# The Lagrange rigid body motion

Annales de l'institut Fourier (1982)

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

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topRatiu, 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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- [8] J. MARSDEN. Geometric methods in mathematical physics, CBMS-NSF. Regional Conference Series, No. 37, SIAM (1981). Zbl0485.70001MR82j:58046
- [9] J. MARSDEN, A. WEINSTEIN, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130. Zbl0327.58005MR53 #6633
- [10] P. van MOERBEKE, D. MUMFORD, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154. Zbl0502.58032MR80e:58028
- [11] T. RATIU, Involution theorems, Springer Lecture Notes, n° 775 (1980), 219-257. Zbl0435.58014MR82f:58047
- [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] E. WHITTAKER, Analytical dynamics, fourth edition, Cambridge University Press (1965).

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