The Geometrical Description of the Nonlinear Dynamics of a Multiple Pendulum

A system of two coupled pendula is an example of a Hamiltonian system with two degrees of freedom. Rott considered such a system in the presence of 1:2 resonance and observed a phase-locking phenomenon. In this paper this phenomenon is explained by means of Duistermaat's method of bringing a Ha...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 1995-12, Vol.55 (6), p.1753-1763
Main Author: Zharnitsky, V.
Format: Article
Language:English
Subjects:
Center of gravity
Circles
Critical points
Degrees of freedom
Hypersurfaces
Mass
Mathematics
Orbits
Pendulums
Periodic orbits
Phase portrait
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Summary:A system of two coupled pendula is an example of a Hamiltonian system with two degrees of freedom. Rott considered such a system in the presence of 1:2 resonance and observed a phase-locking phenomenon. In this paper this phenomenon is explained by means of Duistermaat's method of bringing a Hamiltonian system with two degrees of freedom into standard form. His theory is based on Mather's theory of stability of differentiable mappings. A transparent geometrical picture is obtained in the phase space. In particular, a criterion of the phase-locking phenomenon is provided based on this picture.
ISSN:0036-1399
1095-712X
DOI:10.1137/S0036139993256606