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This question already has an answer here:

What's so special about second order equations in classical mechanics? I have a basic understanding of the Lagrangian and Hamiltonian formulations of classical mechanics, so I'm not looking for answers like 'because Newton's second law is a second order ODE' or 'because Euler-Lagrange equations act on a first time derivative of position'. I'm looking for a deeper physical reason - in the same sense that Energy conservation is not fundamental, it results from time translation invariance. I realise two boundary conditions are required to solve for the dynamics of a given system, but I see that more as a result of the equations being second order than the cause of them being second order. Is there a more fundamental organising principle that I am not aware of?

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marked as duplicate by Qmechanic May 31 '14 at 19:46

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  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/18588/2451 , physics.stackexchange.com/q/4102/2451 and links therein, which e.g. mention Ostrogradski. $\endgroup$ – Qmechanic May 31 '14 at 18:19
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    $\begingroup$ Ostragradski's theorem says higher order derivative terms leads to energy being unbounded below (see, e.g., section 2 of arXiv:astro-ph/0601672 for a good review). $\endgroup$ – Alex Nelson May 31 '14 at 18:23
  • $\begingroup$ Thanks @Qmechanic for pointing me to the duplicates. I must admit I'm having a little trouble understanding this negative/unbounded energy argument. For example there could be something wrong with using the Legendre transform to obtain the Hamiltonian for higher order systems. Surely one could define other transformations which lead to well behaved Hamiltonians? In any case it seems like more of an effect of classical mechanics being second order than a cause. Is there any explanation in terms of degrees of freedom, e.g. one d.o.f. due to time and one d.o.f. due to space? $\endgroup$ – Dennis Addison Jun 1 '14 at 10:16
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I'm not sure there is a more fundamental organising principle. The basic rationale behind second order equations of motion is that the state of real physical systems always seems to be specified by position and velocity/momentum alone, with no additional data.

In the context of quantum mechanics however, the Hamiltonian formulation is fundamental, and that inevitably* leads to second order equations of motion in configuration space.

*If there are no constraints/auxiliary fields etc.

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Galilean (or poincare, if relativistic) invariance mixed with a minimalistic approach (or a accepting an effective low energy paradigm to construct the dynamics) could be a partial answer. The action must indeed invariant under these spacetime symmetry. The only term in the lagrangian with the lowest number of derivatives is clearly $\vec{v}^2$, and therefor the dynamical equations, at least at low energy, must be of second order.

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