Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 257Now, a generalized vector–field v is a generalized infinitesimal symmetry ofa system S of differential equations∆ r [u] = ∆ r (x, u (n) ) = 0,(r = 1, ..., l),iffpr v[∆ r ] = 0for every smooth solution m u = f(x) [Olver (1986)].For example, consider the heat equation∆[u] = u t − u xx = 0.The generalized vector–field v = u x∂∂uThushas prolongation∂pr v = u x∂u + u ∂ ∂ ∂xx + u xt + u xxx + ...∂u x ∂u t ∂u xxpr v(∆) = u xt − u xxx = D x (u t − u xx ) = D x ∆,and hence v is a generalized symmetry of the heat equation.3.9.3.1 Noether SymmetriesHere we present some results about Noether symmetries, in particular forthe first–order Lagrangians L(q, ˙q) (see [Batlle et. al. (1989); Pons et. al.(2000)]). We start with a Noether–Lagrangian symmetry,δL = ˙ F ,and we will investigate the conversion of this symmetry to the Hamiltonianformalism. Definingwe can writeG = (∂L/∂ ˙q i ) δq i − F,δ i L δq i + Ġ = 0, (3.74)where δ i L is the Euler–Lagrangian functional derivative of L,δ i L = α i − W ik ¨q k ,