Generalized Noether’s theorem for continuous mechanical systems

In this paper, the generalized Noether’s theorem for continuous mechanical systems is formulated by analogy with Vujanović and Djukić’s (Acta Mech 23:17–27, 1975) and Vujanović’s (Int J Nonlin Mech 13:185–197, 1978) generalized Noether’s theorem in analytical mechanics of particles. This generalization, unlike by the mentioned authors, is directly performed, i.e., starting from the total variation of the action for continuous systems and the corresponding general Lagrangian equations. The thus formulated generalized Noether’s theorem allows for finding such transformations of the field and time functions that yield integrals (or constants) of motion, and it is also applicable to nonconservative systems, when energy integrals are obtained in a broader sense. In addition, the corresponding pseudoconservative continuous systems are introduced, analogous to those in analytical mechanics (Mušicki in Acta Mech 223:2117–2133, 2012), whereby a second, complementary approach to this problem is made. The analysis of the obtained results particularly focuses on the nature of energy integrals (in both ordinary and broader sense), which are here closely associated with impulses and differ essentially from energy integrals in the mechanics of particles.