Modern Approach to Classical Integrable Systems
For many years, classical mechanics has been the driver of the development of various areas of mathematics. Currently, various applications such as robotics, cryptography, cryptocurrency, computer graphics, computer vision, and machine learning are actively influencing the development of algebraic geometry, differential geometry, Riemannian geometry, number theory, etc. The new tools obtained, or new computer implementations of the known tools, allow not only a new look at the classical integrable systems of the 18th-19th century, but also get a number of previously unknown results.
The goal of this course is to give students an overview of modern concepts and results in classical mechanics.
The goal of this course is to give students an overview of modern concepts and results in classical mechanics.
Lecturer
Date
4th November, 2022 ~ 13th January, 2023
Website
Prerequisite
The course is designed for an audience that has previously attended general course of classical or analytical mechanics and courses of higher mathematics: mathematical analysis and differential equations theory.
Syllabus
Week 1: Euler’s top and elliptic curves
1.1. Earth’s nutation
1.2. Euler’s equations for rigid body motion and Chandler’s wobble
1.3. Euler's equations for arbitrary Lie algebra (Steklov, Poincare and Anol’d)
1.4. Euler’s equations on so(3) and Abel’s quadrature on elliptic curve
1.5. Addition law on elliptic curve
1.6. Multiplication law, division polynomials and mining of Bitcoins
1.7. Discretization of the Euler’s equations of motion via mining
1.8. Discretization of the Euler’s equations of motion using Lax matrices
1.9. Bobenko-Suris and Fedorov discretizations of Euler’s top
1.10. Hirota-Kimura integrable map
Week 2: Arithmetic on hyperelliptic curves
2.1. Group of divisors on plane curves
2.2. Principal and equivalent divisors
2.3. Picard group and Picard scheme
2.4. Hyperelliptic curves (real and imagine), Weierstrass theorem
2.5. Intersection divisors and reduction of divisors
2.6. Examples of Abel’s calculations
2.7. Jacoby’s polynomials and Mumford’s coordinates
2.8. Cantor’s algorithm and its examples
2.9. Different forms of the curves
2.10. Different coordinates of divisors
...
1.1. Earth’s nutation
1.2. Euler’s equations for rigid body motion and Chandler’s wobble
1.3. Euler's equations for arbitrary Lie algebra (Steklov, Poincare and Anol’d)
1.4. Euler’s equations on so(3) and Abel’s quadrature on elliptic curve
1.5. Addition law on elliptic curve
1.6. Multiplication law, division polynomials and mining of Bitcoins
1.7. Discretization of the Euler’s equations of motion via mining
1.8. Discretization of the Euler’s equations of motion using Lax matrices
1.9. Bobenko-Suris and Fedorov discretizations of Euler’s top
1.10. Hirota-Kimura integrable map
Week 2: Arithmetic on hyperelliptic curves
2.1. Group of divisors on plane curves
2.2. Principal and equivalent divisors
2.3. Picard group and Picard scheme
2.4. Hyperelliptic curves (real and imagine), Weierstrass theorem
2.5. Intersection divisors and reduction of divisors
2.6. Examples of Abel’s calculations
2.7. Jacoby’s polynomials and Mumford’s coordinates
2.8. Cantor’s algorithm and its examples
2.9. Different forms of the curves
2.10. Different coordinates of divisors
...
Reference
We will see new material that is not in any book, and explore open problems and directions of current research.
The material is self-sufficient but basic definitions and their discussions may be found in
[1] Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge University Press, Cambridge, 2003.
[2] Reyman A.G., Semenov-Tian-Shansky V.A., Integrable Systems, The Computer Research Institute Publishing, Moscow-Izhvek, 2003.
[3] Handbook of Elliptic and Hyperelliptic Curve Cryptography, editors H. Cohen and G. Frey, Chapman and Hall/CRC, 2006.
[4] D. Eisenbud, J. Harris, 3264 and all that: A second course in algebraic geometry, Cambridge University Press, 2016.
[5] Costello C., A gentle introduction to isogeny-based cryptography, Tutorial at SPACE 2016. December 15, 2016. CRRao AIMSCS, Hyderabad, India 2016.
[6] Helgasson S., Differential Geometry, Lie Groups and Symmetric Spaces, (Graduate studies in Mathematics, vol.34), AMS, Providence, Rhode Island, 2001.
[7] Miranda R., Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, vol 5., American Mathematical Society; UK ed., 1995.
[8] Mumford D., Tata Lectures on Theta, Birkhauser, 1984.
The material is self-sufficient but basic definitions and their discussions may be found in
[1] Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge University Press, Cambridge, 2003.
[2] Reyman A.G., Semenov-Tian-Shansky V.A., Integrable Systems, The Computer Research Institute Publishing, Moscow-Izhvek, 2003.
[3] Handbook of Elliptic and Hyperelliptic Curve Cryptography, editors H. Cohen and G. Frey, Chapman and Hall/CRC, 2006.
[4] D. Eisenbud, J. Harris, 3264 and all that: A second course in algebraic geometry, Cambridge University Press, 2016.
[5] Costello C., A gentle introduction to isogeny-based cryptography, Tutorial at SPACE 2016. December 15, 2016. CRRao AIMSCS, Hyderabad, India 2016.
[6] Helgasson S., Differential Geometry, Lie Groups and Symmetric Spaces, (Graduate studies in Mathematics, vol.34), AMS, Providence, Rhode Island, 2001.
[7] Miranda R., Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, vol 5., American Mathematical Society; UK ed., 1995.
[8] Mumford D., Tata Lectures on Theta, Birkhauser, 1984.
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Andrey Tsiganov currently works at the Department of Computational Physics, Saint Petersburg State University, Russia. His main research interests are integrable and superintegrable systems in classical and quantum mechanics, nonholonomic and vakonomic mechanics, geometry and topology of dynamical systems, see profile at https://www.researchgate.net/profile/Andrey-Tsiganov. He is one of the organizers of the BIMSA Integrable System Seminar, see https://researchseminars.org/seminar/BIMSA-ISS and https://sites.google.com/view/bimsa-iss.