Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 259This important property is valid for any conserved quantity associatedwith a Noether symmetry. Observe that G H is determined up to the additionof linear combinations of the primary constraints. Substitution of thisresult in (3.76) gives[ ( )]W ik δq k − FL ∗ ∂GH= 0,∂p kand so the brackets enclose a null vector of W ik :( )δq i − FL ∗ ∂GH= r µ γ i∂pµ, (3.77)ifor some r µ (t, q, ˙q).We shall investigate the projectability of variations generated by diffeomorphismsin the following section. Assume that an infinitesimal transformationδq i is projectable:Γ µ δq i = 0.If δq i is projectable, so must be r µ , so that r µ = FL ∗ (r µ H). Then, using(3.75) and (3.77), we see that( ∂(GHδq i = FL ∗ + r µ H φ )µ).∂p iWe now redefine G H to absorb the piece r µ H φ µ, and from now on we willhave( )δq i = FL ∗ ∂GH.∂p iDefineˆp i = ∂L∂ ˙q i ;after eliminating (3.76) times ¨q i from (3.74), we get( ∂L∂q i − ∂ ˆp )˙qk i∂q k FL ∗ ( ∂G H) + ˙q i ∂∂p i ∂q i FL∗ (G H ) + FL ∗ ∂ t G H = 0,which simplifies to∂L∂q i FL∗ ( ∂G H∂p i) + ˙q i FL ∗ ( ∂G H∂q i ) + FL∗ ∂ t G H = 0. (3.78)