Reduction theory and the Lagrange-Routh equations by Marsden J.E., Ratiu T.S., Scheurle J.

By Marsden J.E., Ratiu T.S., Scheurle J.

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Here, the θa are cyclic variables and the momentum map constraint reads µa = gaα x˙ α + gab θ˙b . In this case, the components of the mechanical connection are Aaα = g ab gbα , the locked inertia tensor is Iab = gab , and the Routhian is Rµ = 12 gαβ − gaα g ab gbβ x˙ α x˙ β − Vµ (x), where the amended potential is Vµ (x) = V (x) + 12 g ab µa µb . 9 ∂Aaβ ∂Aaα − . ∂xα ∂xβ Examples The Rigid Body. In this case, the Lagrange–Routh equations reduce to a coadjoint orbit equation and simply state that the equations are Hamiltonian on the coadjoint orbit.

Littlejohn and Reinch [1997] (and other recent references as well) have carried on this work in a very interesting way. Landsman [1995, 1998] also uses reduction theory in an interesting way. Multisymplectic Geometry and Variational Integrators. There have been significant developments in multisymplectic geometry that have led to interesting integration algorithms, as in Marsden, Patrick and Shkoller [1998] and Marsden and Shkoller [1999]. There is also all the work on reduction for discrete mechanics which also takes a variational view, following Veselov [1988].

A, 133, 134–139. , J. E. Marsden, I. Stewart and M. Dellnitz [1995], The constrained Liapunov Schmidt procedure and periodic orbits, Fields Inst. , 4, 81–127. Golubitsky, M. and D. Schaeffer [1985], Singularities and Groups in Bifurcation Theory. Vol. 1, Applied Mathematical Sciences, 69, Springer-Verlag. Golubitsky, M. and I. Stewart [1987], Generic bifurcation of Hamiltonian systems with symmetry, Physica D, 24, 391–405. , I. Stewart and D. Schaeffer [1988], Singularities and Groups in Bifurcation Theory.

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