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
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Additional details
- Eprint ID
- 19857
- Resolver ID
- CaltechAUTHORS:20100910-080520844
- F49620-87-C0118
- Air Force Office of Scientific Research AFOSR/DARPA
- 2-DJA-544
- Air Force Office of Scientific Research (AFOSR)
- 2-DJA-771
- Air Force Office of Scientific Research (AFOSR)
- MSI/Cornell University
- Created
-
2010-09-15Created from EPrint's datestamp field
- Updated
-
2019-10-03Created from EPrint's last_modified field
- Series Name
- Comtemporary Mathematics
- Series Volume or Issue Number
- 97