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?