Is it a happy coincidence that second degree differential equations approximate reality or a necessity? They describe how a system will evolve from one state to the next, but surely the ultimate laws of nature will not depend on two prior initial conditions. Surely reality can compute its next state from the current state alone.
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$\begingroup$ Of course,such as the Newton's second law. $\endgroup$– JackCommented Mar 14, 2017 at 6:18
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1$\begingroup$ But surely the ultimate laws of nature will not depend on two prior initial conditions. Why? What do you mean? $\endgroup$– Qmechanic ♦Commented Mar 14, 2017 at 6:28
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1$\begingroup$ Why 2nd derivatives? 0th and 1st derivatives are common. 4th derivatives occur in elasticity theory. ... Reality does not compute anything. We do all the computations. $\endgroup$– sammy gerbilCommented Mar 14, 2017 at 7:04
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1$\begingroup$ Possible duplicates: physics.stackexchange.com/q/18588/2451 , physics.stackexchange.com/q/4102/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Mar 14, 2017 at 13:33
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Actually if you think of classical (and non-relativistic) physics, the canonical equations of motion of the Hamiltonian formalism $$ \dot p=-\frac{\partial H}{\partial q}\, ,\qquad \dot q=\frac{\partial H}{\partial p} $$ are first order. In addition, the Hamilton-Jacobi equation $$ H+\frac{\partial S}{\partial t}=0\, ,\qquad p=\frac{\partial S}{\partial q} $$ is also of first order.
It is true that in Lagrangian mechanics the equations of motions are of second order, but my personal appreciation is that the Hamiltonian formalism - because the equations of motions are expressed in a more symmetrical way amenable to symplectic transformations and because it leads to deeper results such as Liouville's theorem and other associated invariants - is more fundamental.
answered Mar 14, 2017 at 4:02
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$\begingroup$ Are you saying the Hamiltonian formalism retains more symmetry than the Lagrangian formalism? I thought it is the other way around. Certainly, if a theory contains a symmetry it would be a symmetry of the Lagrangian. $\endgroup$ Commented Mar 14, 2017 at 4:20
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$\begingroup$ @flippiefanus Good point. I clarified my answer following your observation. The Hamiltonian formulation of classical mechanics of point particles at least is an example where the equations need not be 2nd order. Granted from field theory perspective the Lagrangian formulation is much more flexible. $\endgroup$ Commented Mar 14, 2017 at 4:33