Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences,
I would like to tell you how Pavel Andreevich has influenced me, my development as a researcher and how he taught us. I will mostly explain my personal impressions, but I am sure that many of those who had the good luck to listen to his lectures, would agree with me and understand me very well.
Pavel Andreevich was an extraordinary, very entire person. He lived a very intense internal life, such as real poets and artists live. His whole existence was filled with a passion for Science, a wish to understand, at least partially, Nature and an admiration of its beauty. I think it gave a unique sense to his way of life, where there were mixed the love of understanding, admiration of creation, love towards his close persons, towards his students, his ability to feel deeply poetry and art, his non-indifference to the world, sometimes causing him pain due to its imperfections. This intense internal life was like a fire which burnt him, but which also gave light and warmth to his being and to the existence of people around him.
Pavel Andreevich was a Visionary Philosopher, he was against people imposing their will or ideas on others. At the same time his passionate way of being, the force of his feeling and thought were changing the world in a natural and beautiful way. Pavel Andreevich was a selfless teacher, giving himself without limits, and generous with his energy, time, and ideas. Once, speaking about teaching, he said that the most important thing is to love one's students. Himself he gave us his love with a free hand:
And there are those who give and know not pain in giving, nor do they seek
joy, nor give with mindfulness of virtue;
They give as in yonder valley the myrtle breathes its fragrance into space.
Through the hands of such as these God speaks, and from behind their eyes He smiles upon the earth.
(Khalil Gibran. The Prophet)
And his students loved him. We felt joy when he entered the classroom, his eyes lit up, he appeared ten years younger and with a passion started his lecture. We enjoyed his dry sense of humour, his reflections together with us, his excursions from the subject, which always were so interesting.
I first made my acquaintance with Pavel Andreevich 15 years ago as a first year student at the Polytechnical Institute (Technical University). The very first lecture by Pavel Andreevich on rational mechanics completely enchanted me. In the introduction to the course he taught us some material from tensor algebra: direct vector and tensor calculus, in particular, the rotation (turn) tensors. His course was rigorous, and at the same time very clear. Abstract mathematical objects appeared real and physical. And they were so, because "Nature speaks in the language of tensors" (P.A. Zhilin). His lectures were easy to understand despite their sophisticated content. I would not be wrong in saying that, not only for me, mathematically minded, but also for all the students in our group, the lectures by Pavel Andreevich were perhaps the most clear and interesting, though we had many other excellent professors. I remember one of the first examples -- introduction of a "circular vector", a physical object related to rotation, which did not change when changing the orientation of space, and to which then the axial vector was put in correspondence.
Many things which Pavel Andreevich taught us, he had discovered or formulated himself, but he never mentioned this. Questions of priority were not important for him, he did not have a false pride. His lectures were a process of creation, and he involved us in this. He discussed with us the subject of the lecture, never avoiding or hiding problems, never simplifying. In teaching, science, and life he was completely honest. Often after the lecture we took with us more questions than answers, and this was marvellous. We learnt to think; Mechanics appeared alive for us, crowded with most interesting and unsolved problems. Pavel Andreevich went to the very foundations of mechanics, developing them and sharing with us his deep understanding of science.
Pavel Andreevich has opened a new world for me and for many others. These words seem banal, but it is difficult to say it in another way. Before knowing him I was rather indifferent to the laws of Nature. I liked to solve clearly stated mathematical problems, to look for nice solutions. However, I was afraid of the real world, its incomprehensibility and variety. The physics I knew (or imagined I knew) seemed to me not rigorous and not having unity. For this reason I did not accept physics in my soul, and, instead of trying to resolve its contradictions, I simply was not really interested in it, despite my excellent marks. I almost did not know how to model and did not like it, and this is the most creative and difficult step in solving a problem. All this changed bit by bit, when I started to work with Pavel Andreevich, and the feeling, which I experienced, was probably similar to the feeling of a colourblind person who by a miracle started to distinguish all the variety of colours.
This did not happen in one day. First, fascinated by the lectures of Pavel Andreevich, in the first year I asked him that we work together. His answer was "Elena, it is still too early for you to work on concrete problems. Grow, learn, gain more knowledge and intuition, and later we shall talk about joint work". Six months later I approached him again with the same words. This time Pavel Andreevich asked "Do you have any problem of your own to solve?" Not expecting such a question, I replied no. "Very bad" -- said Pavel Andreevich -- "it is necessary that you yourself would have something of interest for you". This reaction was very typical for Pavel Andreevich. Unlike many scientific advisers, he was not interested in only solving problems interesting to him or in obtaining concrete results of use for his work. He was interested first of all in the development of a student to become a researcher, and here he, as usual, thought not about his profit, but of the student's. "Well" -- he said -- "let us then try to describe the motion of such a wheel rolling in a circular trajectory, under the condition the dry friction acts between the plane and the wheel".
