zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Hamilton formalism with fractional derivatives. (English) Zbl 1104.70012
Summary: Recently the traditional calculus of variations has been extended to be applicable to systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton’s equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, and the results obtained are in exact agreement with Agrawal formalism.

MSC:
70H05Hamilton’s equations
26A33Fractional derivatives and integrals (real functions)
WorldCat.org
Full Text: DOI
References:
[1] Riewe, F.: Non-conservative Lagrangian and Hamiltonian mechanics. Phys. rev. E 53, 1890 (1996)
[2] Riewe, F.: Mechanics with fractional derivatives. Phys. rev. E 55, 3581 (1997)
[3] Agrawal, O. P.: A new Lagrangian and a new Lagrange equation of motion for fractionally damped systems. J. appl. Mech. 53, 339 (2001) · Zbl 1110.74310
[4] Rekhriashvili, S. Sh.: The Lagrangian formalism with fractional derivatives in problems of mechanics. Technical phys. Lett. 30, 55 (2004)
[5] Rabei, Eqab M.; Halholy, Tareg S. Al; Taani, A. A.: On Hamiltonian formulation of non-conservative systems. Turkish J. Phys. 28, No. 4, 213 (2004)
[6] Rabei, Eqab M.; Halholy, Tareg S. Al; Rousan, A.: Potentials of arbitrary forces with fractional derivatives. Internat. J. Modern phys. A 19, 3083 (2004) · Zbl 1080.70516
[7] Rousan, A.; Malkawi, E.; Rabei, Eqab M.; Widyan, H.: Applications of fractional calculus of gravity. Fract. calc. Appl. anal. 5, 155 (2002) · Zbl 1039.86007
[8] O.P. Agrawal, An Analytical Scheme for Stochastic Dynamics Systems Containing Fractional Derivatives, ASME Design Engineering Technical Conferences, 1999
[9] Agrawal, O. P.: Formulation of Euler -- Lagrange equations for variational problems. J. math. Anal. appl. 272, 368 (2002) · Zbl 1070.49013
[10] Baleanu, D.; Muslih, S.: Lagrangian formulation of classical fields within Riemann -- Liouville fractional derivatives. Phys. scripta 27, 105 (2005) · Zbl 1122.70360
[11] Dreisi-Gmeyer, David W.; Yoang, P. M.: Non-conservative Lagrangian mechanics: A generalized functional approach. J. phys. A 36, 3297 (2003)
[12] Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czechoslovak J. Phys. 52, 1247 (2002) · Zbl 1064.70013
[13] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[14] Goldstein, H.: Classical mechanics. (1980) · Zbl 0491.70001
[15] Muslih, S.; Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann -- Liouville fractional derivatives. J. math. Anal. appl. 304, 599 (2005) · Zbl 1149.70320