torsdag 10 september 2009

Conservation of Momentum or Newton's 2nd Law?

                                          Incremental or Conservation Party?

Continuing the discussion from the previous post, let us note that the Navier-Stokes equation
expressing conservation of momentum, alternatively can be expressed as Newton's 2nd law 

                                                             F = ma = m dv/dt

with F the force acting on an element of fluid of mass m and acceleration a = dv/dt. We can view these formulations to be equivalent from analytical mathematical point of view, but we may ask if they also are equivalent physically, or computationally? 

Of course they are equivalent, you may say, because mathematics rules the game, but it is not so simple and clear if we recall that the Navier-Stokes equations cannot be solved exactly analytically, only approximately digitally by computers. The equivalence is then not so clear anymore.

So which formulation is most suitable to computation?  Newton's 2nd law because it can be solved by time-stepping moving forward in time with small increments of time: The force F gives the acceleration dv/dt = F/m which tells the change of velocity which tells the change in position, from one time level to the next. 

On the other hand, conservation of momentum is not directly ready for time-stepping, since it just expresses that something is conserved, namely momentum.

We are thus led to prefer an interpretation of a law of nature, which is most accessible to computation.  We may prefer such an interpretation also from physical point of view, if we view real physics as some form of analog computation, as discussed in the knol Is the World a Computation?

Light refraction is a result of the wavelike nature of light as propagating electromagnetic waves. Light refraction can alternatively be described as shortest time of travel of light rays. Wave propagation can be time-stepped, while shortest time of travel is a global minimization problem, for which computational solution is less direct. We are led to view light as waves from physical and computational point of view, rather than as rays of particles. 

An equilibrium states may be described as a state of balance of forces without any net force driving change. To find an equilibrium state of a system, we may time-step the system starting from some out-of-balance non-equilibrium state, with the hope that the system by itself approaches equilibrium. A physical law could then express the dynamics of a system computable by time-stepping, rather than a balance of forces at equilibrium, since this balance may not be directly computable. 

A minimization principle in physics, like minimal time of travel of light, would then not qualify as a physical law unless augmented by e.g. time-stepping into computable form.

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