Donderequations for sections r of T ∗ Q → R are written asr ∗ (u⌋dΞ L ) = 0 (5.115)where u is an arbitrary vertical vector–field on T ∗ Q → R [Lopez and Marsden(2003)].Let the Legendre map ĤL : J 1 (R, Q) −→ Z L be a submersion. Thena section c of J 1 (R, Q) −→ R is a solution of the Cartan equations (5.114)iff ĤL ◦ c is a solution of the Hamilton–de Donder equations (5.115), i.e.,Cartan and Hamilton–de Donder equations are quasi–equivalent [Gotay(1991a); Lopez and Marsden (2003)].5.6.8 Time–Dependent Hamiltonian DynamicsLet the Legendre bundle V ∗ Q → R be provided with the holonomic coordinates(t, q i , ˙q i ). Relative to these coordinates, the canonical 3–form Ω(5.101) and the canonical Poisson structure on V ∗ Q readΩ = dp i ∧ dq i ∧ dt, (5.116){f, g} V = ∂ i f∂ i g − ∂ i g∂ i f, (f, g ∈ C ∞ V ∗ Q). (5.117)The corresponding symplectic foliation coincides with the fibration V ∗ Q →R. The symplectic forms on the fibres of V ∗ Q → R are the pull–backsΩ t = dp i ∧ dq i of the canonical symplectic form on the typical fibre