Experimental facts

  • V. I. Arnold
Part of the Graduate Texts in Mathematics book series (GTM, volume 60)


In this chapter we write down the basic experimental facts which lie at the foundation of mechanics: Galileo’s principle of relativity and Newton’s differential equation. We examine constraints on the equation of motion imposed by the relativity principle, and we mention some simple examples.


Mechanical System Experimental Fact World Line Acceleration Vector Inertial Coordinate System 
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  1. 2.
    All these “experimental facts” are only approximately true and can be refuted by more exact experiments. In order to avoid cumbersome expressions, we will not specify this from now on and we will speak of our mathematical models as if they exactly described physical phenomena.Google Scholar
  2. 3.
    The reader who has no need for the mathematical formulation of the assertions of Section 1 can omit this paragraph.Google Scholar
  3. 4.
    Formerly, the universe was provided not with an affine, but with a linear structure (the geocentric system of the universe).Google Scholar
  4. 5.
    Recall that the direct product of two sets A and B is the set of ordered pairs (a, b), where aA and bB. The direct product of two spaces (vector, affine, euclidean) has the structure of a space of the same type.Google Scholar
  5. 6.
    That is, there is a one-to-one mapping of one to the other preserving the galilean structure.Google Scholar
  6. 7.
    The graph of a mapping f: AB is the subset of the direct product A × B consisting of all pairs (a, f(a)) with aA Google Scholar
  7. 8.
    Under certain smoothness conditions, which we assume to be fulfilled. In general, a motion is determined by Equation (1) only on some interval of the time axis. For simplicity we will assume that this interval is the whole time axis, as is the case in most problems in mechanics.Google Scholar
  8. 9.
    In formulating the principle of relativity we must keep in mind that it is relevant only to closed physical (in particular, mechanical) systems, i.e., that we must include in the system all bodies whose interactions play a role in the study of the given phenomena. Strictly speaking, we should include in the system all bodies in the universe. But we know from experience that one can disregard the effect of many of them: for example, in studying the motion of planets around the sun we can disregard the attractions among the stars, etc. On the other hand, in the study of a body in the vicinity of earth, the system is not closed if the earth is not included ; in the study of the motion of an airplane the system is not closed if it does not include the air surrounding the airplane, etc. In the future, the term “mechanical system” will mean a closed system in most cases, and when there is a question of non-closed systems this will be explicitly stated (cf., for example, Section 3).Google Scholar
  9. 10.
    This is the so-called second cosmic velocity υ2. Our equation does not take into account the attraction of the sun. The attraction of the sun will not let the stone escape from the solar system if the velocity of the stone with respect to the earth is less than 16.6 km/sec.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • V. I. Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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