-
Previous Article
Geometry of Routh reduction
- JGM Home
- This Issue
-
Next Article
A geometric perspective on the Piola identity in Riemannian settings
Linear phase space deformations with angular momentum symmetry
Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn-Str. 4, 24118 Kiel, Germany |
Motivated by the work of Leznov-Mostovoy [
References:
[1] |
A. Borel,
Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. U. S. A., 40 (1954), 1147-1151.
doi: 10.1073/pnas.40.12.1147. |
[2] |
O. M. Boyarskyi and T. V. Skrypnik,
Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds, Ukrainian Math. J., 49 (1997), 1003-1015.
doi: 10.1007/BF02528745. |
[3] |
C. Chevalley and S. Eilenberg,
Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124.
doi: 10.2307/1990637. |
[4] |
A. Fialowski, Deformations of Lie algebras, Mat. Sb. (N. S.), 127 (1985), 476-482. |
[5] |
D. M. Fradkin,
Three-dimensional isotropic harmonic oscillator and $SU_3$, Am. J. Phys., 33 (1965), 207-211.
doi: 10.1119/1.1971373. |
[6] |
M. Gerstenhaber,
On the deformation of rings and algebras, Ann. Math., 79 (1964), 59-103.
doi: 10.2307/1970484. |
[7] |
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. |
[8] |
_____, Variations on a Theme by Kepler, Colloquium Publications, Vol. 42, American Mathematical Soc., 2006. |
[9] |
P. W. Higgs,
Dynamical symmetries in a spherical geometry I, J. Phys. A, 12 (1979), 309-323.
|
[10] |
G. Hochschild and J.-P. Serre,
Cohomology of Lie algebras, Ann. Math., 57 (1953), 591-603.
doi: 10.2307/1969740. |
[11] |
R. Howe,
Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570.
doi: 10.2307/2001418. |
[12] |
E. Inonu and E. P. Wigner,
On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 510-524.
doi: 10.1073/pnas.39.6.510. |
[13] |
D. Kazhdan, B. Kostant and S. Sternberg,
Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31 (1978), 481-507.
doi: 10.1002/cpa.3160310405. |
[14] |
A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Soc., 2004.
doi: 10.1090/gsm/064. |
[15] |
W. Lenz, Über den Bewegungsverlauf und die Quantenzustände der gestörten Keplerbewegung, Z. Phys., 24 (1924), 197-207. |
[16] |
M. Levy-Nahas,
Deformation and contraction of Lie algebras, J. Math. Phys., 8 (1967), 1211-1222.
doi: 10.1063/1.1705338. |
[17] |
A. Leznov and J. Mostovoy,
Classical dynamics in deformed spaces, J. Phys. A, 36 (2003), 1439-1449.
doi: 10.1088/0305-4470/36/5/317. |
[18] |
S. P. Novikov,
The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49.
|
[19] |
A. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhäuser, 1990.
doi: 10.1007/978-3-0348-9257-5. |
[20] |
A. Reyman and M. A. Semenov-Tian-Shansky,
Group-theoretical methods in the theory of finite-dimensional integrable systems, Dynamical Systems VII, Springer Berlin Heidelberg, 16 (1994), 116-225.
doi: 10.1007/978-3-662-06796-3_7. |
[21] |
C. A. Weibel, An Introduction to Homological Algebra, Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
doi: 10.1017/CBO9781139644136. |
[22] |
J. Wolf, Representations associated to minimal co-adjoint orbits, Differential Geometrical Methods in Mathematical Physics II., Springer Berlin Heidelberg, (1978), 329-349. |
show all references
References:
[1] |
A. Borel,
Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. U. S. A., 40 (1954), 1147-1151.
doi: 10.1073/pnas.40.12.1147. |
[2] |
O. M. Boyarskyi and T. V. Skrypnik,
Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds, Ukrainian Math. J., 49 (1997), 1003-1015.
doi: 10.1007/BF02528745. |
[3] |
C. Chevalley and S. Eilenberg,
Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124.
doi: 10.2307/1990637. |
[4] |
A. Fialowski, Deformations of Lie algebras, Mat. Sb. (N. S.), 127 (1985), 476-482. |
[5] |
D. M. Fradkin,
Three-dimensional isotropic harmonic oscillator and $SU_3$, Am. J. Phys., 33 (1965), 207-211.
doi: 10.1119/1.1971373. |
[6] |
M. Gerstenhaber,
On the deformation of rings and algebras, Ann. Math., 79 (1964), 59-103.
doi: 10.2307/1970484. |
[7] |
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. |
[8] |
_____, Variations on a Theme by Kepler, Colloquium Publications, Vol. 42, American Mathematical Soc., 2006. |
[9] |
P. W. Higgs,
Dynamical symmetries in a spherical geometry I, J. Phys. A, 12 (1979), 309-323.
|
[10] |
G. Hochschild and J.-P. Serre,
Cohomology of Lie algebras, Ann. Math., 57 (1953), 591-603.
doi: 10.2307/1969740. |
[11] |
R. Howe,
Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570.
doi: 10.2307/2001418. |
[12] |
E. Inonu and E. P. Wigner,
On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 510-524.
doi: 10.1073/pnas.39.6.510. |
[13] |
D. Kazhdan, B. Kostant and S. Sternberg,
Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31 (1978), 481-507.
doi: 10.1002/cpa.3160310405. |
[14] |
A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Soc., 2004.
doi: 10.1090/gsm/064. |
[15] |
W. Lenz, Über den Bewegungsverlauf und die Quantenzustände der gestörten Keplerbewegung, Z. Phys., 24 (1924), 197-207. |
[16] |
M. Levy-Nahas,
Deformation and contraction of Lie algebras, J. Math. Phys., 8 (1967), 1211-1222.
doi: 10.1063/1.1705338. |
[17] |
A. Leznov and J. Mostovoy,
Classical dynamics in deformed spaces, J. Phys. A, 36 (2003), 1439-1449.
doi: 10.1088/0305-4470/36/5/317. |
[18] |
S. P. Novikov,
The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49.
|
[19] |
A. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhäuser, 1990.
doi: 10.1007/978-3-0348-9257-5. |
[20] |
A. Reyman and M. A. Semenov-Tian-Shansky,
Group-theoretical methods in the theory of finite-dimensional integrable systems, Dynamical Systems VII, Springer Berlin Heidelberg, 16 (1994), 116-225.
doi: 10.1007/978-3-662-06796-3_7. |
[21] |
C. A. Weibel, An Introduction to Homological Algebra, Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
doi: 10.1017/CBO9781139644136. |
[22] |
J. Wolf, Representations associated to minimal co-adjoint orbits, Differential Geometrical Methods in Mathematical Physics II., Springer Berlin Heidelberg, (1978), 329-349. |
[1] |
Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399 |
[2] |
Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10. |
[3] |
Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685 |
[4] |
Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239 |
[5] |
Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453 |
[6] |
Miguel Mendes. A note on the coding of orbits in certain discontinuous maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 369-382. doi: 10.3934/dcds.2010.27.369 |
[7] |
Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343 |
[8] |
Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69 |
[9] |
Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629 |
[10] |
Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289 |
[11] |
Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 |
[12] |
Alain Chenciner. The angular momentum of a relative equilibrium. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1033-1047. doi: 10.3934/dcds.2013.33.1033 |
[13] |
Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13 |
[14] |
Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 589-597. doi: 10.3934/dcds.2014.34.589 |
[15] |
Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523 |
[16] |
K. A. Ariyawansa, Leonid Berlyand, Alexander Panchenko. A network model of geometrically constrained deformations of granular materials. Networks & Heterogeneous Media, 2008, 3 (1) : 125-148. doi: 10.3934/nhm.2008.3.125 |
[17] |
Gabriel P. Paternain. On two noteworthy deformations of negatively curved Riemannian metrics. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 639-650. doi: 10.3934/dcds.1999.5.639 |
[18] |
Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420 |
[19] |
Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023 |
[20] |
James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237 |
2017 Impact Factor: 0.561
Tools
Metrics
Other articles
by authors
[Back to Top]