Multidimensional dynamical systems that admit normal shift

Abstract

Abstract. The dynamical systems of the formr = F(r,ṙ) in R n accepting the normal shift are considered. The concept of weak normality for them is introduced. The partial differential equations for the force field F(r,ṙ) of the dynamical systems with weak and complete normality are derived.

Newtonian Dynamical Systems Admitting Normal Blow-Up of Points

“…The matter is that the equations (3.32) are exactly the same as weak normality equations, which arose in considering Newtonian dynamical systems admitting the normal shift of hypersurfaces. For the case M = R 2 they were first derived in [1], then in [3] they were generalized for the case M = R n . In form of (3.32) corresponding to the case of arbitrary Riemannian manifold these equations were derived in [6] (see also Chapter V in thesis [17]).…”

“…This became the origin for the theory of dynamical systems admitting the normal shift. It was developed in the series of papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. On the base of these papers two theses were prepared: thesis for the degree of Doctor of Sciences in Russia [17] and thesis for the degree of Candidate of Sciences in Russia [18].…”

“…In [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] we restricted ourselves to the case of smooth hypersurfaces S t with no singular points. However, in the process of shifting f t : S → S t some singular points can appear (they are called caustics).…”

“…Papers[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] are arranged here in the order they were written. However, the order of publication not always coincides with the order of writing.2 For the convenience of reader we give direct reference to archive file.…”

Global geometric structures associated with dynamical systems admitting normal shift of hypersurfaces in Riemannian manifolds

“…Definitions 1.1 and 1.2 form the base of the theory of dynamical systems admitting the normal shift. This theory was constructed in papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The results of these papers were used in preparing theses [19] and [20].…”