Abstract
In this chapter, we describe Lagrange and Hamilton’s geometric viewpoint of mechanics. We will mainly use Lagrangian methods in this book because their covariance is automatic in relativistic field theories and they allow a more conceptual treatment of symmetries. The chapter starts with a review of Lagrangian mechanics. We then give an introduction to symplectic and Poisson manifolds, that we use to describe the dynamics of a Hamiltonian system. We then explain the relation with Lagrangian variational problems, and describe the Hamilton-Jacobi equations. We finish by discussing Poisson reduction in its classical and homotopical flavors (BRST formalism).
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Paugam, F. (2014). Lagrangian and Hamiltonian Systems. In: Towards the Mathematics of Quantum Field Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-04564-1_8
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DOI: https://doi.org/10.1007/978-3-319-04564-1_8
Publisher Name: Springer, Cham
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