Lagrangian Multiform Theory and Pluri-Lagrangian Systems

创建时间:  2023/07/17  谭福平   浏览次数:   返回

Date: October 22 – October 27, 2023

Venue: Institute for Advanced Study in Mathematics (IASM), Hangzhou, China

Organizers: Frank Nijhoff (University of Leeds), Linyu Peng (Keio University), Yang Shi (Flinders University), Da-jun Zhang (Shanghai University)

Sponsors: BIRS and IASM

Workshop webpage at BIRS; IASM-BIRS Workshops in 2023


The workshop deals with a fast-developing novel field of research that was initiated a decade ago with the pioneering paper by Lobb/Nijhoff of 2009, and that has seen quite an impressive development since with various groups making major contributions. Since there have never been specific workshops dedicated to this particular area yet, the time is now ripe for such a specific workshop to be held. Thus, the objectives of this workshop are as follows:

1. to bring together the main experts in this field in order to consolidate the results obtained so far, and to identify the outstanding problems and challenges and formulate a strategy for moving the subject forward;

2. to bring together experts from various cognate areas (such as differential geometry, theory of invariants, special function theory and from quantum physics and loop quantum gravity) whose research exhibits common ground with the subject of the workshop; to bring them up to speed with the developments and discuss potential applications.

The following areas will be covered during the workshop:

1. foundations of the theory: the basic principles and the derivation of multiform Euler-Lagrange equations;

2. emergence of (higher) variational symmetries, conservation laws and (multiform) Noether theorem;

3. symmetry reductions (e.g. to integrable maps and Painleve hierarchies) and connections to differential and difference invariant theory;

4. classification problem of Lagrangian multiforms and pluri-Lagrangian systems - multicomponent and matrix systems, higher dimensions;

5. discrete and continuous differential geometric descriptions and the variational bicomplex; 

6. the quantum formulation of the theory and Feynman path integrals;

7. connections with topological field theory and fundamental theories of physics (e.g., loop quantum gravity).

Confirmed participants

Vincent Caudrelier

University of Leeds


Xiangke Chang

Chinese Academy of Sciences


Jarmo Hietarinta

University of Turku


Andrew Kels

University of New South Wales


Yuji Kodama

Shandong University of Science and Technology, 

China & Ohio State University, USA

Shihao Li

Sichuan University


Xing Li

Jiangsu Normal University


Qingping Liu

China University of Mining and Technology


Frank Nijhoff

University of Leeds


Yusuke Ono

Keio University


Linyu Peng

Keio University


Nicolai Reshethikin

Tsinghua University


Jacob Richardson

University of Leeds


Yang Shi

Flinders University


Ying Shi

Zhejiang University of Science and Technology


Yingying Sun

University of Shanghai for Science and Technology


Yuri Suris

Technical University of Berlin


Mats Vermeeren

Loughborough University


Pavlos Xenitidis

Liverpool Hope University


Xiaoxue Xu

Zhengzhou University


Di Yang

University of Science and Technology in China


Sikarin Yoo-Kong

Naresuan University


Cheng Zhang

Shanghai University


Da-jun Zhang

Shanghai University


Dafeng Zuo

University of Science and Technology of China


Online participants (confirmed)

Bianca Dittrich

Perimeter Institute Canada


Adam Doliwa

University of Warmia and Mazury


Anton Dzhamay

The University of Northern Colorado


Joao Faria Martins

University of Leeds


Evgeny Ferapontov

Loughborough University


Wei Fu

East China Normal University


Felix Gunther

Technische Universität Berlin


Irina Kogan

North Carolina State University


Nobutaka Nakazono

Tokyo University of Agriculture and Technology


Peter Olver

University of Minnesota -- Twin Cities


Changzheng Qu

Ningbo University


Reinout Quispel

La Trobe University


Anup Anand Singh

University of Leeds


Songlin Zhao

Zhejiang University of Technology



Related papers

First papers:

[1]  S. Lobb, F.W. Nijhoff, Lagrangian multiforms and multidimensional consistency, J. Phys. A: Math Theor., 42 (2009) 454013

[2]  S. Lobb, F.W. Nijhoff, R. Quispel, Lagrangian multiform structure for the lattice KP system, J. Phys. A: Math Theor., 42 (2009) 472002

