Ivancevic_Applied-Diff-Geom

258 Applied Differential Geometry: A Modern IntroductionwhereW ik ≡∂2 L∂ ˙q i ∂ ˙q k and α i ≡ − ∂2 L∂ ˙q i ∂q k˙qk + ∂L∂q i .We consider the general case where the mass matrix, or Hessian (W ij ),may be a singular matrix. In this case there exists a kernel for the pull–backFL ∗ of the Legendre map, i.e., fibre–derivative FL, from the velocity phase–space manifold T M (tangent bundle of the biodynamical manifold M) tothe momentum phase–space manifold T ∗ M (cotangent bundle of M). Thiskernel is spanned by the vector–fieldsΓ µ = γ i ∂µ∂ ˙q i ,where γ i µ are a basis for the null vectors of W ij . The Lagrangian time–evolution differential operator can therefore be expressed as:X = ∂ t + ˙q k ∂∂q k + ak (q, ˙q) ∂∂ ˙q k + λµ Γ µ ≡ X o + λ µ Γ µ ,where a k are functions which are determined by the formalism, and λ µ arearbitrary functions. It is not necessary to use the Hamiltonian techniqueto find the Γ µ , but it does facilitate the calculation:( ) ∂φµγ i µ = FL ∗ , (3.75)∂p iwhere the φ µ are the Hamiltonian primary first class constraints.Notice that the highest derivative in (3.74), ¨q i , appears linearly. BecauseδL is a symmetry, (3.74) is identically satisfied, and therefore the coefficientof ¨q i vanishes:We contract with a null vector γ i µ to find thatW ik δq k − ∂G = 0. (3.76)∂ ˙qiΓ µ G = 0.It follows that G is projectable to a function G H in T ∗ Q; that is, it is thepull–back of a function (not necessarily unique) in T ∗ Q:G = FL ∗ (G H ).