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Lagrangian and causality

  1. Jul 2, 2009 #1

    I read somewhere that the second derivatives of coordinates in Lagrangian would violate causality. Why is this so? Does that mean that the whole concept of jerky mechanics violates causality?

  2. jcsd
  3. Jul 2, 2009 #2
    The third order differential equation needs three initial conditions. From mathematical view point, it is OK, not problem. But in the Classical Mechanics we think of two initial conditions available: r(0) and (dr/dt)(0). Where take the third one from? The arbitrary character of the solution means loss of causatily. The (two) initial conditions and a given force do not determine the solution unambiguously.

    In electrodynamics there is a Lorentz-Abraham force proportional to the third derivative d3r/dt3. Such an equation has non-physical self-accelerating solutions.
  4. Jul 2, 2009 #3
    Does "the arbitrary character of the solution" mean the arbitrary arrow of time? Does it manifest itself also in QFT where our Lagrangians contain only fields and first derivatives?
  5. Jul 2, 2009 #4
    No, I don't think so. It is just incomplete fixation of the intergation constants. We need more constants to fix a unique trajectory.

    No, in QFT we have well defined initial (and final) conditions, so the solution is completely determined by them. Consider, as an example, a usual QM scattering problem: everything is OK there.

    Another thing is the QFT divergences (due to badly guessed equations), but it has nothing in common with the differential equation order.
  6. Jul 2, 2009 #5
    I would like to add that this particular third order differential equation can be re-written as an integral equation, and the time integration is carried out over "future" time. So they often say it violates the causality.

    In fact, even in a regular variational principle in mechanics they say: "We know the initial and the final (future) positions", and then they vary the action. But later on, they use only (two) initial conditions, which is quite physical, instead of (two) initial and final positions. Mathematically either way is good but physically we usually do not know final position. It is an unknown datum and it is found by solving the "initial" rather than "boundary" problem.
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