Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published 1989 | Published
Book Section - Chapter Open

A block diagonalization theorem in the energy-momentum method

Abstract

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.

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.

Attached Files

Published - MaSiLePo1989.pdf

Files

MaSiLePo1989.pdf
Files (484.4 kB)
Name Size Download all
md5:dee3e1cf98baeae3034497d36b5989e4
484.4 kB Preview Download

Additional details

Created:
August 19, 2023
Modified:
January 13, 2024