Incidentally, there are some deep reasons for why we only generally care about stuff up to acceleration. If higher derivative motions terms are relevant in the equations of motion for something, then the things Hamiltonian can reach arbitrary energies, which will cause it to spontaneously produce huge amounts of particles and quickly become catastrophic. See here: http://physics.stackexchange.com/questions/4102/why-are-only-derivatives-to-the-first-order-relevant