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Covariant Hamiltonian first-order field theories with constraints, on manifolds with boundary: the case of Hamiltonian dynamics

Volume 110 / 2016

A. Ibort, A. Spivak Banach Center Publications 110 (2016), 87-104 MSC: 70H15, 70H25, 70S05, 53D12. DOI: 10.4064/bc110-0-6


Inspired by problems arising in the geometrical treatment of Yang–Mills theories and Palatini’s gravity, the covariant formulation of Hamiltonian dynamical systems as a Hamiltonian field theory of dimension $1+0$ on a manifold with boundary is presented. After a precise statement of Hamilton’s variational principle in this context, the geometrical properties of the space of solutions of the Euler–Lagrange equations of the theory are analyzed. A sufficient condition is obtained that guarantees that the set of solutions of the Euler–Lagrange equations at the boundary of the manifold fill a Lagrangian submanifold of the space of fields at the boundary. Finally a theory of constraints is introduced that mimics the constraints arising in Palatini’s gravity.


  • A. IbortICMAT and Departamento de Matemáticas
    Universidad Carlos III de Madrid
    Avda. de la Universidad 30
    28911 Leganés, Madrid, Spain
  • A. SpivakDepartment of Mathematics
    University of California at Berkeley
    903 Evans Hall
    CA 94720, USA

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