Should we be astonished by the Principle of “Least” Action?
September 28, 2014 10 Comments
As one goes through more advanced expositions of quantum physics, the concept of action is gradually given more importance, with it being considered a fundamental piece in some introductions to Quantum Field Theory (Zee, 2003) through the use of the path integral approach. The basic idea behind using the action is to assign a number to each possible state of a system. The function that does so is named the Lagrangian function, and it encodes the physics of the system (i.e. how do different parts of the system affect each other). Then, to a trajectory of a system we associate the integral of this number over all the states in the trajectory. This contrasts with the classical Newtonian approach, where we study a system by specifying all the possible ways in which parts of the system exercise forces on each other (i.e. affect each other’s acceleration). Using the action usually results in nicer mathematics, while I’d argue that the Newtonian approach requires less training to feel intuitive.
In many of the expositions of the use of action in physics (see e.g. this one), I perceive an attempt at transmitting wonder about the world being such that it minimizes a function on its trajectory. This has indeed been the case historically, with Maupertuis supposed to have considered action minimization (and the corresponding unification of minimization principles between optics and mechanics) as the most definite proof available to him of the existence of God. However, along the spirit of this stack exchange question, I never really understood why such a wonder should be felt, even setting aside the fact that it assumes that our equations “are” the world, a perspective that Artem has criticized at length before.
The three main reasons that led me to question the wonder expressed over the least action principle are the following:
- As its Wikipedia article points out, the mathematical result behind all of this (The principle of least action) is inaccurately named. A more accurate name would be instead “The principle of stationary action”. I have actually been amazed for a while by the wide acceptance of this inaccurate use. My impression is that this comes from considering stationary but non-minimizing solutions “unphysical”, but I’m not sure what is the data to back that. In any case, this is admittedly my weakest reason, since one could just replace “minimizing” by “stabilizing” in the original arguments about the wonder of a world-minimizing function.
- Given an arbitrary state (or history of states) for a system, it’s trivial to come up with a function that is minimized by that state – the one equal to zero for that state, and one for all other ones. One might answer that this is a contrived function, unlike the action function, which leads to my third argument.
- Modern physics is expressed in terms of relatively simple local properties, through the use differential equations. However, we know because of the work in the calculus of variations originated by the study of the brachistochrone problem, that there is indeed an equivalence relation between local properties of a system and global properties of its trajectory. This equivalence is established via the use of the Euler-Lagrange equations.
A nice derivation of those equations is given here. Basically, one considers the equivalence between a trajectory being stationary with respect to the action, and the first order of the change is zero with respect to any parametrized continuous deformation of the trajectory. Assuming all functions are nice enough, to compute this first order change one can put the derivative for the the first order change inside the integral for the action, finally obtaining an integral of the deformation times a function of the derivatives of the Lagrangian. Since the deformation is arbitrary, for the integral to be always zero the function of the derivatives of the Lagrangian must be zero at all points of the trajectory*. The stationary action condition is then the same as asking for a nice relation between derivatives of the Lagrangian function. In other words, the least action principle is equivalent to a nice differential relation about the physics of our system.
Once this equivalence is considered, it is not surprising that if physics has traditionally focused on describing systems in terms of elegant local properties, we can come up with relatively simple functions in the space of trajectories that are left stationary by the actual trajectory of our systems. This seems to characterize the least action principle not so much as a wondrous property of nature, but, as Artem might say, as an artifact of how we chose to describe nature
Notes and References
*: A fact related to what is going on — that I credit Derbes (1996) for raising attention to — is that if you have a stationary trajectory of a system, every segment of this trajectory is stationary as well.
Derbes, D. (1996). Feynman’s derivation of the Schrödinger equation. American Journal of Physics, 64(7), 881-884.
Zee, A. (2003). Quantum field theory in a nutshell. Princeton University Press.