Session 8. Dynamic Systems with Fractional and Time Scale Derivatives

Helmholtz theorem for Hamiltonian systems on time scales

Frédéric Pierret, Syrte/Observatoire de Paris, France
The talk is based on the joint work with Jacky Cresson
A classical problem in Analysis is the well-known Helmholtz's inverse problem of the calculus of variations: find a necessary and sufficient condition under which a (system of) differential equation(s) can be written as an Euler-Lagrange or a Hamiltonian equation and, in the affirmative case, find all the possible Lagrangian or Hamiltonian formulations. This condition is usually called Helmholtz condition. Generalisation of this problem in the discrete calculus of variations framework has been done in [2] and [7] in the discrete Lagrangian case. For the Hamiltonian case it has been done for discrete calculus of variation in [5] using the framework of [6] and in [4] using a discrete embedding procedure. In this talk we will generalized the Helmholtz theorem for Hamiltonian systems in the case of time-scale calculus using the work of [3] and [1].
References
  1. Loïc Bourdin, Nonshifted calculus of variations on time scales with \(\nabla\)-differentiable \(\sigma\) , Journal of Mathematical Analysis and Applications, 411(2):543--554, 2014.
  2. Loïc Bourdin and Jacky Cresson, Helmholtz's inverse problem of the discrete calculus of variations , Journal of Difference Equations and Applications, 19(9):1417--1436, 2013.
  3. Jacky Cresson, Agnieszka B. Malinowska, and Delfim F.M. Torres, Time scale differential, integral, and variational embeddings of lagrangian systems , Computers and Mathematics with Applications, 64(7):2294--2301, 2012, Recent Developments in Difference Equations.
  4. Jacky Cresson and Frédéric Pierret, Helmholtz theorem for discrete Hamiltonian systems , To appear.
  5. I.D. Albu and D. Opriş Helmholtz type condition for mechanical integrators}, Novi Sad J. Math., 29(3):11--21, 1999, XII Yugoslav Geometric Seminar (Novi Sad, 1998).
  6. J.E. Marsden and M. West, Discrete mechanics and variational integrators , Acta Numer., 10:357--514, 2001.
  7. Peter E. Hydon and Elizabeth L. Mansfield, A variational complex for difference equations , Foundations of Computational Mathematics, 4(2):187--217, 2004.
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