Marsden, J. E. and Simo, J. C. and Lewis, D. and Posbergh, T. A. (1989) A block diagonalization theorem in the energy-momentum method. In: Dynamics and control of multibody systems : proceedings of the AMS-IMS-SIAM joint summer conference. Comtemporary Mathematics (97). American Mathematical Society , pp. 297-313. ISBN 0821851047 http://resolver.caltech.edu/CaltechAUTHORS:20100910-080520844
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We prove a geometric generalization of a block diagonalization theorem first found by the authors for rotating elastic rods. The result here is given in the general context of simple mechanical systems with a symmetry group acting by isometries on a configuration manifold. The result provides a choice of variables for linearized dynamics at a relative equilibrium which block diagonalizes the second variation of an augmented energy these variables effectively separate the rotational and internal vibrational modes. The second variation of the effective Hamiltonian is block diagonal. separating the modes completely. while the symplectic form has an off diagonal term which represents the dynamic interaction between these modes. Otherwise, the symplectic form is in a type of normal form. The result sets the stage for the development of useful criteria for bifurcation as well as the stability criteria found here. In addition, the techniques should apply to other systems as well, such as rotating fluid masses.
|Item Type:||Book Section|
|Additional Information:||© 1989 American Mathematical Society. Research partially supported by AFOSR/DARPA contract F49620-87-COl18 and MSI at Cornell University. Research partially supported by AFOSR contract 2-DJA-S44 and 2-DJA-771. Research partially supported by MSI at Cornell University. We thank Tony Bloch, P.S. Krishnaprasad, Richard Montgomery, George Patrick, and Tudor Ratiu for helpful comments.|
|Classification Code:||AMS Subject Classification: 58P, 70H|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||15 Sep 2010 21:39|
|Last Modified:||26 Dec 2012 12:24|
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