My main research interests are differential (super)geometry and its applications to mathematical physics. I am particularly interested in symplectic, Poisson, contact, Jacobi, and similar geometric structures, as well as their applications to dynamical systems.
In my doctoral dissertation, The geometry of dissipation (September 2024),
I considered different geometric frameworks for modelling non-conservative dynamics, with a special emphasis on the aspects related to the symmetries and integrability of these systems. More specifically, I explored three classes of geometric frameworks modeling dissipative systems: systems with external forces, contact systems, and systems with impacts.
In a broad sense, my recent research has been mainly concerned with coordinates in which different geometric structures have a canonical form. On the one hand, I have been working on a notion of complete integrability for contact Hamiltonian dynamics. Together with L. Colombo, M. de León, and M. Lainz, we proved a Liouville-Arnold theorem for contact Hamiltonian systems, leading to action-angle coordinates in which both the contact form and the Hamiltonian dynamics have a canonical expression. On the other hand, together with J. Grabowski, we have been investigating homogeneous one-forms on graded (super)manifolds, constructing homogeneous coordinates in which the differential form has a canonical expression. Both lines of research are in fact connected, as our approach to completely integrable contact systems is based on the one-to-one correspondence between contact and homogeneous symplectic manifolds.
I plan to continue my research line on the integrability of non-conservative dynamical systems by focusing on almost-symplectic Hamiltonian dynamics (i.e., the differential form on the phase space is non-degenerate but no longer closed). I shall pursue this goal in collaboration with L. Colombo and N. Sansonetto. Our goal is to understand the notions of integrable system, action-angle coordinates, etc. in this framework, which is a natural generalisation of Hamiltonian dynamics on a symplectic manifold. Once we understand these foundations, we would like to apply our theory to studying the integrability of non-holonomic systems. Such systems not only have a rich geometry, but also have remarkable importance in applications to engineering such as drones or robotics.
Our results concerning homogeneous forms on graded (super)manifolds could have applications in several areas of classical and graded differential geometry. In particular, we plan to study their relation with VB-groupoids and their homogeneous cohomology. The structure of a vector bundle (VB) on a manifold can be uniquely determined by a suitable action of the multiplicative group of real numbers on its total space. Roughly speaking, this means endowing the total space with a grading such that the linear coordinates have degree 1, and the basic coordinates have degree 0. In particular, this leads to a quite simple and natural definition of a VB-groupoid. Furthermore, we would like to study the canonical form for other homogeneous geometric structures on graded (super)manifolds, including a Weinstein splitting theorem for homogeneous Poisson manifolds.
The publications in PDF can be downloaded by clicking on the icon .
Preprints
J. Bajo, M. de León, and A. López-Gordón, “Geometric integrators for adiabatically closed simple thermodynamic systems”, Nov. 2025, arXiv:2511.14154 [math-ph]
Journal articles
L. J. Colombo, M. de León, M. E. Eyrea Irazú and A. López-Gordón, “Generalized hybrid momentum maps and reduction by symmetries of simple hybrid forced mechanical systems”, J. Math. Phys.66(6), June 2025, doi: 10.1063/5.0178542
L. Colombo, M. de León, M. Lainz and A. López-Gordón, “Liouville-Arnold theorem for homogenous symplectic and contact Hamiltonian systems”, Geom. Mech.02(03), pp. 275-307, 2025, doi: 10.1142/S2972458925400039.
L. Colombo, M. de León, M. E. Eyrea Irazú and A. López-Gordón, “Hamilton-Jacobi theory for nonholonomic and forced hybrid mechanical systems”, Geom. Mech.01(02), July 2024, doi: 10.1142/S2972458924500059.
M. de León, M. Lainz, A. López-Gordón and J. C. Marrero, “A new perspective on nonholonomic brackets and Hamilton-Jacobi theory”, J. Geom. Phys.198, 105116, Feb. 2024, doi:10.1016/j.geomphys.2024.105116.
J. Gaset, A. López-Gordón and X. Rivas, “Symmetries, conservation and dissipation in time-dependent contact systems”, Fortschr. Phys.71 (8-9), p. 2300048, May 2023, doi: 10.1002/prop.202300048.
