Draft:Włodzimierz Marek Tulczyjew

Włodzimierz Marek Tulczyjew (18 June 1931 – 4 December 2022) was a Polish-Italian physicist and mathematician, known for his contributions to the geometric formulation of classical mechanics and field theory. He was a professor emeritus of mathematical methods of physics at the University of Camerino and a member of the Academy of Sciences of Turin.

== Biography == Tulczyjew was born in Włodawa, a small town in eastern Poland, in 1931. From 1933 to 1943, he lived in Ostrów Lubelski, where his father worked as an accountant in the Municipal Office. In 1943, he was deported with his family to Germany, where he worked in an armaments factory. In June 1945, he returned to Lublin. His father worked in the Provincial Office of Local Industry in Lublin. In 1949, he graduated from the General Education Gymnasium in Lublin and moved to Warsaw. In 1952, he graduated with a diploma as a top student from the State Telecommunication Technical School of the Ministry of Posts and Telegraphs in Warsaw. From 1952 to 1956, he studied in the Faculty of Mathematics and Physics (physics section) at the University of Warsaw. During his studies, Tulczyjew joined a group centered around Leopold Infeld (Jerzy Plebański, Andrzej Trautman, Iwo Białynicki-Birula, Stanisław Bażański, and others). The oldest of this group, Jerzy Plebański, assisted Infeld in directing the scientific work of younger colleagues, including Tulczyjew, who completed his studies with a master's thesis titled “The Mass of Bodies in the 6th Approximation of a New Approximation Method”. He obtained his Ph.D. in 1959 and his D.Sc. in 1965 at the University of Warsaw under the supervision of Andrzej Mariusz Trautman. Then he became an assistant professor there. Tulczyjew's research was highly valued by Infeld, who, in his posthumously published memoirs, referred to Włodek as his most outstanding student. Infeld gave Tulczyjew a lot of freedom, including frequent scientific trips. After Infeld's death (January 15, 1968), the new leadership of the Institute of Theoretical Physics no longer guaranteed him this freedom. Tulczyjew decided to emigrate. He left Poland on September 28, 1968, and through Rome reached Canada, at the University of Calgary.

At the end of the 1980s, Włodek took advantage of the opportunity for early retirement in Canada and moved to his beloved Italy, to Camerino, where he was employed as a professor per chiara fama (outside the competitions) of Mathematical Methods of Physics at the University of Camerino. He soon also began collaborating with Giuseppe Marmo, professor at Federico II University in Naples and the National Institute of Nuclear Physics (Naples branch). His influence expanded significantly, creating new opportunities for collaboration (Fiorella Barone, Antonio De Nicola, and many others).

In the early 1990s, Włodek and Sara (Tulczyjew's wife since 1975) bought and restored a rectory in the small town of Gabicce Monte, on a hill in the middle of a large garden, with a beautiful view of the Adriatic Sea. This became his beloved place. One of his favorite leisure activities was collecting pine cones for the fireplace. “Włodek, the prelate of Gabicce Monte” (as we often called him) was also “Włodek, the Italian,” as this was where he found his place. In addition to his scientific work, Tulczyjew was an excellent teacher. He had a beautiful way of lecturing: calm, interesting, clear. His problem was that he could not refuse to give a lecture, even at the cost of personal inconvenience. He also loved to give spontaneous “little lectures” — clear, precise, beautifully written on a blackboard or a piece of paper. He also had another pedagogical gift: he inspired others with his ideas, rarely imposing a final solution. He directed young people into areas where he sensed beauty and wealth, allowing them to work on the problems that fascinated them.

He retired in 2006, but remained active in research and teaching until his death in 2022.

Scientific work
Tulczyjew's main research interest was the geometry of classical mechanics and field theory, especially the symplectic and multisymplectic structures that underlie the Hamiltonian and Lagrangian formulations, and the Legendre transformation that connects them. He developed several original concepts and methods, such as the Tulczyjew triple, the Tulczyjew symplectic structure, the Tulczyjew-Dedecker differential, the Tulczyjew condition, and the Tulczyjew isomorphism. He also worked on general relativity, gauge theories, quantum mechanics, and differential geometry.

