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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:
70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70H03Lagrange’s equations
70H40Relativistic dynamics