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Relativistic covariance in classical mechanics?

  1. Aug 5, 2009 #1

    Jano L.

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    Gold Member

    Hi friends,

    long time ago I noticed the following interesting similarity between
    classical mechanics and relativity. Consider particle moving in an external field and the
    action defined as a function of actual time t and position q:

    S(t,q) = \int_0^t L(q^r,\dot q^r,t´)dt´

    The motion [tex]q^r[/tex] is the real motion of the particle which finishes at the
    position q at the time t. It is possible to derive Hamilton-Jacobi
    equations for S:

    \frac{\partial S}{\partial t} = -H, \frac{\partial S}{\partial x^i} = p_i.

    Here comes the point. Let us define new variable [tex]x^0 = ct[/tex]. We see that the Hamilton Jacobi equations can be written in the compact form

    \frac{\partial S}{\partial x^\mu} = p_\mu,

    which is relativistically covariant equation! Its solution for free particle is

    S = -Et + \mathbf{p\cdot r}

    How can we get the right STR metric (-1,1,1,1)? We did not assume the Lorentz invariance anywhere! In STR, this equation is still valid, although L and therefore S is different. Is there some reason for this (maybe waves with phase -Et+pr are fundamental?), or is it only a random coincidence? Another interesting point is that this wave behaviour of free particle is similar to de Broglie wave hypothesis, which lead Schroedinger to his wave equation in QM.
    At least, does CM say us that -1,1,1,1 is better than 1,-1,-1,-1 ??? ;-)

    Jano L.
    Last edited: Aug 5, 2009
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