From noether theorem to Laplace-runge-lenz vector

[wdlang]

it is said that each conserved quantity is related to some symmetry of the system

so, what is the symmetry underlying the Laplace-Runge-Lenz vector?

 

[dextercioby]

SO(4) symmetry and other groups containing subgroups isomorphic to SO(4).

 

[tom.stoer]

Have a look at the Poisson brackets defining the symmetry algebra

http://en.wikipedia.org/wiki/Laplace–Runge–Lenz_vector

Remark: there are so called topological conservation laws which are not related to a local symmetry via Noether's theorem. Think about a field exp(iθ(α)) where the angle α is defined on a circle S¹, so α lives in [0,2π]. When α runs once around the circle from 0 to 2π the field θ (which must be periodic on the circle) may run from 0 to 2wπ with w = 0, ±1, ±2, ... The winding number w is a conserved quantity b/c no local deformation or oscillation of the field θ(x) can change this winding number.

 

[paulmasson]

Conservation of the Runge-Lenz vector does not correspond to a symmetry of the Lagrangian itself. It arises from an invariance of the integral of the Lagrangian with respect to time, the classical action integral. Some time ago I wrote up a derivation of the conserved vector for any spherically symmetric potential:

http://analyticphysics.com/Runge Vector/The Symmetry Corresponding to the Runge Vector.htm

The derivation is at the level of Goldstein and is meant to fill in the gap left by its omission from graduate-level classical mechanics texts.