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THE CARATHEODORY APPROACH TO OPTIMAL
Extremal Fields and HamiltonJacobi
The Partial Differential Equations
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algebra Brayton-Moser calculus of variations called canonical transformation Caratheodory Cartan form Chapter characteristic curve classical conditions are satisfied configuration space Consider constraints control problem control theory coordinate system coordinate-free defined denote diffeomorphism differential form differential operator dimension dual space economics example Exercise extremal field fiber space following conditions following form following formula geometric global H-J equation Hamilton equations Hamilton-Jacobi equation Hamiltonian function Hence Hessian form implies invariant integral Lagrange equations Lagrange multiplier Lagrangian submanifold linear map linear subspace linearly independent manifold mathematical matrix mechanical system non-degenerate extreme point notation one-form one-one open subset optimal control orbit parameterized Pareto physics point of f Poisson bracket positive definite Proof properties prove quadratic form quantization quantum mechanics real numbers real-valued function Remark Section solution symmetric tangent vector Td(Q Theorem 3.1 tion valued function variational problem vector field vector space