Ivancevic_Applied-Diff-Geom

854 Applied Differential Geometry: A Modern IntroductionNote that, with respect to the coordinates S i α and F i α (5.150), the Lagrangian(5.142) readsL = 1 2 a ijF i F j + c ′ ,while the Lagrangian constraint space is given by the reducible constraintsR i = p i − a ij σ jk0 p k = 0.Given the linear map σ and the connection Γ as defined above, let usconsider the affine Hamiltonian mapΦ = ̂Γ + σ : V ∗ Q −→ J 1 (R, Q), Φ i = Γ i + σ ij p j , (5.153)and the Hamiltonian formH = H Φ + Φ ∗ L = p i dq i − [p i Γ i + 1 2 σ 0 ij p i p j + σ 1 ij p i p j − c ′ ]dt(5.154)= (R i + P i )dq i − [(R i + P i )Γ i + 1 2 σij 0 P iP j + σ ij1 p ip j − c ′ ]dt.In particular, if σ 1 is non–degenerate, so is the Hamiltonian form H.The Hamiltonian forms of the type H, parameterized by connections Γ,are weakly associated with the Lagrangian (5.142) and constitute a completeset. Then H is weakly associated with L. Let us write the correspondingHamiltonian equations (5.121) for a section r of the Legendre bundleV ∗ Q −→ R. They areċ = (̂Γ + σ) ◦ r, c = π Q ◦ r. (5.155)Due to the surjections S and F (5.150), the Hamiltonian equations (5.155)break in two partsS ◦ ċ = Γ ◦ c, ṙ i − σ ik (a kj ṙ j + b k ) = Γ i ◦ c, (5.156)F ◦ ċ = σ ◦ r, σ ik (a kj ṙ j + b k ) = σ ik r k . (5.157)Let c be an arbitrary section of Q −→ R, e.g., a solution of the Lagrangianequations. There exists a connection Γ such that the relation (5.156) holds,namely, Γ = S ◦ Γ ′ , where Γ ′ is a connection on Q −→ R which has c as anintegral section.If σ 1 = 0, then Φ = Ĥ and the Hamiltonian forms H are associated withthe Lagrangian (5.142). Thus, for different σ 1 , we have different completesets of Hamiltonian forms H, which differ from each other in the termυ i R i , where υ are vertical vector–fields (5.145). This term vanishes on the