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Invariants (sorry i didn't know where to put it)

  1. Oct 8, 2006 #1
    That's the question, if we can interpretate mathematically using some kind of "invariant" theory for this situation:

    - Using Newton's second Law for gravity then we find that:

    [tex] \ddot r = -GM/r [/tex] (1)

    M is the mass of the earth, and m is the mass of the particle, amazingly using the equation (1) above [tex] r(t) \rightarrow r(m,t) = r(t) [/tex] , then the trajectories of the system of a particle under the gravity is invariant under mass change of the particle (if we make [tex] m' = m + \delta m [/tex] as an infinitesimal mass change of the particle the trajectories r(t) are the same)

    - Using the metric in GR we also find that the metric:

    [tex] g_{ab}=N(t)dt^{2} - \sum_{i,j}H_{ij}dx^{i}dx^{j} [/tex]

    where i,j=x,y,z

    then the solutions of the metric are invariant under infinitesimal changes of the function [tex] N(t)+ \delta N(t) [/tex]

    -And the last example, if we define the "charge" by:

    [tex] Q=e|\Phi|^{2} [/tex] you can check that charge is invariant under

    transformation of the Quantum wave function of the form [tex] \Phi\rightarrow e^{ia(x)}\Phi [/tex]

    My question involving math is if we can define some "conserved" quantities or properties of the system from these "peculiar" invariance of the system under m (mass of the particle) N(t) (any function of t) or invariance under "phase" transformation, hopes the problem seems clear.
  2. jcsd
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