Algebraic Topology, Poznan 1989 by Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

By Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

As a part of the medical job in reference to the seventieth birthday of the Adam Mickiewicz collage in Poznan, a world convention on algebraic topology was once held. within the ensuing lawsuits quantity, the emphasis is on large survey papers, a few provided on the convention, a few written hence.

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10». 7 Now consider the group E(3) of rigid body motion in three-dimensional Euclidean space and its Lie algebra e(3). It is obvious that dimE(3) = 6. Choose 3 The Motion of a Rigid Body 31 some orthonormal frame attached to the body with the origin at some point O. Let VI, V2, V3 be the left-invariant vector fields on £(3) which correspond to the permanent rotations with unit angular velocity about the three orthogonal axes of this frame. Analogously, to the body motions with the velocity vector of the origin 0 parallel to these axes there correspond the vector fields UI, U2, U3.

4) must be considered together with the equation 8 Some Problems of Mathematical Physics 49 of continuity I = 0. 6) + TJcos(Ar) + ( . Here 1/1', ~, TJ, ( are still unknown functions of x, y. The motion of fluid particles is described by the Hamilton equations x. If == H')" y= H -H~; = 1/1' - EX cos(Ar) . 7) can be easily integrated by quadratures. 5) into an identity. o Y = . AY + const. 11) where f:::, is the Laplace operator. 1I1 '1" xxy . 11) is fulfilled. 10) can be checked in a similar way.

5) conserves the standard measure in 1R6 = {w, Yl. As noticed in [44], if this condition is not fulfilled, then for b = this system does not possess even an absolutely continuous (with respect to the Lebesgue measure in 1R6 = {w, y)) invariant measure. Therefore, we suppose that the vector a is generally directed along one of the principal inertia axes; without loss of generality we put a = (0,0, 1). 5) were solved in [109] for the case when a and b are orthogonal. Now consider the case b = ca, £ i= 0.

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