Newton’s Principle of Determinacy

All motions of a system are uniquely determined by their initial positions and initial velocities.

One textbook provides the formula below:

r(t) = R(t, t_0, r(t_0), v(t_0))

When you give a function a constant, the result is a constant, so r(t_0) and v(t_0) are numbers.

Assume we are a point moving along the x-axis. If our journey is a straight line, it makes sense to put us on an axis to simplify the calculation.

  • We want to know what the time was when we started our journey
  • We want to know what our location was when we started our journey
  • We want to know what our velocity was when we started our journey

Let time be seconds and position be meters, then velocity will be meters per second. If I feed you the three values needed to answer the above three questions, it then only remains for someone to specify t and you can calculate the position.

What is puzzling is, the author of the textbook we used to get r(t) = R(t, t_0, r(t_0), v(t_0)) goes on to do math that mentions acceleration. If r(t) depends on acceleration, shouldn’t it be listed in R?

After reading several different authors, it appears, what the idea is trying to say, is that it can’t tell you anything until you tell it x_0 and v_0. We must assume that acceleration is a constant and that this constant is known. So far, the math we see when reading about Newtonian Determinacy is consistent with this. Well…

We find an author saying this:

Newton’s principle of determinacy: The initial state of a mechanical system (the totality of the positions and velocities of its points at some moment of time) uniquely determines all of its motion.

Every point in the system is going to be accelerated by every other point in the system when the clock starts running. It can be calculated, in principle. Damn, those are magic words.

Appendix A

Perhaps the best thing to say in Newton’s defense is that he probably never intended a single sentence to carry the weight of the entire idea. You may come across textbooks that talk about position and momentum rather than position and velocity. Keep in mind, momentum is velocity with mass attached (p=mv). Also, if we ever find the original article, if he said that all particles have the same mass, then there would be no need to declare each individual mass.

Appendix B

We should try to find a verbatim quote where Newton says “My principle of determinacy is that …”

We keep seeing mentions of the name Arnold, and it appears a journal article is quoting Arnold Arnold when it says

The initial positions and velocities of all the particles of a mechanical
system uniquely determine all of its motion.

Appendix C

A student suggested the following:

We can make this real. We could ask, “For the July, 1969” mission that sent Neil Armstrong to the moon, then brought him back, Was it a three point calculation (spaceship is a point mass, Earth is a point mass, moon is a point mass)? Did they need to account for the sun, making it a four point problem?

The prior paragraph discusses something that is real, practical. Newton’s points moving on paths that depend only on their initial locations and velocities is a cartoon. It might be fun to look for a progression that starts with Newton’s Principle of Determinacy and ends with Lunar Missions.