Efficient calculation of actions

Dullin, Holger R.; Wittek, Andreas

doi:10.1088/0305-4470/27/22/019

Cited by 6 publications

(10 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The second problem was to nd a complete set of independent closed paths around each torus. This could be solved by means of an algorithm that was recently devised by Dullin and Wittek 1993 ] . It is based on the constructive part of Arnold's proof for Liouville's theorem on the existence of action-angle variables 1978 ] .…”

Section: Introductionmentioning

confidence: 99%

Action Integrals and Energy Surfaces of the Kovalevskaya Top

Dullin

Juhnke

Richter

1994

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

The different types of energy surfaces are identified for the Kovalevskaya problem of rigid body dynamics, on the basis of a bifurcation analysis of Poincaré surfaces of section. The organization of their foliation by invariant tori is qualitatively described in terms of Poincaré-Fomenko stacks. The individual tori are then analyzed for sets of independent closed paths, using a new algorithm based on Arnold’s proof of the Liouville theorem. Once these paths are found, the action integrals can be calculated. Energy surfaces are constructed in the space of action variables, for six characteristic values of energy. The data are presented in terms of color graphs that give an intuitive access to this highly complex integrable system.

show abstract

Section: Introductionmentioning

confidence: 99%

Action Integrals and Energy Surfaces of the Kovalevskaya Top

Dullin

Juhnke

Richter

1994

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

show abstract

“…Namely, a topological continuation scheme may be implemented , and it will be interesting to relate it to [23] where a continuation scheme for finding the action angle coordinates is found.…”

Section: Discussionmentioning

confidence: 99%

Foliations of isonergy surfaces and singularities of curves

Radnović

Rom‐Kedar

2008

Regul. Chaot. Dyn.

View full text Add to dashboard Cite

Abstract. It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level. We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.

show abstract

“…At an unstable orbit which is always accompanied by a separatrix the energy surface has a singular curvature at the critical point. It is natural that one action is zero at a stable orbit [3]. This can be achieved by a proper choice of fundamental paths on the invariant tori.…”

Section: Example: Coupled Rotorsmentioning

confidence: 99%

“…Hamiltonians of this type have been frequently used as physical models, e.g., for energy transfer in triatomic molecules; see [17] and references therein. However, energy surfaces have only been presented for a special case with two degrees of freedom [3]. We here show the energy surfaces for a broader class of isolated-island systems with N ≥ 2 degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

“…"Simple systems" with trivial foliation like uncoupled harmonic oscillators have globally continuous and smooth energy surfaces. For many of the non-simple systems considered so far [1,2,3,4,5,6,7,8,9], the energy surfaces are nonsmooth at separatrices, but, nevertheless, continuous or can be made continuous by a reduction of the system's discrete symmetries. In [10,11,9] such systems are referred to as "one-component systems".…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Resonance Zones in Action Space

Wiersig

2001

Zeitschrift Für Naturforschung A

View full text Add to dashboard Cite

The classical and quantum mechanics of isolated, nonlinear resonances in integrable systems with N ≥ 2 degrees of freedom is discussed in terms of geometry in the space of action variables. Energy surfaces and frequencies are calculated and graphically presented for invariant tori inside and outside the resonance zone. The quantum mechanical eigenvalues, computed in the semiclassical WKB approximation, show a regular pattern when transformed into the action space of the associated symmetry reduced system: eigenvalues inside the resonance zone are arranged on N -dimensional cubic lattices, whereas those outside are, in general, non-periodically distributed. However, N -dimensional triclinic (skewed) lattices exist locally. Both kinds of lattices are joined smoothly across the classical separatrix surface. The statements are illustrated with the help of two and three coupled rotors.

show abstract

Efficient calculation of actions

Cited by 6 publications

References 3 publications

Action Integrals and Energy Surfaces of the Kovalevskaya Top

Action Integrals and Energy Surfaces of the Kovalevskaya Top

Foliations of isonergy surfaces and singularities of curves

Resonance Zones in Action Space

Contact Info

Product

Resources

About