Abstract
The paper proposes, for the first time, a closed form of the Baker–Campbell–Hausdorff–Dynkin (BCHD) formula in the particular case of the Lie algebra of rigid body displacements. For this purpose, the structure of the Lie group of the rigid body displacements S E ( 3 ) and the properties of its Lie algebra s e ( 3 ) are used. In addition, a new solution to this problem in dual Lie algebra of dual vectors is delivered using the isomorphism between the Lie group S E ( 3 ) and the Lie group of the orthogonal dual tensors.
Highlights
The BCHD theorem—named after the British mathematician Henry Frederick Baker (1866–1956), the Irish mathematician John Edward Campbell (1862–1924), the German mathematician FelixHausdorff (1868–1942) and the Soviet, than American mathematician Eugene Borisovich Dynkin (1924–2014)—is well known as one of the most interesting outcome of the theory of groups of transformations
This paper proves the existence of the closed form of the Baker–Campbell–Hausdorff–Dynkin formula for the Lie algebra of rigid body displacement
Dynkin indicated in 1947 a general procedure to determine the expansion, this formula is of difficult implementation for the case of non-nilpotent operators
Summary
The BCHD theorem—named after the British mathematician Henry Frederick Baker (1866–1956), the Irish mathematician John Edward Campbell (1862–1924), the German mathematician Felix. For the Lie group of rigid body displacements SE(3) a closed form BCHD formula was first given by the authors of this work in the conference paper [35]. This paper proves the existence of the closed form of the Baker–Campbell–Hausdorff–Dynkin formula for the Lie algebra of rigid body displacement. The structure of the paper is as follows: in chapter two a new locally closed form solution is given for the BCHD formula in the case of the rotation group SO(3). A new closed form BCHD formula for Lie algebra so(3) will be given in the following theorem. The author’s searches in the literature did not report the existence of other closed form of the BCHD formula for the Lie algebra of the rigid body displacements. The closed form BCHD formula for Lie algebra se(3) is given by the following theorem.
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