M. de León, M. Lainz, A. López-Gordón and X. Rivas, “Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems”, J. Geom. Phys., 187, Feb. 2023, doi: 10.1016/j.geomphys.2023.104787.
L. J. Colombo, M. de León and A. López-Gordón, “Contact Lagrangian systems subject to impulsive constraints”, J. Phys. A: Math. Theor., 55(42), p. 425203, Oct. 2022, doi: 10.1088/1751-8121/ac96de.
M. de León, M. Lainz and A. López-Gordón, “Discrete Hamilton–Jacobi theory for systems with external forces”, J. Phys. A: Math. Theor., 55(20), p. 205201, Mar. 2022, doi: 10.1088/1751-8121/ac6240.
M. de León, M. Lainz and A. López-Gordón, “Geometric Hamilton–Jacobi theory for systems with external forces”, J. Math. Phys., 63(2), p. 022901, Feb. 2022, doi: 10.1063/5.0073214.
M.
de León, M. Lainz and A. López-Gordón, "Symmetries, constants of the motion,
and reduction of mechanical systems with external forces”, J. Math. Phys., 62(4), p. 042901, Apr. 2021, doi: 10.1063/5.0045073.
Conference papers
L. Colombo, M. E. Eyrea Irazú, M. E. García, A. López-Gordón, and M. Zuccalli, "Reduction of hybrid Hamiltonian systems with non-equivariant momentum maps", in Geometric Science of Information, F. Nielsen and F. Barbaresco, Eds, Cham: Springer Nature Switzerland, Oct. 2025, pp. 303–310. doi: 10.1007/978-3-032-03924-8_31.
L. Colombo, M. de León, M. E. Eyrea Irazú, and A. López-Gordón, "Homogeneous bi-Hamiltonian structures and integrable contact systems", in Geometric Science of Information, F. Nielsen and F. Barbaresco, Eds, Cham: Springer Nature Switzerland, Oct. 2025, pp. 30–39. doi: 10.1007/978-3-032-03924-8_4.
A. López-Gordón and L. J. Colombo, “On the integrability of hybrid Hamiltonian systems”, 8th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2024, IFAC-PapersOnLine, vol. 58, no. 6, pp. 83–88, 2024, doi: 10.1016/j.ifacol.2024.08.261.
M. de León, M. Lainz, A. López-Gordón and J. C. Marrero, “Nonholonomic brackets: Eden revisited”, Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol. 14072. Springer, Cham, 2023, pp. 105-112, doi: 10.1007/978-3-031-38299-4_12.
A. Anahory Simoes, A. López-Gordón, A. Bloch and L. Colombo, “Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage”, 2023 American Control Conference (ACC), San Diego, CA, USA, 2023, pp. 4587-4592, doi: 10.23919/ACC55779.2023.10156020.
A. López-Gordón, L. Colombo and M. de León, “Nonsmooth Herglotz variational principle”, 2023 American Control Conference (ACC), San Diego, CA, USA, 2023, pp. 3376-3381, doi: 10.23919/ACC55779.2023.10156228.
M.
E. Eyrea Irazú, A. López-Gordón, L. J. Colombo and M. de León, “Hybrid Routhian
reduction for simple hybrid forced Lagrangian systems”, 2022 European Control Conference (ECC), London, United Kingdom, 2022, doi: 10.23919/ECC55457.2022.9838077.
Theses
“The geometry of dissipation”, PhD Thesis, Universidad Autónoma de Madrid, September 2024, arXiv:2409.11947 [math.ph].
“The geometry of Rayleigh dissipation”, Master's Thesis, Universidad Autónoma de Madrid, July 2021, arXiv:2107.03780 [physics.class-ph].
“Study of the Entanglement Entropy of the XX Model”, Bachelor's Thesis, Universidad Complutense de Madrid, July 2020, doi: 10.13140/RG.2.2.24773.88809.
Collaborators
Some colleagues I currently collaborate or have collaborated with are:
Jaime Bajo Da Costa (master's student at ETH Zürich, Switzerland)
arXiv
Google Scholar
MathSciNet MR Author Id: 1431337
ORCID: 0000-0002-9620-9647
Scopus Author Identifier: 57222113719
Web of Science ResearcherID: AAN-4932-2021
ResearchGate
Zotero
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