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From Lagrangian products to toric domains via the Toda lattice

Part of: Hamiltonian and Lagrangian mechanics Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Symplectic geometry, contact geometry

Published online by Cambridge University Press:  19 June 2025

Yaron Ostrover ,
Vinicius G. B. Ramos  and
Daniele Sepe Open the ORCID record for Daniele Sepe [Opens in a new window]
Yaron Ostrover
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel ostrover@tauex.tau.ac.il
Vinicius G. B. Ramos
Affiliation:
Institute for Advanced Study, 1 Einstein Dr, Princeton, NJ 08540, USA and Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, RJ 22460–320, Brazil vgbramos@impa.br
Daniele Sepe
Affiliation:
Escuela de Matemáticas, Universidad Nacional de Colombia sede Medellín, Calle 59A Norte #63-20, Edifício 43, Medellín, Colombia and Instituto de Matemática e Estatística, Departamento de Matemática Aplicada (GMA), Universidade Federal Fluminense, Campus Gragoatá, Rua Prof. Marcos Waldemar de Freitas Reis, s/n, São Domingos, Niterói, RJ 24210–201, Brazil dsepe@unal.edu.co
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Abstract

In this paper we use the periodic Toda lattice to show that certain Lagrangian product configurations in the classical phase space are symplectically equivalent to toric domains. In particular, we prove that the Lagrangian product of a certain simplex and the Voronoi cell of the root lattice $A_n$ is symplectically equivalent to a Euclidean ball. As a consequence, we deduce that the Lagrangian product of an equilateral triangle and a regular hexagon is symplectomorphic to a Euclidean ball in dimension 4.

Keywords

53D20symplectic embeddingsperiodic Toda lattice

MSC classification

Primary: 53D20: Momentum maps; symplectic reduction
Secondary: 37J35: Completely integrable systems, topological structure of phase space, integration methods 70H06: Completely integrable systems and methods of integration
Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 2 , June 2025 , pp. 365 - 384
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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References

