Divergent formulas (conservation laws) in the differential geometry of plane curves

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Abstract: 

In this paper, it is discovered that in the differential geometry of arbitrary smooth plane curves there exists a solenoidal vector field S*, i.e., the field with a property div S* = 0 (in a certain area D), having the following geometric meaning. Let {Lτ} be a set of arbitrary smooth non-intersecting plane curves Lτ with the Frene unit vectors τ = τ (x, y), υ = υ (x, y) (τ  is the unit tangent vector, υ is the unit normal of a curve Lτ), and {Lυ} be a set of orthogonal to them curves Lυ with the Frene unit vectors υ and η = –τ. The set {Lτ} fills by the continuous image some area D and satisfies some general conditions. The vector field S* is expressed in terms of the Frene unit vectors τ, υ of the curves Lτ and equals the sum of the curvature vector dτ / ds = (τ•∇)τ = kυ of the curve Lτ ∈ {Lτ} and the curvature vector dυ / dsυ = (υ•∇)υ = kυη = –kυτ of the curve Lυ ∈ {Lυ}. Here s and sυ are natural parameters, i.e., the length of a curve being counted from its certain point along Lτ and Lυ, respectively, k = k(x, y) and kυ = kυ(x, y) are curvatures of the curves Lτ and Lυ, respectively. The symbol (a•∇)a denotes a derivative of the vector a in the direction a. This property can be interpreted as existence in the differential geometry of plane curves of a conservation law (for the vector field S* or for vector fields of unit vectors). A number of equivalent representations of the field S* and equivalent forms of a conservation law div S* = 0 is obtained.

Pages: 
29-40
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