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Double groupoids and the symplectic category
A note on the normalization of generating functions
Previnet S.p.A., Via E. Forlanini, 24, Preganziol (TV), Italy |
In the present note, I will propose some insights on the normalization of generating functions for Lagrangian submanifolds. From the literature (see, for example [
(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.
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. |
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