Recently obtained results linking several constants of motion to one (non-Noetherian) symmetry to the problem of geodesic motion in Riemannian space-times are applied. The construction of conserved quantities in geodesic motion as well as the deduction of geometrical statements about Riemannian space-times are achieved.

A variational principle is presented, by means of which the equation of motion of the damped harmonic oscillator is found. Starting from this variational principle a systematic reformulation of the classical mechanics leads us to a Hamilton–Jacobi equation with an additional term, which is proportional to the action. The quantization of this Hamilton–Jacobi equation, following a method originally due to Schrödinger, gives the Langevin–Schrödinger equation.

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A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density |detg| 1 2 G a b , associated to the Einstein tensor G a b of the regularized metric, be a distribution with support on a submanifold of codimension of at most one and (ii) the regularized metric be a continuous metric which coincides with the singular one everywhere except on this support. In this paper, the curvature and Einstein tensors of the geometries associated to point sources in the 2 + 1-dimensional gravity and the Schwarzschild spacetime are considered. In both examples the regularized metrics are shown to be continuous regular metrics, as defined by Geroch and Traschen, with well defined distributional curvature tensors at all the intermediate steps of the calculation. The limit in which the support of these curvature tensors tends to the singular region of the original spacetime is studied and the results are contrasted with the ones obtained in previous works.

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