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

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


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.


Hamiltonian System Euler Equation Tangent Plane Eikonal Equation Integral Surface 
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  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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