Abstract:
The generalized variational principle of Herglotz defines the functional whose extrema are sought by a differential equation rather than an integral. It reduces to the classical variational principle under classical conditions. The Noether theorems are not applicable to functionals defined by differential equations. For a system of differential equations derivable from the generalized variational principle of Herglotz, a first Noether-type theorem is proven, which gives explicit conserved quantities corresponding to the symmetries of the functional defined by the generalized variational principle of Herglotz. This theorem reduces to the classical first Noether theorem in the case when the generalized variational principle of
Herglotz reduces to the classical variational principle. Applications of the first Noether-type theorem are shown and specific examples are provided. A second Noether-type theorem is proven, providing a non-trivial identity corresponding to each infinite-dimensional symmetry group of the functional defined by the generalized variational principle of Herglotz. This theorem reduces to the classical second Noether theorem when the generalized variational principle of Herglotz reduces to the classical variational principle. A new variational principle with several independent variables is defined. It reduces to Herglotz's generalized variational principle in the case of one independent
variable, time. It also reduces to the classical variational principle with several
independent variables, when only the spatial independent variables are present. Thus, it generalizes both. This new variational principle can give a variational description of processes involving physical fields. One valuable characteristic is that, unlike the classical variational principle with several independent variables, this variational principle gives a variational description of nonconservative processes even when the Lagrangian function is independent of time. This is not possible with the classical variational principle. The equations providing the extrema of the functional defined by this generalized variational principle are derived. They reduce to the classical Euler-Lagrange equations (in the case of several independent variables), when this new variational principle reduces to the classical variational principle with several independent variables. A first Noether-type theorem is proven for the generalized variational principle with several independent variables. One of its corollaries provides an explicit procedure for finding the conserved quantities corresponding to symmetries of the functional defined by this variational principle. This theorem reduces to the classical first Noether theorem in the case when the generalized variational principle with several independent variables reduces to the classical variational principle with several independent variables. It reduces to the first Noether-type theorem for Herglotz generalized variational principle when this generalized variational principle reduces to Herglotz's variational principle. A criterion for a transformation to be a symmetry of the functional defined by
the generalized variational principle with several independent variables is proven.
Applications of the first Noether-type theorem in the several independent variables
case are shown and specific examples are provided.