Topological Methods in Nonlinear Analysis

Constants of motion for non-differentiable quantum variational problems

Jacky Cresson, Gastão S. F. Frederico, and Delfim F. M. Torres

Full-text: Open access

Abstract

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schrödinger equation.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 33, Number 2 (2009), 217-231.

Dates
First available in Project Euclid: 27 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461785634

Mathematical Reviews number (MathSciNet)
MR2549615

Zentralblatt MATH identifier
1188.49023

Citation

Cresson, Jacky; Frederico, Gastão S. F.; Torres, Delfim F. M. Constants of motion for non-differentiable quantum variational problems. Topol. Methods Nonlinear Anal. 33 (2009), no. 2, 217--231. https://projecteuclid.org/euclid.tmna/1461785634


Export citation

References

  • F. Ben Adda and J. Cresson, Quantum derivatives and the Schrödinger equation , Chaos Solitons Fractals, 19(2004), 1323–1334 \ref\key 2 ––––, Fractional differential equations and the Schrödinger equation , Appl. Math. Comput., 161(2005), 323–345 \ref\key 3
  • J. Cresson, Non-differentiable variational principles , J. Math. Anal. Appl., 307(2005), 48–64 \ref\key 4 ––––, Non-differentiable deformations of $\Bbb R^n$ , Int. J. Geom. Methods Mod. Phys., 3(2006), 1395–1415 \ref\key 5
  • J. Cresson and S. Darses, Théorème de Noether stochastique , C. R. Math. Acad. Sci. Paris, 344(2007), 259–264 \ref\key 6
  • G. S. F. Frederico and D. F. M. Torres, Constants of motion for fractional action-like variational problems , Internat. J. Appl. Math., 19(2006), 97–104 \ref\key 7 ––––, Nonconservative Noether's theorem in optimal control , Internat. J. Tomogr. Stat., 5(2007), 109–114 \ref\key 8 ––––, A formulation of Noether's theorem for fractional problems of the calculus of variations , J. Math. Anal. Appl., 334(2007), 834–846 \ref\key 9 ––––, Fractional conservation laws in optimal control theory , Nonlinear Dynam., 53 (2008), 215–222 \ref\key 10
  • P. D. F. Gouveia and D. F. M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control , Comput. Methods Appl. Math., 5(2005), 387–409 \ref\key 11
  • P. D. F. Gouveia, D. F. M. Torres and E. A. M. Rocha, Symbolic computation of variational symmetries in optimal control , Control Cybernet., 35(2006), 831–849 \ref\key 12
  • L. Nottale, The theory of scale relativity , Internat. J. Modern Phys. A, 7(1992), 4899–4936 \ref\key 13 ––––, The scale-relativity program , Chaos Solitons Fractals, 10(1999), 459–468 \ref\key 14
  • L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, Selected works, 4 , Translated from the Russian by K. N. Trirogoff, Translation edited by L. W. Neustadt, Reprint of the 1962 English translation, Gordon & Breach, New York (1986) \ref\key 15
  • E. A. M. Rocha and D. F. M. Torres, Quadratures of Pontryagin extremals for optimal control problems , Control Cybernet., 35(2006), 947–963 \ref\key 16
  • D. F. M. Torres, On the Noether theorem for optimal control , European J. Control, 8(2002), 56–63 \ref\key 17 ––––, The role of symmetry in the regularity properties of optimal controls , Symmetry in nonlinear mathematical physics. Part 1–3, 1488–1495, Natsī onal. Akad. Nauk Ukraï ni, \B Inst. Mat., Kiev (2004) \ref\key 18 ––––, Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations , Comm. Pure Appl. Anal., 3(2004), 491–500 \ref\key 19 ––––, A Noether theorem on unimprovable conservation laws for vector-valued optimization problems in control theory , Georgian Math. J., 13(2006), 173–182