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.
Highlights
For variational systems the relation between symmetries of the Lagrangian function and conservation laws was known from the classical Noether result [1]
We show that the correspondence between symmetries and conservation laws in terms of Lagrange identity is equivalent to the one using the Noether identity and leads to the same conservation laws
In [6], the approach based on the Noether identity was applied to quasi-Noether systems possessing infinite symmetries involving arbitrary functions of all independent variables, in order to generate an extension of the Second Noether theorem for systems that may not have well-defined Lagrangian functions
Summary
For variational systems the relation between symmetries of the Lagrangian function and conservation laws was known from the classical Noether result [1]. The Noether operator identity has been shown to provide a Noether-type relation between symmetries and conservation laws for Lagrangian systems, and for a large class of differential systems that may not have a well-defined variational functional, see [4,5]. In this approach, variational symmetries of quasi-Lagrangians are only sub-symmetries, and need not be symmetries. We compare the invariant submanifolds of quasi-Noether systems to those of Hamiltonian systems and show that they satisfy opposite containments To address this fact we introduce the notion of a critical point of a conserved quantity, and demonstrate examples of the compatibility of the critical point condition with the time evolution of the PDE system
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