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

A note on the normalization of generating functions

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.

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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