Ivancevic_Applied-Diff-Geom

620 Applied Differential Geometry: A Modern Introductionof T Q. It is described by a second–order dynamical equationon Q which preserves the subbundle (4.151), i.e.,¨q α = Ξ α (q µ , ˙q µ ) (4.152)( ˙q α ∂ α + Ξ α ˙∂α )(g µν ˙q µ ˙q ν − 1) = 0, ( ˙∂ α = ∂/∂ ˙q α ).This condition holds if the r.h.s. of the equation (4.152) takes the formΞ α = Γ α µν ˙q µ ˙q ν + F α ,where Γ α µν are Christoffel symbols of a metric g, while F α obey the relationg µν F µ ˙q ν = 0. In particular, if the dynamical equation (4.152) is a geodesicequation,¨q α = K α µ ˙q µwith respect to a (non-linear) connection on the tangent bundle T Q → Q,this connections splits into the sumK = dq α ⊗ (∂ α + K µ α ˙∂ µ ),K α µ = Γ α µν ˙q ν + F α µ (4.153)of the Levi–Civita connection of g and a soldering formF = g λν F µν dq µ ⊗ ˙∂ α , F µν = −F νµ .As was mentioned above, the momentum phase–space of RM on Q isthe cotangent bundle T ∗ Q provided with the symplectic form Ω (4.144).Let H be a smooth real function on T ∗ Q such that the map˜H : T ∗ Q −→ T Q, ˙q µ = ∂ µ H (4.154)is a bundle isomorphism. Then the inverse image N = ˜H −1 (W g ) of thesubbundle of hyperboloids W g (4.151) is a one-codimensional (consequently,coisotropic) closed imbedded subbundle of T ∗ Q given by the constraintH T = 0 (4.145). We say that H is a relativistic Hamiltonian if the Poissonbracket {H, H T } vanishes on N. This means that the Hamiltonian vector–fieldγ = ∂ α H∂ α − ∂ α H∂ α (4.155)