Sniffing the Path (On the Fascination of Classical Mechanics)

Newton’s law has been superseded by relativity and quantum mechanics, and our universe is strange and compelling from a philosophical perspective. Classical Mechanics is dull.

I do not believe that.

The fundamentals of Newtonian Mechanics can be represented in a way that is different from well-known Force = Mass Times Acceleration – being mathematically equivalent, but providing a different philosophical twist. I consider this as fascinating as the so-called spooky action-at-a-distance of quantum mechanics.

The standard explanation is this:

  • There are forces described by respective laws (e.g.: The gravitational force)
  • Forces act on matter and result in the acceleration of particles.
  • In every point of time, the path of the particle can  be calculated based on its acceleration if you know its location and its velocity in the point of time before.
  • Thus step by step, the particle explores its path and the final trajectory is composed of all these tiny steps.

This is why the classical world seems to be deterministic. (Yes, this explanation lacks the interdependence of space(time) and masses and the limitations imposed by quantum mechanics.)

The deterministic laws can be stated in terms of The Principle of Least Action

  • Consider the point in space where the particle starts off and the end point of the journey. Thus we look at the path in hindsight: We demand that the particle needs to travel from A to B, and we also fix the points of time.
  • Now we evaluate all possible trajectories the particle might travel from A to B.
  • For every path we calculate a number: This is called the “Action” (In simple particle mechanics this is equivalent to integration over the difference of kinetic and potential energy – the “action” isn’t something you can easily “feel” like the force or the momentum). Note that the total energy needs to be conserved, thus e.g. there should be no friction. But at a microscopic level, all forces are conservative anyway.
  • Above all: Note that a single number is assigned to a full path which consists of all the points in space the particle traverses.
  • The path that is actually traversed / realized is the path that us assigned the least action.

Thus it seems that the particle sniffs all the paths ((c) Richard Feynman) and selects a path distinguished by a particular property. In addition, we have replaced the necessity to know the initial location and velocity by the knowledge of the location at the start point and the end point.

It seems we are (nature is) working backwards.

Actually, the particle is really sort of sniffing the path: This is a minimum, exactly: an extremum. Near an extremum the slope of a function is nearly zero. Thus the particle sniffs the neighboring paths and checks for changes in the action. The apparent contradiction between working forward and backwards is resolved if The Principle of Least Action is applied to smaller and smaller pieces of the trajectory. Since the principle holds for any path, it also needs to true for infinitesimal parts of a path. For these infinitesimal paths, the principle boils down (mathematically) to Newton’s law.

A mathematical derivation might not be satisfactory from a philosophical point of view. Probably the following may serve as an explanation: Working with the Principle of Least Action we do not know or do not need to know the velocity at the start time. Thus we need some other information instead. By the way, also data as the total energy, momentum or angular momentum may be used as a substitute of the total information about the initial conditions in terms of position and velocity.

Using the Principle, we know only where we are heading for. Since we do not know the initial velocity – the tangent to our path at start time – we need more guidance. The Principle provides such guidance and allows the particle for sniffing for other paths in order to determine that tangent at every point of time.

Former mill on the island of La Palma.

Leave a Comment

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.