Algebraic and Analytic Geometry by Amnon Neeman

By Amnon Neeman

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3) does not exhaust all equations considered in first order Lagrangian theory. 17) i ). 2 Cartan and Hamilton–De Donder Equations 31 n E L : J 1 J 1 Y → T ∗ J 1 Y ∧ (∧ T ∗ X ), E L = [(∂i L − dλ πiλ + ∂i π λj (yλ − yλ ))dy i + ∂iλ π μj (yμj − yμj )dyλi ] ∧ ω, j j μ i ∂i . 18) j yλ )∂i π λj = 0. 3) on integrable sections of J 1 Y → X . These equations are equivalent if a Lagrangian is regular. 20) which is assumed to hold for all vertical vector fields u on J 1 Y → X . 21) ΞY = pω + piλ dy i ∧ ωλ .

62) and its jet prolongation J 1 H : J 1 Π −→ J 1 J 1 Y, Y i (yμi , yλi , yλμ ) ◦ J 1 H = (∂μi H , yλi , dλ ∂μi H ), ν j ∂ν . 1). Then L −1 is a Hamiltonian map. 57) associated to a Hamiltonian map L −1 . 3) for a Lagrangian L. 65). 63)). It follows that, in the case of hyperregular Lagrangians, covariant Hamiltonian formalism is equivalent to the Lagrangian one. Let now L be an arbitrary Lagrangian on a jet manifold J 1 Y . 66) ∗ H = HH + H L . 9). Accordingly, H ◦ L is the projector from J 1 Y onto H (N L ).

8 enables us to relate the Euler–Lagrange equation for an almost regular Lagrangian L with the covariant Hamilton equation for Hamiltonian forms weakly associated to L [54, 117, 118]. 61) for a Hamiltonian form H weakly associated to a semiregular Lagrangian L. 19). Proof Put s = H ◦ r . Since r (X ) ⊂ N L , then r = L ◦ s, J 1r = J 1 L ◦ J 1 s. If r is a classical solution of the covariant Hamilton equation, an exterior form E H vanishes at points of J 1r (X ). Hence, the pull-back form E L = (J 1 L)∗ E H vanishes at points J 1 s(X ).

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