Practical Introduction to Action-Dependent Field Theories

Action-dependent field theories are systems where the Lagrangian or Hamiltonian depends on new variables that encode the action. They model a larger class of field theories, including non-conservative behavior, while maintaining a well-defined notion of symmetries and a Noether theorem. This makes them especially suited for open systems. After a conceptual introduction, a quick presentation of a new mathematical framework is made for action-dependent field theory: multicontact geometry. The formalism is illustrated with a variety of action-dependent Lagrangians, some of which are regular and others singular, derived from well-known theories whose Lagrangians have been modified to incorporate action-dependent terms. Detailed computations are provided, including the constraint algorithm for the singular cases, in both the Lagrangian and Hamiltonian formalisms. These are the one-dimensional wave equation, the Klein–Gordon equation and the telegrapher equation, Maxwell's electromagnetism, Metric-affine gravity, the heat equation and Burgers' equation, the Bosonic string theory, and $(2+1)$-dimensional gravity and Chern–Simons equation.