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Semi-global symplectic invariants of the spherical pendulum
- Source :
- Journal of Differential Equations. 254:2942-2963
- Publication Year :
- 2013
- Publisher :
- Elsevier BV, 2013.
-
Abstract
- We explicitly compute the semi-global symplectic invariants near the focus-focus point of the spherical pendulum. A modified Birkhoff normal form procedure is presented to compute the expansion of the Hamiltonian near the unstable equilibrium point in Eliasson-variables. Combining this with explicit formulas for the action we find the semi-global symplectic invariants near the focus-focus point introduced by Vu Ngoc 2003. We also show that the Birkhoff normal form is the inverse of a complete elliptic integral over a vanishing cycle. To our knowledge this is the first time that semi-global symplectic invariants near a focus-focus point have been computed explicitly. We close with some remarks about the pendulum, for which the invariants can be related to theta functions in a beautiful way.<br />Comment: 27 pages, 2 figures
- Subjects :
- Pure mathematics
Nonlinear Sciences - Exactly Solvable and Integrable Systems
Applied Mathematics
Spherical pendulum
Mathematical analysis
Classical Physics (physics.class-ph)
FOS: Physical sciences
Theta function
Physics - Classical Physics
Dynamical Systems (math.DS)
37J35, 37J15, 37J40, 70H06, 70H08, 37G20
Action (physics)
Mathematics - Symplectic Geometry
FOS: Mathematics
Vanishing cycle
Symplectic Geometry (math.SG)
Elliptic integral
Point (geometry)
Mathematics - Dynamical Systems
Exactly Solvable and Integrable Systems (nlin.SI)
Analysis
Hamiltonian (control theory)
Mathematics
Symplectic geometry
Subjects
Details
- ISSN :
- 00220396
- Volume :
- 254
- Database :
- OpenAIRE
- Journal :
- Journal of Differential Equations
- Accession number :
- edsair.doi.dedup.....f43ec099e7f6014020260262a687f8fb