Adler, M., van Moerbeke, P. and Vanhaecke, P., Algebraic integrability, Painlevé geometry and Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, vol. 47 (Springer, Berlin, 2004).10.1007/978-3-662-05650-9CrossRefGoogle Scholar
Arnold, V. I., Mathematical methods of classical mechanics , Graduate Texts in Mathematics, vol. 60 (Springer Science & Business Media, 2013).Google Scholar
Artstein-Avidan, S. and Ostrover, Y., Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not. IMRN 2014 (2014), 165–193.CrossRefGoogle Scholar
Artstein-Avidan, S., Karasev, R. and Ostrover, Y., From symplectic measurements to the Mahler conjecture, Duke Math. J. 163 (2014), 2003–2022.Google Scholar
Balitskiy, A., Equality cases in Viterbo’s conjecture and isoperimetric billiard inequalities, Int. Math. Res. Not. IMRN 2020 (2020), 1957–1978.CrossRefGoogle Scholar
Bogoyavlensky, O. I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201–209.10.1007/BF01617919CrossRefGoogle Scholar
Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups , Grundlehren der mathematischen Wissenschaften, vol. 290 (Springer, New York, 1988).Google Scholar
Conway, J. H. and Sloane, N. J. A., Low-dimensional lattices. VI. Voronoĭ reduction of three-dimensional lattices, Proc. Roy. Soc. Lond. Ser. A 436 (1992), 55–68.10.1098/rspa.1992.0004CrossRefGoogle Scholar
Flaschka, H., The Toda lattice. I. Existence of integrals, Phys. Rev. B 9 (1974), 1924–1925.10.1103/PhysRevB.9.1924CrossRefGoogle Scholar
Flaschka, H., On the Toda lattice II, Prog. Theor. Phys. 51 (1974), 703–716.CrossRefGoogle Scholar
Flaschka, H. and McLaughlin, D. W., Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Prog. Theor. Phys. 55 (1976), 438–456.CrossRefGoogle Scholar
Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.10.1007/BF01388806CrossRefGoogle Scholar
Haim-Kislev, P., On the symplectic size of convex polytopes, Geom. Funct. Anal. 29 (2019), 440–463.10.1007/s00039-019-00486-4CrossRefGoogle Scholar
Hénon, M., Integrals of the Toda lattice, Phys. Rev. B 9 (1974), 1921–1923.10.1103/PhysRevB.9.1921CrossRefGoogle Scholar
Henrici, A. and Kappeler, T., Global action-angle variables for the periodic Toda lattice, Int. Math. Res. Not. IMRN 2008 (2008), rnn031.10.1093/imrn/rnn031CrossRefGoogle Scholar
Henrici, A. and Kappeler, T., Global Birkhoff coordinates for the periodic Toda lattice, Nonlinearity 21 (2008), 2731–2758.10.1088/0951-7715/21/12/001CrossRefGoogle Scholar
Kozlov, V. V. and Treschëv, D. V., Billiards, in A genetic introduction to the dynamics of systems with impacts [Translated from the Russian by J. R. Schulenberger, Translations of Mathematical Monographs, vol. 89 (American Mathematical Society, Providence, RI, 1991).CrossRefGoogle Scholar
Latschev, J., McDuff, D. and Schlenk, F., The Gromov width of 4-dimensional tori, Geom. Topol. 17 (2013), 2813–2853.10.2140/gt.2013.17.2813CrossRefGoogle Scholar
Mahler, K., Ein Übertragungsprinzip für konvexe Körper, Casopis Pyest. Mat. Fys. 68 (1939), 93–102.CrossRefGoogle Scholar
Manakov, S. P., Complete integrability and stochastization of discrete dynamical systems, Sov. Phys. JETP 40 (1974), 269–274.Google Scholar
McDuff, D., Blowups and symplectic embeddings in dimension 4, Topology 30 (1991), 409–421.CrossRefGoogle Scholar
McDuff, D., Symplectic embeddings of 4-dimensional ellipsoids, J. Topol. 2 (2009), 1–22.CrossRefGoogle Scholar
Moser, J. and Zehnder, E. J., Notes on dynamical systems , Courant Lecture Notes in Mathematics, vol. 12 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Ostrover, Y., When symplectic topology meets Banach space geometry, in Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II (Kyung Moon Sa, Seoul, 2014), 959–981.Google Scholar
Ostrover, Y. and Ramos, V. G. B., Symplectic embeddings of the $\ell_p$ -sum of two discs, J. Topol. Anal. 14 (2022), 793–821.10.1142/S1793525321500242CrossRefGoogle Scholar
Pelayo, Á. and Vũ Ngoc, S., Hofer’s question on intermediate symplectic capacities, Proc. Lond. Math. Soc. (3) 110 (2015), 787–804.10.1112/plms/pdu066CrossRefGoogle Scholar
Ramos, V. G. B., Symplectic embeddings and the Lagrangian bidisk, Duke Math. J. 166 (2017), 1703–1738.10.1215/00127094-0000011XCrossRefGoogle Scholar
Ramos, V. G. B. and Sepe, D., On the rigidity of Lagrangian products, J. Symplectic Geom. 17 (2019), 1447–1478.10.4310/JSG.2019.v17.n5.a7CrossRefGoogle Scholar
Rudolf, D., Viterbo’s conjecture for Lagrangian products in ${\mathbb{R}}^4$ and symplectomorphisms to the Euclidean ball, Preprint (2022), arXiv:https://arxiv.org/abs/2203.022942203.02294.Google Scholar
Schlenk, F., Packing symplectic manifolds by hand, J. Symplectic Geom. 3 (2005), 313–340.CrossRefGoogle Scholar
Schlenk, F., Embedding problems in symplectic geometry, De Gruyter Expositions in Mathematics, vol. 40 (Walter de Gruyter Verlag, Berlin, 2005).10.1515/9783110199697CrossRefGoogle Scholar
Toda, M., Vibration of a chain with nonlinear interaction, J. Phys. Soc. Jpn. 22 (1967), 431–436.CrossRefGoogle Scholar
Toda, M., Wave propagation in anharmonic lattices, J. Phys. Soc. Jpn. 23 (1967), 501–506.10.1143/JPSJ.23.501CrossRefGoogle Scholar
Traynor, L., Symplectic packing constructions, J. Differential Geom. 41 (1995), 735–751.CrossRefGoogle Scholar
van Moerbeke, P., The spectrum of Jacobi matrices, Invent. Math. 37 (1976), 45–81.10.1007/BF01418827CrossRefGoogle Scholar
Viterbo, C., Metric and isoperimetric problems in symplectic geometry, J. Am. Math. Soc. 13 (2000), 411–431.10.1090/S0894-0347-00-00328-3CrossRefGoogle Scholar
Živaljević, R. T., Illumination complexes, $\triangle$ -zonotopes, and the polyhedral curtain theorem, Comput. Geom. 48 (2015), 225–236.CrossRefGoogle Scholar