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...The calculus of variations m bendersky december these notes are partly based on a course given by jesse douglas contents introduction typical problems some preliminary results lemmas afirst necessary condition for weak relative minimum euler lagrange dierential equation consequences weierstrass erdmann corner conditions examples extension to vector function y x s in parametric form theory more rst variation an integral i t j rx f dx application transversality fields extremals and hilbert invariant legendre conjugate points focal envelope theorems jacobi or strong geo metric derivation review preview sucient smooth imbedding lemma fundamental suciency second other proof one fixed variable end point both variational with multiple integrals functionals involving higher derivatives constraints functions dimensions n equations canonical hamilton field characteristic curves first theorem poisson bracket use principles mechanics further topics...