Quasi-Noether Systems and Quasi-Lagrangians

Abstract

We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green-Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries, not true symmetries. We finally introduce the critical point condition for evolution equations with a conserved integral, demonstrate examples of its compatibility, and compare the invariant submanifolds of quasi-Lagrangian systems with those of Hamiltonian systems.be the i-th total derivative, 1 ≤ i ≤ p, the sum extending over all (unordered) multi-indices J = (j 1 , j 2 , ..., j k ) for k ≥ 0 and 1 ≤ j k ≤ p. Two conservation laws K andK are equivalent if they differ by a trivial conservation law [11]. A conservation law D i P i . = 0 is trivial if a linear combination of two kinds of triviality