Department of Mathematics and Statistics,

California State University,

Chico, CA 95929-0525,

USA

E-mail: vrosenhaus@csuchico.edu

**Infinite Conservation Laws for Infinite Symmetries**

**Abstract:**

We will consider differential equations possessing infinite symmetry
algebras and infinite sets of local conservation laws.

Infinite symmetry algebras with arbitrary functions of independent variables
and their corresponding conservation laws were studied in V.Rosenhaus,
"Infinite Symmetries and Conservation Laws", *J. Math. Phys.* **43**,
6129-6150 (2002)]. It was shown that these symmetries may lead to a finite
number of local conservation laws that are determined by a specific form
of boundary conditions. Conservation laws corresponding to infinite symmetries
were calculated for a number of interesting equations, e.g.: V.Rosenhaus,
"On conservation laws for the equation of non-stationary transonic gas
flows", *J. Dyn. Sys. Geom. Theor.* *1*, 95-107 (2002), "On conservation
laws and boundary conditions for short waves equation", *Rep. Math. Phys.*
**51**, 71-86 (2003), "On infinite symmetries and essential conservation
laws for Navier-Stokes equations", *Proc. XXIV Intern. Conf. Group Theor.
Methods in Physics* (Paris, July, 2002), pp.737-740.

The present work is an extension of this approach for Lagrangian differential equations whose symmetry algebras contain arbitrary functions of dependent variables and their derivatives. We will show that these symmetries lead to an infinite number of local conservation laws. We will discuss classification, boundary conditions and give examples.