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)
On the fractional Hamilton and Lagrange mechanics. (English) Zbl 1256.35192
Summary: The fractional generalization of Hamiltonian mechanics is constructed by using the Lagrangian involving fractional derivatives. In this paper the equation of projectile motion with air friction using fractional Hamiltonian mechanics and equation for current loop involving electric source, a resistor, an inductor and a capacitor has been obtained. Furthermore, fractional optics has been introduced.

MSC:
35R11Fractional partial differential equations
35Q70PDEs in connection with mechanics of particles and systems
WorldCat.org
Full Text: DOI
References:
[1] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New York (1974) · Zbl 0292.26011
[2] Miller, K.S., Ross, B.: An Introduction to the Fractional Integrals and Derivatives-Theory and Application. Wiley, New York (1993) · Zbl 0789.26002
[3] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon & Breach, New York (1993) · Zbl 0818.26003
[4] Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997) · Zbl 0917.73004
[5] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) · Zbl 0924.34008
[6] Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[7] Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Connecticut (2006)
[8] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos). World Scientific, Singapore (2012) · Zbl 1248.26011
[9] Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890--1899 (1996) · doi:10.1103/PhysRevE.53.1890
[10] Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581--3592 (1997) · doi:10.1103/PhysRevE.55.3581
[11] Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368--379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[12] Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72, 119--121 (2005) · Zbl 1122.70360 · doi:10.1238/Physica.Regular.072a00119
[13] Muslih, S., D, B.: Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl. 304, 599--606 (2005) · Zbl 1149.70320 · doi:10.1016/j.jmaa.2004.09.043
[14] Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A, Math. Gen. 39, 10375--10384 (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008
[15] Tarasov, V.E.: Fractional variation for dynamical systems: Hamilton and Lagrange approaches. J. Phys. 39(26), 8409--8425 (2006) · Zbl 1122.70013
[16] Rabei, E.M., Nawafleh, K.I., Hiijawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891--897 (2007) · Zbl 1104.70012 · doi:10.1016/j.jmaa.2006.04.076
[17] Klimek, K.: Lagrangian and Hamiltonian fractional sequential mechanics. Czechoslov. J. Phys. 52, 1247--1253 (2002) · Zbl 1064.70013 · doi:10.1023/A:1021389004982
[18] Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135--3145 (2000) · Zbl 0948.81595 · doi:10.1103/PhysRevE.62.3135
[19] Naber, M.: Time fractional Schrodinger equation. J. Math. Phys. 45, 3339--3352 (2004) · Zbl 1071.81035 · doi:10.1063/1.1769611
[20] Tarasov, V.E.: Continuous medium model for fractal media. Phys. Lett. A 336, 167--174 (2005) · Zbl 1136.81443 · doi:10.1016/j.physleta.2005.01.024
[21] Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461--580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[22] Tarawneh, K.M., Rabei, E.M., Ghassib, H.B.: Lagrangian and Hamiltonian formulations of the damped harmonic oscillator using Caputo fractional derivative. J. Dyn. Syst. Geom. Theories 8(1), 59--70 (2010) · Zbl 05907481 · doi:10.1080/1726037X.2010.10698578
[23] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[24] West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, Berlin (2003)
[25] Golmankhaneh, A.K., Golmankhaneh, A.K., Baleanu, D., Baleanu, M.C.: Fractional odd-dimensional mechanics. Adv. Differ. Equ. 2011, 526472 (2011) · Zbl 05847752 · doi:10.1155/2011/526472
[26] Baleanu, D., Alireza, K., Golmankhaneh, A.K., Golmankhaneh, A.L., Nigmatullin, R.R.: Newtonian law with memory. Nonlinear Dyn. 60, 81--86 (2010) · Zbl 1189.70002 · doi:10.1007/s11071-009-9581-1
[27] Baleanu, D., Golmankhaneh, A.K., Nigmatullin, R.R., Golmankhaneh, A.K.: Fractional Newtonian mechanics. Cent. Eur. J. Phys. 8, 120--125 (2010) · doi:10.2478/s11534-009-0085-x
[28] Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140--1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027