# Algebraic Topology, Gottingen 1984 by L. Smith

By L. Smith

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**Example text**

It is easy to verify that all the conditions indicated above are valid. Just for the sake of simplicity, consider the case '3 = O. 7) tum into Let 8 tend to zero. Then Eqs. 8) form a closed system defining the restricted problem of the rotation of a heavy rigid body with a fixed point. The meaning of this restricted problem is the following. As 8 ~ 0, the rigid body degenerates into a line segment rotating around the fixed point according to the spherical pendulum law. The well-known picture of spherical pendulum motion gives us a clear idea of the nutation and the precession phenomena in rigid body dynamics.

Note that Eqs. c takes place. 13) is given by the expression H = Ho + vH I + o(v) , Ho = TJ2/2+cos~, HI = sin~sin(J2ht). 14) For v = 0, we come back to the integrable problem-the mathematical pendulum. 16) are absent: B = 0. We can put 2L = (fw, w) + (1- l v, v) , where 1,1 are positive definite symmetric operators, and introduce the variables w, P = 1- l v. 14) take the following form IriJ = Iw x w + P x 1p, p = P x w. 15) These can be interpreted as the Euler-Poinsot equations governing rigid body motion in a force field with potential (1 p, p) /2.

8. 50 I Hamiltonian Mechanics Fig. 8. 3 In the paper of Contopoulos [60] dealing with models of galaxies some Hamiltonian systems in a neighborhood of an equilibrium position were considered. These systems were assumed to admit resonance relations between frequencies. The simplest system that has such a property and describes the star motion in a galaxy with cylindrical symmetry is defined by the Hamiltonian H 2 2 2 2 = 2:1 (2 PI + P2 + ql + q2 + 2ql q2 2 3) . 12) This system was numerically investigated in detail by Henon and Heiles [99].