Immediately I did not like that it was not clear what was given and what we had to find out. Pavel Andreevich answered my question, "think yourself what is interesting to find out here, and what we need to know for this, state the problem yourself". In a couple of days I wrote down, as it seemed to me, the solution. I made a brave hypothesis that the vertical reaction is distributed homogeneously along the wheel thickness, wrote corresponding dry friction forces and made some derivations. Pavel Andreevich picked my "solution" to pieces. "From where, Elena, have you concluded, that these forces are distributed and directed in this way?"-- he asked -- "And where is the torque, acting upon the wheel from the side of the rod?" He gave many other criticisms of my "solution". I remember my disappointment. It seemed to me, the result is obtained, why should we ask ourselves strange questions, to look how the forces are directed, when anyway it is clear. However, listening to Pavel Andreevich, I had to agree with him and to understand, that not only in mathematics we have to act rigorously. Pavel Andreevich taught us to account for each step of modelling, to understand what follows from fundamental laws, what are the consequences of the conditions of the problem, for instance, of its symmetry, and what could be our possibly wrong or particular hypothesis on the nature of acting forces or the character of the solution. The habit of such a responsibility, if one may say, to scientific neatness, in my opinion, is obligatory for a researcher, it is a necessary condition of a clear creative mentality without prejudice. Further our discussion led to our realising neither of us knew how to write the Coulomb law of friction for a rigid body (and not a mass point) on a plane. Pavel Andreevich suggested to suppose that a general rigid body touches the plane in three points, at each one the classical Coulomb law acts. We wanted to deduce from such a model, practically applicable, perhaps an approximate law for the force and torque of dry friction, depending on the translational and angular velocities of the body.
A turning-point of my scientific development was caused by this problem. I could obtain no such reasonable law. It was difficult and painful to learn to think, to state the problem and to accept failure. Despite everything, I think that it was the most beneficial time for me. Working, I was led to the conclusion that the solution of the dynamical equations for such a rigid body does not always exist, and sometimes can be non-unique. Pavel Andreevich listened to me and said: "Such situations appear. This is a Painlevé paradox". Only later after I had listened to a brilliant course of lectures by the recently passed away Le Suan An, who was telling us about these paradoxes and their possible solutions. At that time I struggled against them alone. In fact, I had not achieved any apparent success with this problem. However, it was an important result. I think that the main result was not even a deeper understanding of dry friction problems and the Painlevé paradoxes. The main result was an understanding of the fact that the process of cognition, an attempt of creative work is much more important than the success of this attempt. I am sure now, that at the moment Pavel Andreevich left me on purpose to struggle alone against a difficult, perhaps, insoluble problem, on purpose did not insure me by a simple problem, clear for him, because he saw that I was able to overcome if not the problem itself, but the psychological situation, and he wanted me to develop.
Pavel Andreevich taught us not to look for too easy a goal, external success, not to compete between us by little achievements. Once I mentioned one person as a best student of his course. The reaction of Pavel Andreevich was characteristic: "Why compare -- who is smarter, me, Petya or Vasya. What an importance does it have? For some reason one never asks oneself, whether one is more intelligent than Leonhard Euler. Because Euler is far, and Vasya is near. All this is small-minded. One has to think of great people and to aim to achieve their level, and it is not important whether you can do this or not". And, though Pavel Andreevich taught us not to be afraid of difficulties and to work independently, he, not sparing his time, was always ready to help us. Independently of his state of health or if they were paying his salary, he was spending long hours with us discussing scientific problems. Once I felt desperate because I could not find a mistake in my derivations for one month. Pavel Andreevich sat down by my side and looked for the mistake, and, of course, he found it. Well, it was an exceptional case. To those students who wanted and could work independently, he gave a complete freedom, a full confidence, never asked about the results achieved, never tried to control, and at the same time was always ready to help as a colleague and as a teacher.
For instance, a part of my Master thesis problem I already suggested myself, and, formally speaking, I was working independently on my PhD thesis, because I decided that "to navigate is necessary, to live is not so necessary", as ancient Greeks said. Pavel Andreevich understood and approved that, and he said this when I read my PhD thesis. "I did not control Elena on purpose, did not impose her my help. If it would be otherwise, perhaps, her thesis would have two or three secondary results more, but the main results would not be achieved -- her converting into an independent researcher". Of course, I felt extremely happy when listening to his words. However, my "independent work" was not really independent. Both my Master and my PhD theses dealt with polar media -- gyrocontinua, whose particles possessed dynamical spin. We named this gyrocontinuum "Kelvin's medium", since, apparently, the idea to consider such a medium belonged to Kelvin.
