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.


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
Full Text: DOI arXiv


[1] DOI: 10.1017/CBO9780511599941
[2] Noether A. E., Gottingen Math. Phys. Kl. 235
[3] DOI: 10.1142/1729
[4] DOI: 10.1007/BF02741369
[5] DOI: 10.1063/1.528084 · Zbl 0668.53050
[6] DOI: 10.1016/B978-0-08-012315-8.50007-0
[7] Stephani H., Exact Solutions of Einstein’s Field Equations (2002)
[8] DOI: 10.1063/1.1385175 · Zbl 1009.83011
[9] Bokhari A. H., Int. J. Theor. Phys. 45 pp 1063–
[10] DOI: 10.1007/s10773-007-9390-6 · Zbl 1128.83006
[11] DOI: 10.1007/s10714-007-0501-8 · Zbl 1136.83010
[12] Qadir A., SIGMA 3 pp 103–
[13] Gelfand I. M., Calculus of Variations (2000) · Zbl 0964.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.