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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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

Figure 1.  The two paths $\alpha+\beta$ and $\gamma+\delta$
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