Abstract
This chapter is devoted to the discussion of the theory of integral invariants, which reveals an interesting structure of the general solution to Hamiltonian equations. We discuss the basic Ponicaré-Cartan and Ponicaré integral invariants that represent line integrals of a special vector field defined on extended phase space. The integrals retain the same value for any closed contour taken on a given two-dimensional surface formed by solutions to the Hamiltonian equations. As will be discussed in Sect. 5.1.3, this property could be taken as a basic principle of mechanics, instead of the principle of least action. Besides their applications in mechanics, integral invariants are widely used in the theory of differential equations, see [1, 4].
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Notes
- 1.
All the results of this section remain true for the functions Q, P with the manifest dependence on time.
References
F.R. Gantmacher, Lectures on Analytical Mechanics (MIR, Moscow, 1970)
E. Cartan, Leçons sur les Invariants Intégraux (Hermann, Paris, 1922)
V.P. Maslov, M.V. Fedoruk, Semiclassical Approximation in Quantum Mechanics (D. Reidel Publishing Company, Dordrecht, 1981)
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© 2010 Springer-Verlag Berlin Heidelberg
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Deriglazov, A. (2010). Integral Invariants. In: Classical Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14037-2_5
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DOI: https://doi.org/10.1007/978-3-642-14037-2_5
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