Investigation of the equations of motion

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


In most cases (for example, in the three-body problem) we can neither solve the system of differential equations nor completely describe the behavior of the solutions. In this chapter we consider a few simple but important problems for which Newton’s equations can be solved.


Potential Energy Angular Momentum Phase Plane Radius Vector Central Field 
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  1. 11.
    Here we assume for simplicity that the solution φ is defined on the whole time axis .Google Scholar
  2. 12.
    For a definition, see, e.g., p. 155 of Ordinary Differential Equations by V. I. Arnold, MIT Press, 1973.Google Scholar
  3. 13.
    The only exception is the case when the period does not depend on the energy.Google Scholar
  4. 15.
    With the usual limitations.Google Scholar
  5. 16.
    Including reflections.Google Scholar
  6. 17.
    Let a drop of tea fall into a glass of tea close to the center. The waves collect at the symmetric point. The reason is that, by the focal definition of an ellipse, waves radiating from one focus of the ellipse collect at the other.Google Scholar
  7. 18.
    By planets we mean here points in a central field.Google Scholar
  8. 19.
    This problem is taken from V. V. Beletskii’s delightful book, “Notes on the Motion of Celestial Bodies,” “Nauka,” 1972.Google Scholar
  9. 20.
    The case M = 0 is left to the reader.Google Scholar
  10. 21.
    The moment of force is also called the torque [Trans. note].Google Scholar
  11. 22.
    Here we are assuming that U does not depend on m. In the field of gravity, the potential energy U is proportional to m, and therefore the acceleration does not depend on the mass m of the moving point.Google Scholar
  12. 23.
    J. M. Smith, Mathematical Ideas in Biology, Cambridge University Press, 1968.CrossRefGoogle Scholar
  13. 24.

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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