Dynamical Systems X: General Theory of Vortices by Victor V. Kozlov

By Victor V. Kozlov

The English train mechanics as an experimental technological know-how, whereas at the Continent, it has constantly been thought of a extra deductive and a priori technological know-how. certainly, the English are correct. * H. Poincare, technology and speculation Descartes, Leibnitz, and Newton As is celebrated, the fundamental ideas of dynamics have been acknowledged through New­ ton in his well-known paintings Philosophiae Naturalis Principia Mathematica, whose e-book in 1687 used to be paid for through his good friend, the astronomer Halley. In essence, this booklet used to be written with a unmarried objective: to turn out the equivalence of Kepler's legislation and the belief, recommended to Newton through Hooke, that the acceleration of a planet is directed towards the heart of the solar and reduces in inverse percentage to the sq. of the space among the planet and the solar. For this, Newton had to systematize the rules of dynamics (which is how Newton's recognized legislation seemed) and to country the "theory of fluxes" (analysis of features of 1 variable). the primary of the equality of an motion and a counteraction and the inverse sq. legislation led Newton to the idea of gravitation, the interplay at a distance. moreover, New­ ton mentioned numerous difficulties in mechanics and arithmetic in his booklet, corresponding to the legislation of similarity, the idea of effect, specific vari­ ational difficulties, and algebraicity stipulations for Abelian integrals. nearly every thing within the Principia for that reason grew to become vintage. during this connection, A. N.

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17), 82£* I I 8v2 v . v > 0. 18) Therefore, this quadratic form is positive definite on the tangent plane at each point of the indicatrix. The Euler formula for a homogeneous function 8L* -- · v 8v = 2L* implies that the derivatives 8L* j8vk are first-degree functions homogeneous in v. Applying the Euler formula again, we obtain 8 2 L* 8v2 v = 8L* 8v ' and hence 8 2L* --v · v 8v 2 8L* 2£ * > 0. 18) is valid for vectors v transversal to the indicatrix. 18) is positive definite in the entire JR 3 and hence = -- · v = By virtue of the relations and L = 1, the function x(t) is an extremal of the variational problem We introduce the canonical momentum Y = 8L* = L8L 8x 8x · 36 Chapter 1.

It should however be observed that the converse of this theorem-namely the theorem that the solution of a partial differential equation such as Hamilton's depends on the solution of a set of ordinary differential equations (the differential equations of the characteristics), which in the case are of the Hamiltonian form, had been discovered by Pfaff and Cauchy (completing the earlier work of Lagrange and Monge) before Hamilton and Jacobi approached the subject from the dynamical side" (Whittaker [76]).

7) The function L is called the Lagrange function or Lagrangian. We now see that motions of the mechanical system in a potential field coincide with extremals of the variational problem 1 t2 8 Ldt=O, h "It is astonishing that in Lagrange's work this statement may be read only between the lines. This explains the strange situation that this relation-mainly through Jacobi's influence--is generally known in Germany, and therefore also in France, as Hamilton's principle. In England no one understands this expression; there this equation is known under the correct if intuitive name of principle of stationary action" (F.

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