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From Lagrangian products to toric domains via the Toda lattice

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

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 , February 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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