Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Here we assume for simplicity that the solution φ is defined on the whole time axis ℝ.
For a definition, see, e.g., p. 155 of Ordinary Differential Equations by V. I. Arnold, MIT Press, 1973.
The only exception is the case when the period does not depend on the energy.
With the usual limitations.
Including reflections.
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.
By planets we mean here points in a central field.
This problem is taken from V. V. Beletskii’s delightful book, “Notes on the Motion of Celestial Bodies,” “Nauka,” 1972.
The case M = 0 is left to the reader.
The moment of force is also called the torque [Trans. note].
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.
J. M. Smith, Mathematical Ideas in Biology, Cambridge University Press, 1968.
Ibid.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1978 Springer Science+Business Media New York
About this chapter
Cite this chapter
Arnold, V.I. (1978). Investigation of the equations of motion. In: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1693-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-4757-1693-1_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-1695-5
Online ISBN: 978-1-4757-1693-1
eBook Packages: Springer Book Archive