Non-Noether symmetries and their influence on phase space geometry. (English) Zbl 1056.70010
The paper connects certain non-Noether symmetries, not only to a number of conservation laws that under some conditions ensure integrability, but also to certain objects from the geometry of phase space: Lax pairs, bi-Hamiltonian structures, bicomplexes, Fröhlicher-Nijenhuis operators. Example of KdV suggests that some mysterious mathematical constructions (for instance, Lenard recursion operator, Lax pair and bi-Hamiltonian structure of KdV equation) that often carry no direct physical content, could be regarded as manifestations of non-Noether symmetries.
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 70G10 | Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics |
| 53Z05 | Applications of differential geometry to physics |
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