Non-classical Lagrangian dynamics and potential maps

  • Authors:

  • Constantin Udriste

  • Affiliations:

  • University Politehnica of Bucharest, Faculty of Applied Sciences, Department of Mathematics, Bucharest, Romania

  • Venue:
  • WSEAS Transactions on Mathematics
  • Year:
  • 2008

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Abstract

The basic theory regarding Nonclassical Lagrangian Dynamics and Potential Maps was announced in [7]. Since its mathematical impact is now at large vogue, we reinforce some arguments. Section 1 extends the theory of harmonic and potential maps in the language of differential geometry. Section 2 defines a generalized Lorentz world-force law and shows that any PDE system of order one (in particular, p-flow) generates such a law in a suitable geometrical structure. In other words, the solutions of any PDE system of order one are harmonic or potential maps, i.e., they are solutions of Euler-Lagrange prolongation PDE system of order two built via Riemann-Lagrange structures and a least squares Lagrangian. Section 3 formulates open problems regarding the geometry of semi-Riemann manifolds (J1(T,M), S1), (J2(T,M), S2). Section 4 shows that the Lorentz-Udriste world-force law is equivalent to certain covariant Hamilton PDEs on (J1(T,M), S1). Section 5 describes the maps determining a continuous group of transformations as ultra-potential maps.