American Institute of Mathematical Sciences

Advanced
June 2018, 10(2): 209-215. doi: 10.3934/jgm.2018008

A note on the normalization of generating functions

Simone Vazzoler

Previnet S.p.A., Via E. Forlanini, 24, Preganziol (TV), Italy

Received  December 2016 Revised  December 2017 Published  May 2018

In the present note, I will propose some insights on the normalization of generating functions for Lagrangian submanifolds. From the literature (see, for example [4], [6], [7], [3] and [1]), it is clear that a problem exists concerning the nonuniqueness of generating functions and, in particular, of the generating functions quadratic at infinity (GFQI). This problem can be avoided introducing a normalization on the whole set of generating functions that will allow us to

(ⅰ) choose an unique GFQI for Lagrangian submanifolds of the form $\varphi(L)$, where $L$ is a Lagrangian submanifold and $\varphi$ is an Hamiltonian isotopy;

(ⅱ) compare the critical values $c(α, S_1)$ and $c(α, S_2)$ of two GFQI generating the submanifolds, $\varphi_1(L)$ and $\varphi_2(L)$, where $\varphi_1$ and $\varphi_2$ are Hamiltonian isotopies relative to two Hamiltonians $H_1$ and $H_2$, respectively.

Keywords: Symplectic geometry, Lagrangian submanifolds, generating functions.
Mathematics Subject Classification: Primary: 53D05; Secondary: 53D12.
Citation: Simone Vazzoler. A note on the normalization of generating functions. Journal of Geometric Mechanics, 2018, 10 (2) : 209-215. doi: 10.3934/jgm.2018008
References:
[1]

F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Mathematical Journal, 144 (2008), 235-284. doi: 10.1215/00127094-2008-036.

[2]

F. Cardin, Elementary Symplectic Topology and Mechanics, Springer, 2015. doi: 10.1007/978-3-319-11026-4.

[3]

A. Monzner, N. Vichery and F. Zapolsky, Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization, J. Mod. Dyn., 6 (2012), 205-249, arXiv: 1111.0287. doi: 10.3934/jmd.2012.6.205.

[4]

D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology and its Applications, 96 (1999), 249-266. doi: 10.1016/S0166-8641(98)00049-2.

[5]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.

[6]

C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 439-459, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 217, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-4266-3_10.

[7]

C. Viterbo, Symplectic Homogenization, 2014, arXiv: 0801.0206v3.

show all references

References:
[1]

F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Mathematical Journal, 144 (2008), 235-284. doi: 10.1215/00127094-2008-036.

[2]

F. Cardin, Elementary Symplectic Topology and Mechanics, Springer, 2015. doi: 10.1007/978-3-319-11026-4.

[3]

A. Monzner, N. Vichery and F. Zapolsky, Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization, J. Mod. Dyn., 6 (2012), 205-249, arXiv: 1111.0287. doi: 10.3934/jmd.2012.6.205.

[4]

D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology and its Applications, 96 (1999), 249-266. doi: 10.1016/S0166-8641(98)00049-2.

[5]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.

[6]

C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 439-459, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 217, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-4266-3_10.

[7]

C. Viterbo, Symplectic Homogenization, 2014, arXiv: 0801.0206v3.
Figure 1.  The two paths $\alpha+\beta$ and $\gamma+\delta$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811
[2]

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

Włodzimierz M. Tulczyjew, Paweł Urbański. Regularity of generating families of functions. Journal of Geometric Mechanics, 2010, 2 (2) : 199-221. doi: 10.3934/jgm.2010.2.199
[4]

Ely Kerman. Displacement energy of coisotropic submanifolds and Hofer's geometry. Journal of Modern Dynamics, 2008, 2 (3) : 471-497. doi: 10.3934/jmd.2008.2.471
[5]

Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99
[6]

Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229
[7]

Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1
[8]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291
[9]

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

[10]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
[11]

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[12]

Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $ S_n$-symmetry. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823
[13]

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
[14]

Dae San Kim. Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups. Advances in Mathematics of Communications, 2009, 3 (2) : 167-178. doi: 10.3934/amc.2009.3.167
[15]

Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial & Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693
[16]

Nir Avni, Benjamin Weiss. Generating product systems. Journal of Modern Dynamics, 2010, 4 (2) : 257-270. doi: 10.3934/jmd.2010.4.257
[17]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
[18]

Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253
[19]

H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361
[20]

Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233

2017 Impact Factor: 0.561

Tools

Metrics

  • PDF downloads (40)
  • HTML views (197)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2018 American Institute of Mathematical Sciences