To those who needed guidance in a more detailed way, he helped at each step. But most importantly it was not guidance "from above to below", but a collaboration and co-creativity. It was the same during his lectures. For Pavel Andreevich it was boring to repeat the same course. Each year he changed his courses, developed them, and some of us were going to his lectures again already after graduating from the fifth year. Each time it was a new enthralling story. Often Pavel Andreevich improvised during the lecture, and this was especially interesting, because the solutions and proofs were being born before our very eyes, and we felt as if we took part in this. Once at the beginning of the lecture Pavel Andreevich looked at us with a frown and reproached us: "How could you! I was proving to you something all the last lecture, and everything was wrong. And you, you just sat there and listened. Not a single person noticed my mistake!" Indeed, Pavel Andreevich encouraged us to a real understanding, to collaboration. And this was appreciated by his students. Once, in the fourth year, our group was alone in a classroom, for some reason the lecture was cancelled. Pavel Andreevich had a class in a neighbouring room, but being distracted, entered our lecture room, and, seeing familiar faces, took a piece of a chalk and approached the blackboard. Hesitating, he asked us "Listen, what have I to lecture to you today?" The group replied in chorus "Whatever you like!" Pavel Andreevich realised the situation and tried to go, but we did not want to let him go and asked him to give a lecture on any topic. During his lectures, being captivated with the subject, he often did not hear the bell. If it was the last lecture, we did not make him rush. Once he continued the lecture for half of an hour after the time was over, and in the whole room nobody interrupted him. Almost everyone went to his lecture, though Pavel Andreevich did not pay any attention to the attendance, and in general never forced in any way the students to study. For example, Pavel Andreevich, giving us homework tasks or exam questions, often asked: "Do you want a difficult problem, a medium one, or a simpler one?", and the mark did not depend on the kind of problem you chose. Pavel Andreevich had a big difficulty to put a mark less than "good", and most of them were "excellent". Once our classmate got "satisfactory". We surrounded this classmate and with indignation asked him, "What have you told him?!" Usually Pavel Andreevich's lectures were so clear and neat, that we all really learnt the material well enough. However, if a student felt unsure or was failing the exam, Pavel Andreevich discussed with him what was not clear in the subject, and finally often explained the question himself. Each exam was rather a discussion on a scientific topic. When we were the students of the first year, Pavel Andreevich also gave us example classes. I remember that we wrote some tests on tensor algebra, and waited with some awe the practice exam (there were two possible marks, pass or fail, we had to pass it for each exam class to be allowed to pass real exams). We were not sure what we had to do. Pavel Andreevich did the following thing. He called several students and passed them. Then he called several more students, not in alphabetical order, and also passed them, and continued in this way. The group started to make a joyful buzz. Pavel Andreevich looked at us and said: "And those, who have passed, why are you sitting here? Well, go, you disappear!" Of course, everyone has passed the exam. There happened also some other nice episodes. Once Pavel Andreevich denoted by both torque and mass. After some calculations, these quantities appeared in the same formula in numerator and denominator, but Pavel Andreevich continued derivations distinguishing them. Someone told him about this. He answered with a very serious expression: "You know, it is already a long time that all these letters have no significance at all for me. You will work a couple of years more on this subject, and for you it will also be all the same".
Pavel Andreevich taught us not only during the lectures, but also in discussions, both in the University, in the Institute, and here, at School. Those who knew and loved him, feel now emptiness without him. Many people here, probably, remember how passionately he participated in discussions. Often his questions made the subject more interesting and clear than the presentation itself. Heatedly arguing, he never offended people, because the question of discussion was not personal ambition, but scientific truth, and this was obvious to everyone. At one of the first Schools where I participated, our colleague, a friend of our institute, made a presentation on the theory of shells. The chairman said: "Any questions, remarks?" Pavel Andreevich got up and with a very good feeling said: "All that you have told us is completely wrong!" The whole audience, including the speaker, fell about laughing, and then this led to a very interesting discussion on physically and geometrically nonlinear theories. Pavel Andreevich never argued for the sake of it. When he saw someone making a mistake, he always tried to find the true idea and to put it right. Many of the people present here now participated in evening teas, where all questions could be discussed, starting with philosophy and continuing with the electrodynamics, mechanics of granular media and the second law of thermodynamics.
Pavel Andreevich did not accept a mere superficial "understanding" in the sense of "accustom and know how to use", he always wanted to reach the very essence, the base. He was never afraid to seem ridiculous, denying a consensus that he did not accept. Sometimes it was amusing that for this reason some people looked at him a little from above. Perhaps these people were not even able to understand the questions that Pavel Andreevich asked himself, never mind responding to them. Pavel Andreevich preferred to be ("être") rather than to seem ("paraître"). He taught all of us this. Perhaps it was his most important lesson.
I think that if at the end of my life I could know that Pavel Andreevich would feel proud of me as his student, I could not think of a better praise. Probably many of his past pupils would repeat these words.