## söndag 9 mars 2014

### Newton's Laws vs Lagrange's Principle of Least Action

There is a firm belief among physicists that Nature somehow functions so as to find the least action, for example that light finds its way by choosing among all paths connecting two points the quickest path. There is good reason to question this idea on the ground that physics does not have any means to compute, store and compare action integrals, just because mathematicians can do that by writing symbols on a piece of paper.

There are two mathematically equivalent ways of describing a dynamical system of classical mechanics:
1. Newton's laws expressing equilibrium of forces.
2. Lagrange's principle of least action.
Here 1 expresses stationarity (or minimality) of the action integral
• $\int_0^T (K(t) -V(t))\, dt$,
where $K(t)$ and $V(t)$ are kinetic and potential energies as functions of time $t$ over a given time interval $[0,T]$. We see that the dimension of the action is energy x time.

While equilibrium of forces has a direct physical reality in the sense that a dynamical system directly reacts to forces according to Newton's laws, this is less clear for Lagrange's principle of least action since it requires evaluating and comparing different action integrals and then choosing the one statisfying stationarity or minimality as the physical one.

While the kinetic and potential energies have physical representations, the action as the integral of kinetic and potential energies does not seem to have a physical representation. We are thus led to the conclusion that Lagrange's principle of least action does not describe physics, only mathematics, and confusion arises if least action nevertheless is believed to have a physical reality.

This directly connects to the definition of Planck's constant $h$ which has the dimension of energy x time as an integral of energy over time.  The physical meaning of Planck's constant $h$ was a mystery to Planck and the mystery remains today, if you give it a careful thought and not just accept that since so many are speaking about $h$, it must have a definite meaning as an "elementary quantum of action" or something like that…or something according to your own favorite idea...