[3]  S. Lobb, F.W. Nijhoff, Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, J. Phys. A: Math. Theor., 43 (2010) 072003

[4]  A. Bobenko, Yu.B. Suris, On the Lagrangian structure of integrable quad-equations, Lett. Math. Phys., 92 (2010) 17-31

[5]  P. Xenitidis, F.W. Nijhoff, S. Lobb, On the Lagrangian formulation of multidimensionally consistent systems, Proc. Roy. Soc., A467 (2011) 3295-3317

[6]  S. Yoo-Kong, S. Lobb, F.W. Nijhoff, Discrete-time Calogero-Moser system and Lagrangian 1-form structure, J. Phys. A: Math. Theor., 44 (2011) 365203

[7]  J. Atkinson, S.B. Lobb, F.W. Nijhoff, An integrable multicomponent quad equation and its Lagrangian formalism, Theor. Math. Phys., 173 (2012) 1644-1653

[8]  S. Yoo-Kong, F.W. Nijhoff, Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure, arXiv:1112.4576 (Dec. 2011)

[9]  Yu.B. Suris, Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms, J. Geom. Mech., 2013, 5(3): 365-379, arXiv: 1212.3314 (Dec. 2012)

[10] R. Boll, M. Petrera, Yu. B. Suris, Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems, arXiv:1302.7144 (Feb. 2013)

[11] R. Boll, M. Petrera, Yu.B. Suris, Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems, J. Phys A: Math. Theor., 48 (2015) 085203

[12] R. Boll, M. Petrera Yu.B. Suris, What is integrability of discrete variational systems? Proc, Roy Soc. A470 (2014) 20130550

[13] S. Lobb, F.W. Nijhoff, A variational principle for discrete integrable systems, SIGMA, 14 (2018) 041, arXiv: 1312.1440

EL equations and continuum limits:

[14] Yu.B. Suris, Variational fomrtulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms, J. Geom. Mech, 5 (2013) 365--379

[15] M. Vermeeren, Continuum limits of pluri-Lagrangian, J. Int. Syst. 4 (2019) 1-34

[16] M. Vermeeren, A variational perspective on continuum limits of ABS and lattice GD eqations, SIGMA, 15 (2019) 044

The pluri-Lagrangian formulation:

[17] A. Bobenko, Yu.B. Suris, Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems, Comm. Mah. Phys., 336 (2015) 199-215

[18] Yu.B. Suris, Variational symmetries and pluri-Lagrangian systems, in: Dynamical systems, Number Theory and Appls. (Springer Verlag), 2016) 255-266

[19] Yu.B. Suris, M. Vermeeren, On the Lagrangian structure of integrable hierarchies, in: Advances in Discrete Differential Geometry (Springer Verlag, 2016) 347-378

Some recent developments:

Lagrangian multiform formulation of the Lax pair:

[20] D. Sleigh, F.W. Nijhoff, V. Caudrelier, A variational approach to Lax representations, J. Geom. Phys., 142 (2019) 66--79

Role of variational symmetries in multiform context:

[21] Yu.B. Suris, Variational symmetries and pluri-Lagrangian systems, in "Dynamical Systems, Number Theory and Applications, (Springer Verlag, 2016) 255-266

[22] D. Sleigh, F.W. Nijhoff, V. Caudrelier, Variational symmetries and Lagrangian multiforms, Lett. Math. Phys., 110 (2020) 805--826

[23] M. Petrera, M. Vermeeren, Variational symmetries and pluri-Lagrangian strucures for integrable hierarchies of PDEs, Eur. J. Math., 7 (2021) 741-765

Lagrangian 3-form structure for the KP hierarchy:

[24] D. Sleigh, F.W. Nijhoff, V. Caudrelier, Lagrangian multiforms for the Kadomtsev-Petviashvili (KP) and the Gel'fand-Dickey hierarchy, Int. Math. Res. Not., 2021 (2021) 1-41

[25] S. Lobb, F.W. Nijhoff, R. Quispel, Lagrangian multiform structure for the lattice KP system, J. Phys. A: Math. Theor., 42 (2009) 472002

