  In the recent years, with the incorporation of contact geometry, there has
been a renewed interest in the study of dissipative or non-conservative systems
in physics and other areas of applied mathematics. The equations arising when
studying contact Hamiltonian systems can also be obtained via the Herglotz
variational principle. The contact Lagrangian and Hamiltonian formalisms for
mechanical systems has also been generalized to field theories. The main goal
of this paper is to develop a generalization of the Herglotz variational
principle for first-order and higher-order field theories. In order to
illustrate this, we study three examples: the damped vibrating string, the
Korteweg-De Vries equation, and an academic example showing that the
non-holonomic and the vakonomic variational principles are not fully
equivalent.
