Cotangent Bundle Reduction and its Applications to Poincaré-Birkhoff Normal Forms: Analysis of Reactions in Rotating Molecules
Event details
| Date | 01.10.2014 |
| Hour | 16:30 › 17:30 |
| Speaker | Ünver Çiftçi (EPFL) |
| Location |
GR A3 30
|
| Category | Conferences - Seminars |
Geometry and Dynamics Seminar
Abstract: Reaction type dynamics arises due to the presence of bottlenecks in the system's phase space. Such bottlenecks are typically induced by saddle type equilibrium points. In recent years it has been shown that the transport through such bottlenecks is governed by various phase space structures. These phase space structures can be constructed from a Poincaré-Birkhoff normal form. To study reactions in rotating molecules modeled by N-body systems one has to reduce the rotational symmetry and study the phase space bottlenecks induced by the saddle type relative equilibria of the reduced system.
In the first part of the talk, I will outline a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré-Birkhoff normal forms of relative equilibria using standard algorithms. The case of natural mechanical systems with symmetries will be considered in some detail. As examples I will give computations of Poincaré-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum. In the second part, I will talk about how the construction of phase space structures can be generalized to the case of the relative equilibria of a rotational symmetry reduced three-body system. My main example will be the isomerization reaction of the HCN molecule. Time permitting I want to introduce a new type of bottleneck which mediates kinetic rather than configurational changes.
This is a joint work with H. Waalkens and H. Broer.
Abstract: Reaction type dynamics arises due to the presence of bottlenecks in the system's phase space. Such bottlenecks are typically induced by saddle type equilibrium points. In recent years it has been shown that the transport through such bottlenecks is governed by various phase space structures. These phase space structures can be constructed from a Poincaré-Birkhoff normal form. To study reactions in rotating molecules modeled by N-body systems one has to reduce the rotational symmetry and study the phase space bottlenecks induced by the saddle type relative equilibria of the reduced system.
In the first part of the talk, I will outline a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré-Birkhoff normal forms of relative equilibria using standard algorithms. The case of natural mechanical systems with symmetries will be considered in some detail. As examples I will give computations of Poincaré-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum. In the second part, I will talk about how the construction of phase space structures can be generalized to the case of the relative equilibria of a rotational symmetry reduced three-body system. My main example will be the isomerization reaction of the HCN molecule. Time permitting I want to introduce a new type of bottleneck which mediates kinetic rather than configurational changes.
This is a joint work with H. Waalkens and H. Broer.
Practical information
- Expert
- Free
Organizer
- Sonja Hohloch