توصيفگر ها :
مختصات عمل-زاويه , سيستم هميلتوني , انتگرال پذيري ليوويل , نگاشت پراكندگي , مونودرومي پراكندگي
چكيده فارسي :
مساله دو مركز ثابت توسط اويلر در سال 1760 معرفي شده است. به طوري كه هم در مكانيك سماوي و در دنياي ميكروسكوپي و
هم چنين در مساله بيليارد نقش بازي مي كند. در اين پايان نامه، ما مساله فضايي در مورد نيرو هاي دلخواه مركزها (هر دو مثبت و منفي)
را مطالعه مي كنيم. با تركيب روش هاي نظريه پراكندگي و انتگرال پذيري ليوويل، ما نشان مي دهيم كه اين مساله فضايي ديناميك هاي
پراكندگي غير بديهي توپولوژيكي دارد، چنان كه مونودرومي پراكندگي مشخص مي شود. روشي كه ما در اين پايان نامه معرفي مي كنيم،
به طور كلي تر براي سيستم هاي پراكندگي كه در نمايش ليوويل انتگرال پذير هستند، بكار مي رود.
چكيده انگليسي :
The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in
celestial mechanics and in the microscopic world and so billiard game. In the present thesis we study the spatial
problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from
scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering
dynamics, which we identify as scattering monodromy. The approach that we introduce in this thesis applies more
generally to scattering systems that are integrable in the Liouville sense.
In the present work we will be interested in the spatial Euler problem. For us, it will be important that this problem is
a Hamiltonian system with two additional structures: it is a scattering system and it is also integrable in the Liouville
sense. The structure of a scattering system comes from the fact that the potential V (q) → 0, q → ∞, decays at
infinity sufficiently fast (is of long range). It allows one to compare a given set of initial conditions at t = −∞ with
the outcomes at t = +∞.
In fact, we prove the following theorems from the above paper.
Theorem 1: Among all Kepler Hamiltonians only,
Hr1 =
1
2
k p k
2 −
µ1 − µ2
r1
, (8-6)
Hr2 =
1
2
k p k
2 −
µ2 − µ1
r2
.
are reference Hamiltonians of the Euler problem F = (H, Lz, G). In particular, the free Hamiltonian is a reference
Hamiltonian of the Euler problem only in the case µ1 = µ2.
Theorem 2: The scattering monodromy matrices Mi
in the direction γi with respect to Hr1
in SL(3, Z) are conjugated to one of the following matrices,
M1 =
1 0 0
0 1 1
0 0 1
, (9-6)
M2 =
1 0 −1
0 1 1
0 0 1
, (10-6)
69 نامه واژه M3 =
1 0 1
0 1 0
0 0 1
