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?
2 Answers
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.
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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