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New conserved quantities for the spaces of different curvatures. (English) Zbl 1189.70062

Summary: It is known that corresponding to each isometry there exists a conserved quantity. It is also known that the Lagrangian of the line element of a space is conserved. Here we investigate the possibility of the existence of “new” conserved quantities, i.e. other than the Lagrangian and associated with the isometries, for spaces of different curvatures. It is found that there exist new conserved quantities only for the spaces of zero curvature or having a section of zero curvature.

MSC:

70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70H03 Lagrange’s equations
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
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