10. Computer assisted proofs in Hamiltonian dynamics and
celestial mechanics
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Polish supervisor:
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Piotr Zgliczyński
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Cooperating partners:
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Carles Simo (Universidad de Barcelona)
Amadeu Delshams (Univ. Politecnica de Catalunya, Barcelona)
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The goal of this project to develop the tools for rigorous
computer assisted study of the dynamics of Hamiltonian systems
with emphasis on the problems coming from celestial mechanics.
The study of problems arising from celestial mechanics offers an
opportunity to work on the problems in which subtle mathematics
related to the preservation of the symplectic structure by the
flow and the presence of large symmetry group meets very practical
questions of finding a good trajectory for a spacecraft going to
other planets or to their satellites, or the estimation whether a
particular piece of rock travelling through the space will hit the
Earth.
In Hamiltonian systems and in particular in the celestial
mechanics there is a huge gap between what we can observe in the
numerical simulations of and what we can prove rigorously. For
example, there is a lot of simulations of the Solar system or of
the formation of galaxies, which involved many bodies and span
over billion of years. The question is: Can we trust these
results? One possibility to approach this question are the
computer assisted proofs of the existence of invariant objects,
like KAM tori, invariant manifolds etc.
In the project the student may focus more on the theory or on the
computations leading according to his or her preferences. She/he
is expected to work on several topics from the following list
- rigorous computations of Birkhoff normal forms,
- rigorous integrators of ODEs dedicated to N-body problem or to Hamiltonian systems,
- the construction the transition chain for the Arnold diffusion problem,
- normally hyperbolic invariant manifolds,
- versions of KAM theory amenable for computer assisted proofs,
- the existence of choreographies in N-body problem for large N,
- the destruction and bifurcations of invariant curves and invariant tori,
- the influence of resonances on the dynamics,
- the stability/instability of planetary systems.
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