scholarly journals Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory

2005 ◽  
Vol 46 (10) ◽  
pp. 103507 ◽  
Author(s):  
E. G. Kalnins ◽  
J. M. Kress ◽  
W. Miller
Keyword(s):  
Three Dimensional ◽  
Second Order ◽  
Structure Theory ◽  
Conformally Flat ◽  
Classical Structure ◽  
Superintegrable Systems

scholarly journals Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators

2020 ◽  
Author(s):  
Bjorn K. Berntson ◽  
◽  
Ernest G. Kalnins ◽  
Willard Miller ◽  
◽  
...  
Keyword(s):  
Euclidean Space ◽  
Flat Space ◽  
Structure Theory ◽  
Conformally Flat ◽  
3 Dimensional ◽  
Superintegrable Systems ◽  
Complex Euclidean Space ◽  
Spatial Coordinates ◽  
Symmetry Generators

We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and 2-parameter rational potential is the best known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.

scholarly journals Adding potentials to superintegrable systems with symmetry

2021 ◽  
Vol 477 (2248) ◽  
Author(s):  
Allan P. Fordy ◽  
Qing Huang
Keyword(s):  
Kepler Problem ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Potential Functions ◽  
Conformally Flat ◽  
Coordinate Systems ◽  
Separable Potentials ◽  
Superintegrable Systems ◽  
Three Dimensional Coordinate ◽  
Three Dimensional Space

In previous work, we have considered Hamiltonians associated with three-dimensional conformally flat spaces, possessing two-, three- and four-dimensional isometry algebras. Previously, our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the three-dimensional space reduce to 3 or 4 parameter potentials for Darboux–Koenigs Hamiltonians. Other three-dimensional coordinate systems reveal connections between Darboux–Koenigs and other well-known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

THREE-DIMENSIONAL STAGNATION-POINT FLOW OVER A PERMEABLE STRETCHING/SHRINKING SHEET WITH A SECOND ORDER SLIP FLOW

2021 ◽  
Vol 10 (9) ◽  
pp. 3273-3282
Author(s):  
M.E.H. Hafidzuddin ◽  
R. Nazar ◽  
N.M. Arifin ◽  
I. Pop
Keyword(s):  
Differential Equations ◽  
Stagnation Point ◽  
Flow Model ◽  
Nonlinear Partial Differential Equations ◽  
Slip Flow ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Second Order ◽  
Stagnation Point Flow ◽  
Shrinking Sheet

The problem of steady laminar three-dimensional stagnation-point flow on a permeable stretching/shrinking sheet with second order slip flow model is studied numerically. Similarity transformation has been used to reduce the governing system of nonlinear partial differential equations into the system of ordinary (similarity) differential equations. The transformed equations are then solved numerically using the \texttt{bvp4c} function in MATLAB. Multiple solutions are found for a certain range of the governing parameters. The effects of the governing parameters on the skin friction coefficients and the velocity profiles are presented and discussed. It is found that the second order slip flow model is necessary to predict the flow characteristics accurately.

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