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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.

Tomek Dobrzynski
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  • Of course,such as the Newton's second law. – Jack Mar 14 '17 at 06:18
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    But surely the ultimate laws of nature will not depend on two prior initial conditions. Why? What do you mean? – Qmechanic Mar 14 '17 at 06:28
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    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. – sammy gerbil Mar 14 '17 at 07:04
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    Possible duplicates: http://physics.stackexchange.com/q/18588/2451 , http://physics.stackexchange.com/q/4102/2451 and links therein. – Qmechanic Mar 14 '17 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.

ZeroTheHero
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  • 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. – flippiefanus Mar 14 '17 at 04:20
  • @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. – ZeroTheHero Mar 14 '17 at 04:33