Ivancevic_Applied-Diff-Geom

852 Applied Differential Geometry: A Modern Introductionated with L, and H L = ̂L ∗ H N .For any Hamiltonian form H weakly associated with an almost regularLagrangian L, every solution of the Hamiltonian equations which lives in theLagrangian constraint space N L is a solution of the constrained Hamiltonianequations (5.134).Using the equality H L = ̂L ∗ H N , one can show that the constrainedHamiltonian equations (5.134) are equivalent to the Hamilton–de Donderequations (5.115) and are quasi–equivalent to the Cartan equations (5.114)[Giachetta et. al. (1997); Mangiarotti and Sardanashvily (1998); Lopez andMarsden (2003)].5.6.11 Quadratic Degenerate Lagrangian SystemsGiven a configuration bundle Q → R, let us consider a quadratic LagrangianL which has the coordinate expressionL = 1 2 a ij ˙q i ˙q j + b i ˙q i + c, (5.142)where a, b and c are local functions on Q. This property is coordinate–independent due to the affine transformation law of the coordinates ˙q i .The associated Legendre mapp i ◦ ̂L = a ij ˙q j + b i (5.143)is an affine map over Q. It defines the corresponding linear mapL : V Q −→ V ∗ Q, p i ◦ L = a ij ˙q j . (5.144)Let the Lagrangian L (5.142) be almost regular, i.e., the matrix functiona ij is of constant rank. Then the Lagrangian constraint space N L =̂L(J 1 (R, Q)) is an affine subbundle of the bundle V ∗ Q −→ Q, modelled overthe vector subbundle N L (5.144) of V ∗ Q −→ Q. Hence, N L −→ Q has aglobal section. For the sake of simplicity, let us assume that it is the canonicalzero section ̂0(Q) of V ∗ Q −→ Q. Then N L = N L . Therefore, the kernelof the Legendre map (5.143) is an affine subbundle of the affine jet bundleJ 1 (R, Q) −→ Q, modelled over the kernel of the linear map L (5.144). Thenthere exists a connection Γ on the fibre bundle Q −→ R, given byΓ : Q −→ Ker ̂L ⊂ J 1 (R, Q), with a ij Γ j µ + b i = 0.