Ivancevic_Applied-Diff-Geom

374 Applied Differential Geometry: A Modern Introduction(For example, let G = SO(3) and (M, ω) = R 6 , ∑ 3i=1 dp i ∧ dq), i andthe action of G on R 6 is given by φ : (R, (q, p)) ↦→ (R q , R p ). Then themomentum map is the well known angular momentum and for each µ ∈g ∗ ≃ R 3 µ ≠ 0, G µ ≃ S 1 and the reduced phase–space (M µ , ω µ ) is (T ∗ R,ω = dp i ∧ dq i ), so that dim (M µ ) = dim (M) − dim (G) − dim (G µ ). Thisreduction is in celestial mechanics called by Jacobi ’the elimination of thenodes’.The equations of motion: f ˙ = {f, H} ω on M reduce to the equations ofmotion: f ˙ µ = {f µ , H µ } ωµ on M µ (see [Marsden and Ratiu (1999)]).3.12.4 Multisymplectic GeometryMultisymplectic geometry constitutes the general framework for a geometrical,covariant formulation of classical field theory. Here, covariant formulationmeans that space–like and time–like directions on a given space–time be treated on equal footing. With this principle, one can constructa covariant form of the Legendre transformation which associates to everyfield variable as many conjugated momenta, the multimomenta, asthere are space–time dimensions. These, together with the field variables,those of nD space–time, and an extra variable, the energy variable,span the multiphase–space [Kijowski and Szczyrba (1976)]. For arecent exposition on the differential geometry of this construction, see [Gotay(1991a)]. Multiphase–space, together with a closed, nondegeneratedifferential (n + 1)−form, the multisymplectic form, is an example of amultisymplectic manifold. Among the achievements of the multisymplecticapproach is a geometrical formulation of the relation of infinitesimalsymmetries and covariantly conserved quantities (Noether’s Theorem), see[León et. al. (2004)] for a recent review, and [Gotay and Marsden (1992);Forger and Römer (2004)] for a clarification of the improvement techniques(‘Belinfante–Rosenfeld formula’) of the energy–momentum tensor in classicalfield theory. Multisymplectic geometry also gives convenient sets ofvariational integrators for the numerical study of partial differential equations[Marsden et. al. (1998)].Since in multisymplectic geometry, the symplectic two–form of classicalmechanics is replaced by a closed differential form of higher tensordegree, multivector–fields and differential forms have their natural appearance.(See [Paufler and Römer (2002)] for an interpretation of multivector–fields as describing solutions to field equations infinitesimally.) Multivector–