[26] S. Lobb, F.W. Nijhoff, Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, J. Phys. A: Math. Theor., 43 (2010) 072003

[27] F.W. Nijhoff, Lagrangan 3-form structure for the Darboux systems and the KP hierarchy, Lett. Math. Phys., (2023) arxiv: 2206.14338

Connection with classical r-matrices:

[28] V. Caudrelier, M. Stoppato, B. Vicedo, Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies, arXiv:2201.08286

[29] V. Caudrelier, M. Stoppato, Multiform description of the AKNS hierarchy and classical r-matrix, J. Phys. A: Math. Theor., 54 (2022) 235204, arXiv:2010.07163

[30] V. Caudrelier, M, Stoppato, Hamiltonian multiform of an integrable hierarchy, J. Math. Phys., 61 (2020) 123506, arxiv:2004.01164

Lagrangian structure of non-commuting flows:

[31] V. Caudrelier, F. Nijhoff, D. Sleigh, M. Vermeeren, Lagrangian multiforms on Lie groups and non-commuting flows, J. Geom. Phys., 187 (2023) 104807, arxiv:2204.09663

Towards quantum multiform theory:

[32] S.D. King, F.W. Nijhoff, Quantum variational principle and quantum multiform theory, Nucl. Phys. B947 (2019) 114686, arxiv:1702.08709 [math-ph]

[33] T. Kongkoom, S. Yoo-Kong, Quantum integrability: Lagrangian 1-form case, Nucl. Phys. B987 (2023) 116101, arxiv:2203.16914 [math-ph]

[34] C. Puttarprom, W. Piensuk, S. Yoo-Kong, Integrable Hamiltonian hierarchies and Lagangian 1-forms, arxiv:1904.00582

[35] A. Sridhar, Yu.B. Suris, Commutativity in Lagrangian and Hamiltonian Mechanics, J. Geom. Phys., 137 (2019) 154-161, arxiv: 1801.06076 [math-ph]

Connection with Loop Quantum Gravity:

[36] B. Bahr, B. Dittrich, Breaking and restoring of diffeomorphism symmetry in discrete gravity, arxiv:0909.5688 [gr-qc]

[37] B. Bahr, B. Dittrich, Improved and perfect actions in discrete gravity, Phys. Rev. D, 80 (2009) 124030, arxiv: 0907.4323 [gr-qc]

[38] C. Rovelli, On the structure of a background independent quantum theory: Hamilton function, transition amplitudes, classical limit and continuous limit, arxiv:1108.0832 [gr-qc]

[39] S. Ariwahjoedi, J.S. Kosasih, C. Rovelli, F.P. Zen, Degrees of freedom in discrete geometry,  arxiv: 1607.07963 [gr-qc]

[40] C. Rovelli, Discretizing parametrized systems: the magic of Ditt–invariance, Phys. Rev. D 106 (2022) 104062, arxiv:1107.2310 [hep-lat]

Connections with Chern-Simons theory and topological field theory:

[41] L. Martina, Kur. Myrzakulov, R. Myrzakulov, G. Soliani, Deformation of surfaces, integrable systems and Chern-Simons theory, J. Math. Phys., 42 (2001) 1397-1417, arXiv:nlin/0006039

[42] K.J. Costello, Integrable lattice models from four-dimensional field theories, Proc. Symp. Pure Math., 88 (2014) 3-23. arxiv:1308.0370

[43] K. Costello, E. Witten, M. Yamazaki, Gauge theory and integrability, I, ICCM Not. 6 (2018) 46–119, arxiv:1709.09993 [hep-th]

[44] K. Costello, E. Witten, M. Yamazaki, Gauge theory and integrability, II, ICCM Not. 6 (2018) 120–149, arxiv:1802.01579 [hep-th]

[45] K. Costello, M. Yamazaki, Gauge theory and integrability, III, arXiv:1908.02289 [hep-th]

[46] F. Delduc, S. Lacroix, M. Magro, B. Vicedo, A unifying 2d action for integrable σ-models from 4d Chern-Simons theory, Lett. Math. Phys., 110 (2020) 1645–1687

[47] F. Nijhoff, Integrable hierarchies, Lagrangian structures and noncommuting flows, published in Eds. M.J. Ablowitz, B. Fuchssteiner and M. Kruskal, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, pp. 150–181, Signapore, World Scientific Publ. Co., 1987.

