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The class additional to point and tangent transformation groups is formed by groups of “higher-order symmetries,” or by Lie–B¨acklund groups [5, 73, 107]. For formal groups, just as for Lie point and tangent transformation groups, one can introduce the notion of invariants and invariant manifolds. 54). 54). 50) and all transformations of the formal group. A criterion for a manifold to be invariant can also be written with the use of the group operator (see [73, 107]): Xφ(z) φ(z)=0 = 0. 1. 50), a special place is occupied by point and contact groups and by higherorder symmetries.

80) Proof. The invariance condition follows directly from the action of X prolonged to the differentials dt and dq i , i = 1, . . , n: X = ξ(t, q, p) ∂ ∂ ∂ ∂ ∂ +η i (t, q, p) i +ζi (t, q, p) +D(η i )dt . +D(ξ)dt ∂t ∂q ∂pi ∂(dt) ∂(dq i ) An application of this equation gives X(pi dq i − H dt) = ζi q˙i + pi D(η i ) − X(H) − HD(ξ) dt = 0. C OROLLARY. 81) that if the Lagrangian is invariant, then the Hamiltonian is also invariant with respect to the same group. 81). 1. B RIEF INTRODUCTION TO L IE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS lxi Proof.

Moreover, to reduce a given system of equations, one can use differential rather than finite invariants. In this case, differential-invariant solutions are constructed with the use of relations between differential invariants. 13. Contact symmetries of differential equations Contact (tangent) transformations have been widely used in mechanics and the theory of differential equations for a long time. Sophus Lie used the group of contact transformations of the form x∗i = f i (x, u, u1 , a), u∗ = g(x, u, u1 , a), i = 1, 2, .

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