It is well-known that integrable systems are exceptionally rare since most (nonlinear) differential equations exhibit chaotic behaviour and no explicit solutions can be obtained. Informally, integrability is the property of a system that allows the solution to be solved in finite steps of operations (or integrations). In more simply speaking, integrability enables us to solve the set of equations in a closed form or in terms of quadratures (ordinary integrals). In classical mechanics, especially Hamiltonian systems with N degrees of freedom, integrability has a direct connection to action-angle variables. It is well-understood that the choice of coordinates on phase space is not unique since one can transform an old set of coordinates to a new set of coordinates through the canonical transformation preserving the Hamilton’s equations. Of course, action-angle coordinates are special since the Hamiltonian depends solely on action variables, which are constants of motion, and, consequently, the angle variables are cyclic, evolving linearly in time. With this feature, the complexity of the problem will be reduced and explicit solutions can be determined by quadratures. In this sense, the existence of the action-angle variables guarantees the integrability of the system. However, finding action-angle coordinates is not trivial in practice. Then, the notion of integrability is rather defined in terms of existence of the invariants known as Liouville integrability. In this notion, the Hamiltonian system with N degrees of freedom, whose evolution is on 2N-dimensional manifold embedded in phase space, is integrable if there exists N invariants, which are normally treated as Hamiltonians, that are independent and in involution as the Poisson brackets for every pair of invariants vanish. With involution of invariants, all evolutions belong to the same level set and are mutually commute, known as commuting Hamiltonian flows which are considered to be the main feature for integrable Hamiltonian systems. Furthermore, there exists a canonical transformation such that a set of invariants is nothing but a set of action variables. This means that 2N-dimensional manifold can be foliated into a N-dimensional invariant torus in which the angle variables are naturally the periodic coordinates on this torus.
In this QTFT webinar, we will guide you into the lands, called integrable systems, through various routes (notions of integrability) for systems with finite degree of freedom both on discrete-time and continuous-time levels.
Date: Sunday, 13 October 2019
Time: 8:00pm – 10:00pm (BKK)Time
Venue: Online event
- Dr. Sikarin Yoo-Kong, The Institute for Fundamental Study (IF), Naresuan University
- Moderator : Toonyawat Angkhanawin, University of Strathclyde, Scotland
8:00pm – 8:20pm: Networking
8:20pm – 8:30pm: Introduction to QTFT
8:30pm – 9:30pm: Talk
9:30pm – 10:00pm: Q&A
About the speaker :
Sikarin Yoo-Kong obtained his BSc in physics from King Mongkut’s University of Technology Thonburi, Thailand and MSc in physics at Mahidol University, Thailand. Then he went to the University of York, United Kingdom for his MS(by research) in the topic “entanglement measure”. After that he moved to University of Leeds for his PhD in the topic “integrable systems”. He is currently Assistant Professor in theoretical physics at the Institute for Fundamental Study (IF), Naresuan University, Thailand.