Lucía Bua, Ioan Bucataru, Modesto Salgado
In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether Theorem and its converse, in the framework of the $k$-symplectic formalism, using the Fr\"olicher-Nijenhuis formalism on the space of $k^1$ velocities of the configuration manifold. For the $k=1$ case it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether Theorem and its converse. For $k>1$, we provide a new proof that Noether Theorem is true, and hence each Cartan symmetry induces a conservation law. We show that under some assumptions, the converse of Noether Theorem is also true and provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.
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http://arxiv.org/abs/1204.6573
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