Ivancevic_Applied-Diff-Geom

906 Applied Differential Geometry: A Modern IntroductionRecall the following relationship between first–order dynamical equations,connections, multivector–fields and evolution operators on a fibrebundle.(i) Let π : Q → X be a fibre bundle coordinated by (x µ , q a ). As a sectionγ : Q −→ J 1 (X, Q) of the 1–jet bundle J 1 (X, Q) → Q, any connectionγ = dx µ ⊗ (∂ µ + γ a µ∂ a ), (5.287)on Q → X defines the first–order differential operatorD : J 1 (X, Q) → T ∗ X ⊗ V Q, (x µ , q a , q a µ) → (x µ , q a , q a µ − γ a µ(x ν , q b ))(5.288)on Q → X called the covariant differential with respect to γ. The kernel ofthis differential operator is a closed imbedded subbundle of J 1 (X, Q) → X,given by the first–order dynamical equationq a µ − γ a µ(x ν , q b ) = 0 (5.289)on a fibre bundle Q → X. Conversely, any first–order dynamical equationon Q → X is of this type.(ii) Let HQ ⊂ T Q be the horizontal distribution defined by a connectionγ. If X is orientable, there exists a nowhere vanishing global section of theexterior product ∧ n HQ → Q. It is a locally decomposable π−transversen−vector–field on Q. Conversely, every multivector–field of this type onQ → X induces a connection and, consequently, a first–order dynamicalequation on this fibre bundle [Echeverría et. al. (1998)].(iii) Given a first–order dynamical equation γ on a fibre bundle Q → X,the corresponding evolution operator d γ is defined as the pull–back d γ ontothe shell of the horizontal differentiald H = dx µ (∂ µ + q a µ∂ a )acting on smooth real functions on Q. It readsd γ f = (∂ µ + γ a µ∂ a )fdx µ , (f ∈ C ∞ (Q)). (5.290)This expression shows that d γ is projected onto Q, and it is a first–order differentialoperator on functions on Q. In particular, if a function f obeys theevolution equation d γ f = 0, it is constant on any solution of the dynamicalequation (5.289).In Hamiltonian dynamics on Q, a problem is to represent the evolutionoperator (5.290) as a bracket of f with some exterior form on Q.