This talk offers a concise overview of Lie systems, commemorating the 16th anniversary of my PhD defense. I will trace the historical development of Lie systems, beginning with foundational work by Guldberg, Vessiot, and Lie, which had a relevant moment in 1893 with the celebrated Lie theorem. This will be followed by the analysis of the contributions from Winternitz and the CRM school in Canada. My presentation will then explore the geometric approach (Carinena, Marmo, Grabowski) developed at the begining of my PhD. Then, I will explore the developments during my PhD thesis culminating with the coalgebra method (developed with my coworkers), deformation theory, Lie systems related to geometric structures, and other developments that followed after that. I will finish the newer approaches to Lie systems and their generalisations: stochastic and super Lie systems, discrete Lie systems, and Lie groupoid methods in Lie systems. I will also pay special attention to the results by Marmo, Carinena, Grundland, Ballesteros, Herranz, Fernandez-Sainz, Sardon, Campoamor-Stursberg, Carballal, Odzijewicz, Ibragimov, etc.

Zapraszamy na spotkanie o godzinie 15:15

Robet Wolak (Jagiellonian University)

Sasakian geometry and foliations (provisional title)

Zapraszamy na spotkanie o godzinie 15:15

Tomasz Sobczak (KMMF)

Introduction to PDE Lie systems

Zapraszamy na spotkanie o godzinie 15:15

Bartołmiej Bąk (KMMF)

Classical model of particle with spin: geometric structure

The motion of point-like particles with mass and charge in the relativistic regime has been well understood for over a century. However, incorporating rotation proved to be far more challenging than initially expected. The first attempts to describe spinning objects were made independently by Jakov Frenkel [1], Myron Mathisson [2], Jan Weyssenhoff and Antoni Raabe [3], Achilles Papapetrou and Ernesto Corinaldesi [4], Barbara and W lodzimierz Tulczyjew [5], and William G. Dixon [6]. The classical spinning particle model on which I intend to rely is based on the approach proposed by Jerzy Kijowski [7]. He briefly outlines a procedure for deriving the equations of motion fromconservation laws and constructs the canonical (symplectic) structure together with the associated Lagrangian and Hamiltonian (cf. [8]). In my talk, I will focus on the symplectic structure of the theory, which exhibits several non-trivial features. In particular, it depends on acceleration (second derivatives of position with respect to time), which makes it a priori degenerate and leads to second-order Euler–Lagrange equations. To address this issue, the Lagrange multiplier method and a reduction procedure are introduced. If time permits, I will also discuss exact solutions and the Hamiltonian formulation of the theory. Most of the results to be presented are not yet published.References[1] J. Frenkel, Die Elektrodynamik des rotierenden Elektrons, Z. Phys. 37, 243 (1926);[2] M. Mathisson: Die Mechanik des Materieteilchens in der Allgemeinen Relativit¨atstheorie, Z. Phys. 67, 826 (1931); Neue Mechanik Materieller Systemes, Acta Phys. Pol. 6, 163 (1937); Das Zitternde Elektron und Seine Dynamik, Acta Phys. Pol. 6, 218 (1937);[3] J. Weyssenhoff, A. Raabe, Relativistic dynamics of spin-fluids and spin-particles, Acta Phys. Pol. 9, 7 (1947); J. Weyssenhoff: Relativistic dynamics of spin-particles moving with the velocity of light, Acta Phys. Pol. 9, 19 (1947); Further contributions to the dynamics of spin-particles moving witha velocity smaller than that of light, Acta Phys. Pol. 9, 26 (1947); Further contributions to the dynamics of spin-particles moving with the velocity of light, Acta Phys. Pol. 9, 34 (1947); On two relativistic models of Dirac’s electron, Acta Phys. Pol. 9, 46 (1947); [4] A. Papapetrou, Spinning test-particles in general relativity. I, Proc. R. Soc. Lond. A. 209, 248 (1951); E. Corinaldesi, A. Papapetrou, Spinning test-particles in general relativity. II, Proc. R. Soc. Lond. A. 209, 259 (1951);[5] W. Tulczyjew, Motion of Multipole Particles in General Relativity Theory, Acta Phys. Pol. 18, 393 (1959); B. Tulczyjew, W. Tulczyjew, On multipole formalism in general relativity, article in Recent Developments in General Relativity, Pergamon Press, New York, 465 (1962);[6] W.G. Dixon: A Covariant Multipole Formalism for Extended Test Bodies in General Relativity, Nuovo Cimento 34, 317 (1964); Mathisson’s new mechanics:its aims and realisation, Acta Phys. Pol. 1, 27 (2008);[7] J. Kijowski; Hamiltonian description of motion of charged particle with spin, Acta Phys. Pol. 1, 143 (2008);[8] J. Kijowski: Electrodynamics of moving bodies, Gen. Relat. Grav.Journal, 26, 167 (1994); On electrodynamical self-interaction, Acta Phys. Pol. 85, 771 (1994); D. Chruscinski, J. Kijowski, Equations of Motion from Field Equations and a Gauge- invariant Variational Principle for the Motion of Charged Particles, Journal Geom. Phys. 20, 393 (1996); H.P. Gittel, J. Kijowski, E. Zeidler, The Relativistic Dynamics of the Combined Particle-Field System in Nonlinear Renormalized Electrodynamics, Comm. Math. Phys. 198, 711(1998);