On the Subset of Normality Equations

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this case some part of normality equations describe generalized Legendre transformation itself irrespective to that Newtonian dynamical system, for which others are written. In present paper this smaller part of normality equations is studied. 1. Newtonian dynamical systems and generalized Legendre transformation. Let M be smooth manifold of dimension n. We say that the motion of a point p = p(t) of this manifold obeys Newton’s second low if in local chart it is described by the following ordinary differential equations: ẋ = v, v̇ = Φ(x, . . . , x, v, . . . , v). (1.1) Here v, . . . , v are components of velocity vector v of moving point. Its mass is assumed to be equal to unity: m = 1. Therefore functions Φ, . . . , Φ in (1.1) play the role of force vector, though, unlike v, . . . , v, they are not components of tangent vector to M . Not always, but very often differential equations (1.1) are associated with some extremal principle and hence are given implicitly by Euler-Lagrange equations: ẋ = v, d dt ( ∂L ∂vi ) = ∂L ∂xi In this case they can be transformed to Hamiltonian form ẋ = ∂H ∂pi , ṗi = − ∂H ∂xi by means of classical Legendre transformation that relates velocity vector v and momentum covector p according to the following formula: pi = ∂L ∂vi . (1.2) 1991 Mathematics Subject Classification. 53D50, 70G10, 70G45.