For one reason or another, I often find myself driving the Taconic State Parkway at night (which, if you've ever done that, you know what I'm saying). On one of these trips I was heading back up to Vermont after a dinner meeting in New York. It was already late when I left the city, and out on the dark road I was keeping myself awake by thinking of some fiendish physics problems to give to my students. (Such are the revenge fantasies of physics professors.) I thought of this one:
A particle moving in one dimension has the trajectory x(t) = t^4.
(a) What is the velocity of the particle at time t=0?
(b) What is the net force acting on the particle at time t=0?
(c) In view of (a) and (b), why does the particle move at all?
I ran through the answers in my mind: (a): The particle is at rest at time t=0. (b): There is no net force on the particle at time t=0. (c): (c): (c):
My grip on the wheel loosened, and my eyes focused on the far distance, as I realized that I didn't really know the answer to part (c) myself!
The Taconic is basically a giant deer park, especially in the middle of the night, so for safety's sake I had to drop the problem and get back to scanning the verges. But over the next year I thought about the problem from time to time, struggling to clarify my thinking on some subtle issues. Finally I wrote up some of the results, and the resulting article, entitled "Inertia and Determinism," has been accepted for publication in
The British Journal for the Philosophy of Science.
I continue to reflect on the curious trajectory of this project - from its whimsical origins in my teaching practice, to its fruition as a published research paper. And meanwhile, my Year of Isaac Newton continues: this summer I'm writing the solutions for the end-of-chapter problems in my manuscript on Newton's Laws. Hopefully I'll know how to answer my own questions this time....
Just for the sake of interest, I'll end by excerpting some of the less mathematical material from the paper; for references, see the PDF:
From J. Zimba, "Inertia and Determinism," to appear, Brit. J. Phil. Sci.:
"Beginning in the 19th Century with Ernst Mach and Rouse Ball, and continuing on to more recent times, commentators on Newtonian mechanics have universally asserted that the Law of Inertia follows immediately from the Second Law. Does it? If we are willing to insist, as an axiom on a par with the Laws of Motion themselves, that non-Lipschitz forces do not belong to the theory, then the answer is yes. But if we do not wish to make this a priori restriction on the kinds of forces that may appear in the theory, then we can say two things. First, the Law of Inertia itself stands incomplete, or at least ambiguous, until a specific approach to [[problems like the x(t) = t^4 problem]] is selected. And, second, whatever approach is selected, the completed Law of Inertia will no longer follow mathematically from the Second Law. The Law of Inertia would instead act as a boundary condition, selecting the physical trajectory from among the many mathematical solutions to the Second Law differential equation."
"Within the community of mathematicians and physicists, it is often taken for granted that Newtonian mechanics has a deterministic structure. At times, this is even made a matter of definition. Arnold defines classical mechanics as the study of 'the motion of systems whose past and future are uniquely determined by the initial positions and initial velocities of all particles of the system.' Landau and Lifshitz go beyond mere definition, claiming that determinism is in fact an observed feature of classical systems. But such observations could only apply to the small class of non-chaotic systems. And for that matter, to the best of our knowledge, our world is not, in fact, deterministic; so the claim that determinism has ever been observed is open to dispute.
"Mathematicians like Arnold probably want to impose smoothness conditions because doing so makes theorems easier to prove. Then, having imposed the smoothness condition, they don't want to feel that they are leaving out any interesting behaviors; so they define classical mechanics to be the very mathematical object they are studying. Maybe the mathematicians are correct that their theorems are not leaving out any interesting behaviors. But I don't think they can be correct in saying that the smoothness conditions of their treatises are mandated by observed facts about determinism."
"One response to this example might be to attempt to repair [the Law of Inertia], or to strengthen it, leading to a conception of the Law of Inertia so strong that it ensures determinism in all possible situations. But it is unclear whether such a program is mathematically possible. Another approach would be to give up the attempt to complete the Laws of Motion, and simply conclude - despite the prejudices of history - that Newtonian mechanics is, and always has been, an indeterministic picture of the universe."