The Lagrange rigid body motion

Tudor Ratiu; P. van Moerbeke

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 1, page 211-234
  • ISSN: 0373-0956

Abstract

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We discuss the motion of the three-dimensional rigid body about a fixed point under the influence of gravity, more specifically from the point of view of its symplectic structures and its constants of the motion. An obvious symmetry reduces the problem to a Hamiltonian flow on a four-dimensional submanifold of s o ( 3 ) × s o ( 3 ) ; they are the customary Euler-Poisson equations. This symplectic manifold can also be regarded as a coadjoint orbit of the Lie algebra of the semi-direct product group S O ( 3 ) × s o ( 3 ) with its natural symplectic structure. Finally the Lagrange motion is also a Hamiltonian flow on a coadjoint orbit in a kac-Moody Lie algebra; this approach has the virtue that the linearization in terms of elliptic integrals follows at once from a general theorem.

How to cite

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Ratiu, Tudor, and van Moerbeke, P.. "The Lagrange rigid body motion." Annales de l'institut Fourier 32.1 (1982): 211-234. <http://eudml.org/doc/74526>.

@article{Ratiu1982,
abstract = {We discuss the motion of the three-dimensional rigid body about a fixed point under the influence of gravity, more specifically from the point of view of its symplectic structures and its constants of the motion. An obvious symmetry reduces the problem to a Hamiltonian flow on a four-dimensional submanifold of $so(3)\times so(3)$; they are the customary Euler-Poisson equations. This symplectic manifold can also be regarded as a coadjoint orbit of the Lie algebra of the semi-direct product group $SO(3)\times so(3)$ with its natural symplectic structure. Finally the Lagrange motion is also a Hamiltonian flow on a coadjoint orbit in a kac-Moody Lie algebra; this approach has the virtue that the linearization in terms of elliptic integrals follows at once from a general theorem.},
author = {Ratiu, Tudor, van Moerbeke, P.},
journal = {Annales de l'institut Fourier},
keywords = {motion of the three-dimensional rigid body; fixed point under the influence of gravity; symplectic structures; constants of the motion; Hamiltonian flow on a four-dimensional submanifold of ; Euler-Poisson equations; coadjoint orbit of the Lie algebra of the semi-direct product group ; Hamiltonian flow on a coadjoint orbit in a Kac-Moody Lie algebra; elliptic integrals},
language = {eng},
number = {1},
pages = {211-234},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Lagrange rigid body motion},
url = {http://eudml.org/doc/74526},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Ratiu, Tudor
AU - van Moerbeke, P.
TI - The Lagrange rigid body motion
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 1
SP - 211
EP - 234
AB - We discuss the motion of the three-dimensional rigid body about a fixed point under the influence of gravity, more specifically from the point of view of its symplectic structures and its constants of the motion. An obvious symmetry reduces the problem to a Hamiltonian flow on a four-dimensional submanifold of $so(3)\times so(3)$; they are the customary Euler-Poisson equations. This symplectic manifold can also be regarded as a coadjoint orbit of the Lie algebra of the semi-direct product group $SO(3)\times so(3)$ with its natural symplectic structure. Finally the Lagrange motion is also a Hamiltonian flow on a coadjoint orbit in a kac-Moody Lie algebra; this approach has the virtue that the linearization in terms of elliptic integrals follows at once from a general theorem.
LA - eng
KW - motion of the three-dimensional rigid body; fixed point under the influence of gravity; symplectic structures; constants of the motion; Hamiltonian flow on a four-dimensional submanifold of ; Euler-Poisson equations; coadjoint orbit of the Lie algebra of the semi-direct product group ; Hamiltonian flow on a coadjoint orbit in a Kac-Moody Lie algebra; elliptic integrals
UR - http://eudml.org/doc/74526
ER -

References

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  2. [2] M. ADLER, On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV-type equations, Inventiones math., (1979), 219-248. Zbl0393.35058
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  6. [6] A. IACOB, Topological methods in mechanics (in Romanian), Bucharest (1973). 
  7. [7] A. IACOB, S. STERNBERG, Coadjoint structures, solitons, and integrability, Springer Lecture Notes in Physics, No. 120 (1980). 
  8. [8] J. MARSDEN. Geometric methods in mathematical physics, CBMS-NSF. Regional Conference Series, No. 37, SIAM (1981). Zbl0485.70001MR82j:58046
  9. [9] J. MARSDEN, A. WEINSTEIN, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130. Zbl0327.58005MR53 #6633
  10. [10] P. van MOERBEKE, D. MUMFORD, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154. Zbl0502.58032MR80e:58028
  11. [11] T. RATIU, Involution theorems, Springer Lecture Notes, n° 775 (1980), 219-257. Zbl0435.58014MR82f:58047
  12. [12] T. RATIU, Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, American Journal of Math., Vol. 103, No. 3 (1982). Zbl0509.58026
  13. [13] E. WHITTAKER, Analytical dynamics, fourth edition, Cambridge University Press (1965). 

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