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    $\begingroup$ I wonder if this problem goes away if you allow only $C^\infty$ (infinitely differentiable) solutions to the equations of motion, or if a dome can be contrived so that a Bump Function of time is a solution. $\endgroup$
    – Dzamo Norton
    Commented Oct 14, 2012 at 4:36
  • $\begingroup$ Related question on Math.SE: math.stackexchange.com/q/1426755/11127 $\endgroup$
    – Qmechanic ♦
    Commented Sep 8, 2015 at 21:28