ELLIPSOIDAL BILLIARDS WITH ISOTROPIC HARMONIC POTENTIALS

2000 ◽  
Vol 10 (09) ◽  
pp. 2075-2098 ◽  
Author(s):  
JAN WIERSIG
Keyword(s):  
Harmonic Oscillator ◽  
Energy Dependence ◽  
Invariant Tori ◽  
Low Energy ◽  
Action Space ◽  
Harmonic Potential ◽  
Classical Dynamics ◽  
Isotropic Harmonic Oscillator ◽  
Energy Surfaces ◽  
Action Variables

The classical dynamics of the triaxial ellipsoidal billiard with isotropic harmonic potential attracting to the center of the ellipsoid is discussed. The integrability preserving potential introduces an energy dependence to the foliation of energy shells into invariant tori. This foliation and the character of the corresponding motion is described in terms of 13 qualitatively different energy surfaces in the space of the action variables. Frequencies and the location of resonances are calculated. The consequences of the superintegrability of the low-energy case, the isotropic harmonic oscillator, for the energy surfaces in action space are investigated.

scholarly journals Resonance Zones in Action Space

2001 ◽  
Vol 56 (8) ◽  
pp. 537-556
Author(s):  
Jan Wiersig
Keyword(s):  
Degrees Of Freedom ◽  
Lattice Structure ◽  
Invariant Tori ◽  
Action Space ◽  
Cubic Lattices ◽  
Resonance Zone ◽  
Level Statistics ◽  
Energy Surfaces ◽  
Good Agreement ◽  
03.65 Sq

Abstract 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 vari­ ables. 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 sem iclassical 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 iV-dimensional cubic lattices, whereas those outside are, in general, non-periodically distributed. However, TV-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. The energy-level statistics of this system are found numerically to be in very good agreement with the Poisson distribution, despite of the complex lattice structure. PACS: 03.65.Sq, 05.45.-a

Energy Surfaces of Ellipsoidal Billiards

1996 ◽  
Vol 51 (4) ◽  
pp. 219-241 ◽  
Author(s):  
Jan Wiersig ◽  
Peter H. Richter
Keyword(s):  
Perturbation Theory ◽  
Invariant Tori ◽  
Actual Computation ◽  
Different Types ◽  
Energy Surfaces ◽  
Action Variables

Abstract Energy surfaces in the space of action variables are calculated and graphically presented for general triaxial ellipsoidal billiards. As was demonstrated by Jacobi in 1838, the system may be integrated in terms of hyperelliptic functions. The actual computation, however, has never been done. It is found that generic energy surfaces consist of seven pieces, representing topologically different types of invariant tori. The character of the corresponding motion is discussed. Frequencies, winding numbers, and the location of resonances are also determined. The results may serve as a basis for perturbation theory of slightly modified systems, and for semi-classical quantization.

ACTION INTEGRALS AND ENERGY SURFACES OF THE KOVALEVSKAYA TOP

1994 ◽  
Vol 04 (06) ◽  
pp. 1535-1562 ◽  
Author(s):  
HOLGER R. DULLIN ◽  
MARCUS JUHNKE ◽  
PETER H. RICHTER
Keyword(s):  
Liouville Theorem ◽  
Invariant Tori ◽  
Rigid Body Dynamics ◽  
Body Dynamics ◽  
Different Types ◽  
Characteristic Values ◽  
The Individual ◽  
Energy Surfaces ◽  
Poincaré Surfaces Of Section ◽  
Action Variables

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.

Visualizing {110} Surface Structure of Equilibrium-form ZIF-8 Crystals by Low-dose Cs-corrected TEM

Nanoscale ◽  
2021 ◽  
Author(s):  
Xiaocang Han ◽  
Wenqian Chen ◽  
Rui Su ◽  
Yuan Tian ◽  
Pan Liu ◽  
...  
Keyword(s):  
Surface Structure ◽  
Low Dose ◽  
Low Energy ◽  
Zeolitic Imidazolate Frameworks ◽  
Equilibrium Form ◽  
Energy Surfaces

The properties of Zeolitic imidazolate frameworks (ZIFs) crystals highly depend on the structures of the low-energy surfaces, such as {110} of ZIF-8. However, the atomic/molecular configurations of the ZIF-8 {110}...

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