7 Classical mechanics in a nutshell

An exhaustive description of a classical mechanical system at any one time consists of the values of a fixed number N of coordinates and an equal number of momenta. Such a description is usually referred to as the system’s state, and the system is said to have N degrees of freedom. This makes it possible to treat the state of a classical system with N degrees of freedom as a point P in a 2N-dimensional space S known as phase space.

Consider, for example, the classical harmonic oscillator (Fig. 2.6.1). It has one degree of freedom, a 2-dimensional phase space, and its behavior is governed by the equation ma = −kx, where m is the oscillator’s mass, a its acceleration, and k a positive constant.

classical harmonic oscillator
Figure 2.6.1 Phase space trajectory of a harmonic oscillator. Where the magnitude of x is at its maximum, p changes its sign (the oscillator reverses its motion), and where x=0, the magnitude of p is at its maximum (the oscillator moves fastest).

Now for a couple of truisms. For one thing, a physical theory is as good as its predictions, and what a physical theory predicts are measurement outcomes. For another, no real-world experiment ever has an exact outcome. (Otherwise one could experimentally establish whether a physical quantity with a continuous range of possible values has a rational rather than irrational value.) If measurement outcomes are digitally displayed, as may reasonably be assumed, then every measurement has a finite number of possible outcomes. A (successful) measurement is therefore equivalent to a finite number of simultaneously performed elementary tests.

An elementary test is a measurement that has exactly two possible outcomes. An elementary test associated with a continuous physical quantity Q typically answers the question: does the value of Q lie in a given interval I? If the possible outcomes of a measurement are the mutually disjoint (non-overlapping) intervals I1, I2, …, In, then the outcome of one elementary test will be positive, while those of the remaining n − 1 tests will be negative.

In our oscillator example, intervals of the x-axis and intervals of the p-axis are some of the possible outcomes of elementary tests. In the system’s phase space, the former intervals correspond to vertical strips, the latter to horizontal strips (see Fig. 2.6.1). The intersection of a horizontal and a vertical strip corresponds to the outcome of another elementary test, determining the simultaneous truth of the propositions “the value of x lies in the vertical strip” and “the value of p lies in the horizontal strip.” Generalizing for the sake of mathematical simplicity, we allow every subset of the system’s phase space to be the possible outcome of an elementary test. This leads to the following characterization of the probability calculus of classical mechanics.

An elementary test is (represented by) a subset U of a phase space S.

  • The algorithm that serves to assign probabilities to the outcomes of elementary tests is a point P in S.
  • The probability of obtaining a positive outcome for U is 1 if U contains P.
  • The probability of obtaining a positive outcome for U is 0 if U does not contain P.

This probability algorithm is trivial, in the sense that it only assigns trivial probabilities: 0 or 1. P can therefore be thought of as a state in the classical sense of the word: a collection of possessed properties. We are licensed to believe that U represents a physical property (rather than an elementary test or its positive outcome), and that if the probability of finding this property is 1, it is because the system possesses the property, regardless of whether the corresponding elementary test is made.

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