By Victor V. Kozlov
The English train mechanics as an experimental technological know-how, whereas at the Continent, it has continually been thought of a extra deductive and a priori technological know-how. undoubtedly, the English are correct. * H. Poincare, technological know-how and speculation Descartes, Leibnitz, and Newton As is celebrated, the fundamental rules of dynamics have been acknowledged via New ton in his recognized paintings Philosophiae Naturalis Principia Mathematica, whose ebook in 1687 was once paid for by means of his good friend, the astronomer Halley. In essence, this e-book was once written with a unmarried objective: to turn out the equivalence of Kepler's legislation and the idea, recommended to Newton through Hooke, that the acceleration of a planet is directed towards the guts of the sunlight and reduces in inverse percentage to the sq. of the gap among the planet and the sunlight. For this, Newton had to systematize the rules of dynamics (which is how Newton's recognized legislation seemed) and to nation the "theory of fluxes" (analysis of capabilities of 1 variable). the primary of the equality of an motion and a counteraction and the inverse sq. legislations led Newton to the speculation of gravitation, the interplay at a distance. furthermore, New ton mentioned loads of difficulties in mechanics and arithmetic in his ebook, reminiscent of the legislation of similarity, the idea of influence, specific vari ational difficulties, and algebraicity stipulations for Abelian integrals. nearly every little thing within the Principia as a result grew to become vintage. during this connection, A. N.
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Extra info for Dynamical Systems X: General Theory of Vortices
In general, caustics of the surface E are also surfaces; singularities of the orthogonal surfaces Et lie exactly on caustics of E. Caustics divide the space into domains, which are filled by light rays with different multiplicities: the same number of rays pass through each point of such a domain. This is why the eikonal has singularities on caustics; it becomes a multivalued function. A caustic can be described constructively as follows. Let x be a point of the surface E. A plane passing through the normal to the surface E at the point x interects E in a plane curve.
Then its tangent vectors annihilate the 2form ¢, and, in particular, iu¢ = 0 for tangent vectors. 5). 6) holds along a curve 1 for all vector fields u, then the relation iu¢ = 0 holds for all vectors tangent to I· Because the vectors u are arbitrary, we see that 1 is a vortex line. 0 Theorem 9 has the following important consequence. Corollary 1. Vortex lines in the extended phase space coincide with extremals of the action in the class of curves with fixed endpoints. This is the action stationarity principle in the phase space suggested by Helmholtz and Poincare.
As in Sec. 4, we introduce canonical momenta (Yl, ... , Yn) = y by setting aL Yi = ~· 1 :::; UXi i:::; n. 12) r; These covector quantities are elements of the space M dual to the tangent space TxM. 12) becomes n Yi = 2:::aiixi j=l r; and defines the isomorphism of the vector spaces TxM and M. 12) TxM -t T;M surjective? The following two conditions are sufficient: §5. Hamiltonian Form of the Equations of Motion 43 Fig. 15. Legendre transformation 1. the symmetric matrix lla:i:Xj I 2 is positive definite; 2.