By Constantin Udrişte (auth.)
Geometric dynamics is a device for constructing a mathematical illustration of genuine international phenomena, according to the concept of a box line defined in methods: -as the answer of any Cauchy challenge linked to a first-order self sufficient differential method; -as the answer of a undeniable Cauchy challenge linked to a second-order conservative prolongation of the preliminary procedure. the elemental novelty of our e-book is the invention box line is a geodesic of an appropriate geometrical constitution on a given house (Lorentz-Udri~te world-force law). In different phrases, we create a much broader category of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds, making sure that each one trajectories of a given vector box are geodesics. this can be our contribution to an outdated open challenge studied by means of H. Poincare, S. Sasaki and others. From the kinematic point of view of corpuscular instinct, a box line indicates the trajectory via a particle at some extent of the definition area of a vector box, if the particle is delicate to the similar form of box. accordingly, box traces seem in a typical method in difficulties of theoretical mechanics, fluid mechanics, physics, thermodynamics, biology, chemistry, etc.
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Extra info for Geometric Dynamics
Generally, divX = and hence any Killing field is a solenoidal field. Let n;>: 2. Partially differentiating (1) and permuting indices, we find - ' = 0, i = 1 , ... , n, ° a2 X. a2 X. a~a~ a~a~ - - ' + --'- = a2 X 0, --'_. a~a~ a2 X + _ _k_ = a~a~ a2 Xk 0 -- 'a~a~ a2 Xi +-- = a~a~ o. We add the first two equalities and subtract the third from the sum; based on the a2 x. complete integrability conditions, - - ' axjaxk a2 x. a2 X. we find --'- = o.
The vector Vector Fields 20 D Xx d Y(x+tx)1 _ dt /-0 Y = - tangent to D at x is called the covariant derivative of Y with respect to X x . 21) and hence it represents an action of the vector X x on the vector field Y. Fig. 21 Lemma. If Y = YI ul + ... + Y nUn is a vector field of class C I and X x is a tangent vector to D at x, then ul (x) + ... (x). Dxx Y = (Dxx YI ) Proof. We notice that Y(x+tX) = YI(X+tX)UI(x+tX) + ... + Yn(x+tX)Un(x+tX). To differentiate such a vector field at t = 0 means to differentiate its components at t = o.
Let D be an open set of R n and /: D -+ R a scalar field of class C I. Let x = (Xl"" ,xn) ED and Xx a tangent vector to D at the point x. We fix the interval I and tEl such that x + tX E D, where X is the point corresponding to the vector X x' Obviously t -+ x + t X represents the restriction of a straight line, and if / is of class C I, then the composed function t -+ /(x + t X) is also of class CI . ,=o dt x is called the derivative of/with respect to the vector Xx' The derivative of / with respect to the vector Xx represents the action of the vector X x on the function /, indicating qualitatively the modification of f(x) while x is moving in the sense of X x .