Alex,
****Harvey, you can NOT derive F=ma from minima of action integral, it is mathematically ILLOGICAL. Action (= time integral of energy) can not be defined BEFORE enegy is defined. Energy is defined as work, work is defined as force times distance.****
Feynman in his 1948 article ("Space-time approach to non-relativistic quantum mechanics", Rev. Mod. Phys. 20(2), 367 http://cornell.mirror.aps.org/abstract/RMP/v20/i2/p367_1 ) showed was able to derive Newton's and Schrodinger's equations using his transition amplitude (gives us the quantum amplitude for the transition from the quantum state at the time *t'* to the quantum state at the time *t"*). This formulation can be used to derive Newton's F=ma equation. However, this formulation is not completely identical to Newton's formulation because in Feynman's RMP (1948) paper he didn't set the equation equal to zero due to the restrictions of QM.
****So, force should be defined FIRST and then action can be.*
If you want to define force within quantum mechanics, then you cannot use Newton's formulation. This is the advantage of using the quantum action S.
****By the way, F=ma as you like to write many times, is INCORRECT definition of force. Correct one is F=dp/dt.****
That formulation is the more conservative approach, however it is tacitly assumed that the inertial mass *m* is a constant individual property of a body. This, of course, is not true in terms of relativistic scales, but the classical Newtonian equation does not work for relativistic scales. So, why not simplify the equation for everybody since we are talking about classical limits anyway? (unless you know of a classical situation that requires dp/dt?)
****So, minimum of action integral is NOT a fundamental principle, but is a mathematical consequence of the definition of force F=dp/dt.****
Oh my! Does (-dp/dt + F)=0? You know well that this does not work in quantum mechanical applications. This is why the minimum of action integral is more fundamental, it works for both classical and quantum mechanical applications.
****(No need to break a head thinking how nature selects one path out of all possible - it is simply the only one allowed by correct definition of force (and thus energy). Very simple and beautiful.****
You have a very classical perspective of the world. Why do you virtually ignore quantum mechanics?
Warm regards, Harv |