[48] V. Caudrelier, M. Stoppato, B. Vicedo, On the Zakharov-Mkhailov action: 4d Chern-Simons origin and covariant Poisson algebra of the Lax connection, arXiv:2012.04431 [hep-th]

Connection with quantum geometry:

[49] V.V. Bazhanov, A.P. Kels, S.M. Sergeev, Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs, J. Phys. A: Math. Theor., 49 (2016) 464001,  arXiv:1602.07076 [math-ph]

[50] A.P. Kels, M. Yamazaki, Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity, J. Stat. Mech., (2018) 023108, arXiv:1709.07148 [math-ph]

[51] A. Bobenko, F. Gunther, On discrete integrable equations with convex variational principles, Lett. Math. Phys., 102 (2012) 181-202, arXiv:1111.6273 [nlin.SI]  

[52] V.V. Bazhanov, S.M. Sergeev, A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations, Adv. Theor. Math, Phys., 16 (2012) 65-95, arxiv: 1006.0651 [math-ph]

[53] V.V. Bazhanov, V.V. Mangazeev, S.M. Sergeev, Quantum geometry of 3-dimensional lattices and tetrahedron equation, J. Stat. Mech., (2008) P07004, arxiv: 0911.3693 [math-ph]

References on variational bicomplex:

For Differential Equations:

[54] I.M. Anderson, Introduction to the variational bicomplex, in Mathematical Aspects of Classical Field Theory, Contemp. Math. 132 (1992) 5173

[55] I. M. Anderson, The Variational Bicomplex, (book manuscript), Utah State University, 1989.

[56] L.A. Dickey, Soliton Equations and Hamiltonian Systems, 2nd edition, World Scientific, 2003

[57] I. Kogan, P.J. Olver, Invariant Euler-Lagrange equations and the invariant variational Bicomplex, Acta Appl. Math.,76 (2003) 137-193

[58] I.S. Krasil'shchik, A.M. Vinogradov (eds), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Providence, RI: AMS Publications, Monograph, 1999.

[59] P.J. Olver, Applications of Lie Groups to Differential Equations, (2nd edn), New York: Springer-Verlag, 1993.

[60] T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982) 311363.

[61] W.M. Tulczyjew, The Lagrange complex, Bulletin la S. M. F. 105 (1977) 419-431.

[62] W.M. Tulczyjew, The Euler-Lagrange resolution, in Lecture Notes in Mathematics 836, Differential Geometrical Methods in Mathematical Physics (Eds. A. Dold, B. Eckmann), pp 22–48 (Springer-Verlag, New York 1980).

[63] A.M. Vinogradov, A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Sov. Math. Dokl. 19 (1978) 144148.

[64] A.M. Vinogradov, The b-spectral sequence, Lagrangian formalism, and conservation laws. I: The linear theory, J. Math. Anal. Appl., 100 (1984) 1-40.

[65] A.M. Vinogradov, The b-spectral sequence, Lagrangian formalism, and conservation laws. II: The nonlinear theory, J. Math. Anal. Appl., 100 (1984) 41-129.

[66] R. Vitolo, Finite order variational bicomplexes, Mathematical Proceedings of the Cambridge Philosophical Society 125 (1999) 321-333, arXiv:math-ph/0001009

For discrete equations:

[67] P.E. Hydon, E.L. Mansfield, A variational complex for difference equations, Found. Comput. Math., 4 (2004) 187-217.

[68] B.A. Kupershmidt, Discrete Lax Equations and Differential-Difference Calculus, Paris: Asterisque 123, SMF, 1985.

[69] E.L. Mansfield, P.E. Hydon, Diffeence forms, Found. Comput. Math., 8 (2008) 427-467.

[70] L. Peng, From Differential to Difference: The Variational Bicomplex and Invariant Noether's Theorems, Ph.D. Thesis, University of Surrey, 2013.

[71] L. Peng, P.E. Hydon, The difference variational bicomplex and multisymplectic systems, arXiv:2307.13935 [math-ph].

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