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CATEGORIES:DAMTP Astrophysics Seminars
SUMMARY:Adiabatic approximation in studies of mean motion
resonances in celestial mechanics - Prof. Vladisla
v Sidorenko (Keldysh Institute\, Moscow)
DTSTART;TZID=Europe/London:20200309T140000
DTEND;TZID=Europe/London:20200309T150000
UID:TALK136474AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/136474
DESCRIPTION:In the last decades of the XXth century\, it becam
e clear that when studying resonant effects in dyn
amics of celestial bodies\, it is useful to pay at
tention to the behavior of approximate integrals o
f motion called adiabatic invariants (Henrard and
Lemaitre 1983\; Wisdom 1985). The standard scheme
of the adiabatic approximation in the investigatio
ns of mean motion resonances (MMR) as a first step
involves averaging over the fastest dynamic proce
ss\, i.e. over the orbital motion of the objects i
n commensurability. In averaged equations of motio
n one should take a subsystem that describes the p
rocess of “intermediate” time scale - the variatio
n of the resonant angle. This subsystem can be int
erpreted as a Hamiltonian system with one degree o
f freedom\, depending on other variables as slowly
varying parameters. Consequently\, the value of t
he “action” variable for this subsystem will be an
adiabatic invariant (AI). Studying then the prope
rties of level surfaces of AI in the subspace of t
he slowest variables\, we can draw conclusions abo
ut the secular evolution of the orbits of celestia
l bodies in MMR. More delicate situation arises in
the case of nonuniqueness of the resonant modes a
llowed by the system (Sidorenko et al. 2014\; Sido
renko 2018). In particular\, in this case it is ne
cessary to identify regions in the phase space\, w
here resonant modes can coexist\, to compare the p
robabilities of the capture into different modes\,
and to analyze the possibility of a transition be
tween these modes. Fortunately\, the theory of AI
allows to do almost all of this. All phenomena und
er the discussion are illustrated by examples of t
heir possible implementation in the dynamics of re
al celestial bodies.
LOCATION:MR14\, Centre for Mathematical Sciences\, Wilberf
orce Road\, Cambridge
CONTACT:Chris Hamilton
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