Classical Mechanics

1. Broad Overview

In physics mechanics is code for “how things move.” It allows us to describe the motions of particles, like electrons, and agglomerations of particles, like rocket ships, through spacetime. Without qualification by “classical” or “quantum” mechanics means how everything moves, period, full stop. Students typically learn classical before learning quantum.

Classical mechanics should trigger you to think to yourself “non-quantum + non-relativistic mechanics.” There are no probabilities here — everything is certain. There are no wavefunction collapses, because there are no wavefunctions. There is no role for an observer; classical mechanics describes an observerless world. There is no length contraction; everything stays the same length at every speed. There is no time dilation; everything takes the same amount of time no matter its speed. There is no mass enhancement; everything keeps the same mass no matter its speed. It is a world perfectly compatible with what the philosophy of mind calls zombies. And, as I discuss elsewhere, it is the only physics compatible with zombies; the whole philosophy of zombies makes no sense under relativity or quantum mechanics.

There are 3 equivalent descriptions of classical mechanics, all named after the person who first described them. They each do different things and are used for different purposes but they describe the same world. These are:

  1. Newtonian mechanics (named for Isaac Newton)
  2. Hamiltonian mechanics (named for Hamilton)
  3. Lagrangian mechanics (named for Lagrange)

It is crucially important to realize that classical mechanics is not simply Newtonian mechanics, and further to realize that Newtonian mechanics are not the “best” mechanics. They came first and are basically the worst. You should feel like you are “trying” to get to Hamiltonian and Lagrangian mechanics, and that Newtonian is just a stop along the way.

2. Newtonian Mechanics

2.1. Four Basic Quantities

Newtonian physics assumes four basic (starting, fundamental, elementary) quantities. It is conceptually important to realize that you only need these four to create a complete physics:

  • Space
  • Time
  • Mass
  • Force

2.1.1. Space

Newton took space to be absolute space and to be independent from absolute time. He therefore believed in absolute basic units of space although of course you can have basic units in relativistic spacetime.

Space is described with vectors. Vectors can be described either with arrows over their heads or in bold. For example a vector in the x-direction can be written:

\vec{x} or \mathbf{x}.

A unit vector is written: