American Institute of Mathematical Sciences

Advanced
  2014, 21: 80-88. doi: 10.3934/era.2014.21.80

Compactly supported Hamiltonian loops with a non-zero Calabi invariant

Asaf Kislev 1,

1. 

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel

Received  November 2013 Revised  February 2014 Published  May 2014

We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds.
Keywords: Symplectic geometry..
Mathematics Subject Classification: 53D0.
Citation: Asaf Kislev. Compactly supported Hamiltonian loops with a non-zero Calabi invariant. Electronic Research Announcements, 2014, 21: 80-88. doi: 10.3934/era.2014.21.80
References:
[1]

A. Cannas da Silva, Symplectic Toric Manifolds,, 2001. Available from: \url{http://www.math.ist.utl.pt/~acannas/Books/toric.pdf}., ().   Google Scholar

[2]

Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Memoirs Amer. Math. Soc., 141 (1999). doi: 10.1090/memo/0672.  Google Scholar

[3]

D. McDuff, Loops in the Hamiltonian group: A survey, in Symplectic Topology and Measure Preserving Dynamical Systems, Contemp. Math., 512, Amer. Math. Soc., Providence, RI, 2010, 127-148. doi: 10.1090/conm/512/10061.  Google Scholar

[4]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[5]

L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8299-6.  Google Scholar

[6]

L. Polterovich, Hamiltonian loops and Arnold's principle, in Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser 2, 180, Amer. Math. Soc., Providence, RI, 1997, 181-187.  Google Scholar

[7]

S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Top. Anal., 4 (2012), 481-498. doi: 10.1142/S1793525312500215.  Google Scholar

show all references

References:
[1]

A. Cannas da Silva, Symplectic Toric Manifolds,, 2001. Available from: \url{http://www.math.ist.utl.pt/~acannas/Books/toric.pdf}., ().   Google Scholar

[2]

Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Memoirs Amer. Math. Soc., 141 (1999). doi: 10.1090/memo/0672.  Google Scholar

[3]

D. McDuff, Loops in the Hamiltonian group: A survey, in Symplectic Topology and Measure Preserving Dynamical Systems, Contemp. Math., 512, Amer. Math. Soc., Providence, RI, 2010, 127-148. doi: 10.1090/conm/512/10061.  Google Scholar

[4]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[5]

L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8299-6.  Google Scholar

[6]

L. Polterovich, Hamiltonian loops and Arnold's principle, in Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser 2, 180, Amer. Math. Soc., Providence, RI, 1997, 181-187.  Google Scholar

[7]

S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Top. Anal., 4 (2012), 481-498. doi: 10.1142/S1793525312500215.  Google Scholar

[1]

Fiammetta Battaglia and Elisa Prato. Nonrational, nonsimple convex polytopes in symplectic geometry. Electronic Research Announcements, 2002, 8: 29-34.

[2]

Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009
[3]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
[4]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[5]

Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253
[6]

Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239
[7]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[8]

Björn Gebhard. A note concerning a property of symplectic matrices. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2135-2137. doi: 10.3934/cpaa.2018101
[9]

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
[10]

Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211
[11]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407
[12]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[13]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291
[14]

Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002
[15]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[16]

Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
[17]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291
[18]

Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455
[19]

George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215
[20]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105

2020 Impact Factor: 0.929

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences