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The General Theory for One First-Order Equation (Continued)

  • Vladimir I. Arnold
Part of the Universitext book series (UTX)

Abstract

We are considering a general first-order partial differential equation F(x,y,p) = 0, where x = (x 1,...,x n), p = (p 1,...,p n), \( p_i = \frac{{\partial u}} {{\partial x_i }},\) and y = u(x) is an unknown function. The equation determines a 2n-dimensional hypersurface V 2n in the space J 1 of 1-jets of functions of (x 1,... ,x n). Each function has a 1-graph in J 1; it is a solution of the equation if its 1-graph is a submanifold in V 2n.

Keywords

Hamiltonian System Euler Equation Tangent Plane Eikonal Equation Integral Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

  1. 1.
    Arnold, V.I.:Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edition. Springer, New York (1988)CrossRefGoogle Scholar
  2. 2.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edition. Springer, New York (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Vladimir I. Arnold
    • 1
    • 2
  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.CEREMADEUniversité de Paris-DauphineParis Cedex 16France

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