He did not accept partial answers and sought proper solutions. He rejected the arguments of Wolfgang Pauli and Victor Weisskopf about the non-existence of quantum mechanics of bosons, according to which only field-theoretic descriptions of bosons are possible. In his habilitation thesis, Tulczyjew presented an elegant scheme of relativistic quantum mechanics as a scattering theory, where antiparticles are described as particles moving backward in time. In the works comprising this thesis, Tulczyjew appears as a precursor of geometric quantization, developed a few years later by Jean-Marie Souriau and Bertram Kostant. Tulczyjew was also one of the first physicists to recognize the deep connection between Utiyama's theory and Yang-Mills field theory. Seeking the proper formulation of relativistic quantum mechanics led him back to the foundations of classical theories, especially to classical mechanics and the fundamentals of variational calculus (or rather the variational description of physical systems).

The task Tulczyjew set for himself required a broad front of work (as he said: one must look under every stone) and, consequently, numerous collaborators. Already in Warsaw, he inspired younger colleagues, mainly from the Department of Mathematical Methods of Physics at the Faculty of Physics, UW, which was headed by Krzysztof Maurin, drawing their attention to multisymplectic geometry and its applications to the geometrization of variational calculus. The works of Jerzy Kijowski, Wiktor Szczyrba, Jacek Komorowski, and Krzysztof Gawędzki belong to the classics of this subject. In Canada, he continued work started in Warsaw on a gauge-independent description of the dynamics of a charged particle in a Kaluza-Klein-like formulation (Ryszard Kerner in Warsaw, R. J. Torrence in Calgary). He also began work on the inverse problem of variational calculus: finding the necessary and sufficient condition for the existence of a Lagrangian for a given system of partial differential equations, meaning that this system is an Euler-Lagrange system. Tulczyjew solved the problem by constructing a double variational complex. The Euler-Lagrange equations are "exact forms" in this complex. Tulczyjew also proved Poincaré's Lemma for this complex. These results were highly regarded by many mathematicians (André Lichnerowicz, Vinogradov, Paul Dedecker, Willy Sarlet, Michael Crampin). According to Dedecker, it was the most important result in variational calculus in many years. In 1974, Tulczyjew published the work “Hamiltonian Systems, Lagrangian Systems, and the Legendre Transformation” (Symposia Mathematica, 15), in which he gave the full geometric sense of the Legendre transformation as a transition from the Lagrangian to the Hamiltonian description (and vice versa) of a mechanical system's dynamics (and not only) in the language of symplectic relations and their forming objects. This topic was present in Włodek's work for many years and found its culmination in the fundamental work “A Slow and Careful Legendre Transformation for Singular Lagrangians” (Acta Physica Polonica B, 30), which Paweł Urbanski co-authored. The basic geometric structure associated with the Legendre transformation is now known as the Tulczyjew triple. Tulczyjew did not feel well in Canada, so he enthusiastically accepted the proposal to lead the program “Symplectic Geometry in Mechanics and Field Theory” at the Institute of Mathematical Physics, University of Turin. The director of this institute at the time was Dionigi Galletto, who later became a long-time friend of Włodek. The collaboration that began in the second half of the 1970s lasted until the late 1980s. Members of Tulczyjew's group in Turin (Sergio Benenti, Mauro Francaviglia, Marco Ferraris, Giorgio Pidello) were also joined by colleagues from Warsaw, from the Department of Mathematical Methods of Physics (Jerzy Kijowski, Paweł Urbański, Adam Smólski, Stanisław Zakrzewski). The collaboration in Turin resulted in, among other things, a series of works on symplectic relations, the geometric content of Hamilton-Jacobi equations, and the Jacobi method (Sergio Benenti, Włodzimierz Tulczyjew). Włodek's work in Turin was recognized — in 1981 he became a foreign member of the Academy of Sciences in Turin (Accademia delle Scienze di Torino).

Tulczyjew published around 100 scientific papers and several books, including A symplectic framework for field theories (1979), and Geometric Formulation of Physical Theories (1989). He collaborated with many distinguished mathematicians and physicists.

Honors and awards
Tulczyjew was elected as a member of several academies and societies, such as:
 * The Academy of Sciences of Turin (1981)
 * The Polish Academy of Sciences (1990)
 * The Gold Cross of Merit awarded by the President of Poland (2014)
 * The Commander's Cross of the Order of Polonia Restituta awarded by the President of Poland (2015).