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From noether theorem to Laplace-runge-lenz vector

  1. May 4, 2012 #1
    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?
  2. jcsd
  3. May 4, 2012 #2


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    SO(4) symmetry and other groups containing subgroups isomorphic to SO(4).
  4. May 5, 2012 #3


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    Have a look at the Poisson brackets defining the symmetry algebra


    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.
  5. Jan 10, 2014 #4
    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.
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