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.

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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).

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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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