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)
Conservation laws and Hamilton’s equations for systems with long-range interaction and memory. (English) Zbl 1221.70024
Summary: Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether’s theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time–space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.
MSC:
70H05Hamilton’s equations
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70S05Lagrangian formalism and Hamiltonian formalism
References:
[1]Montroll, E. W.; Shlesinger, M. F.: The wonderful world of random walks, Studies in statistical mechanics 11, 1-121 (1984) · Zbl 0556.60027
[2]Bouchaud, J. P.; Georges, A.: Anomalous diffusion in disordered media-statistical mechanisms, models and physical applications, Phys rep 195, 127-293 (1990)
[3]Nigmatullin, R. R.: The generalized fractals and statistical properties of the pore-space of the sedimentary-rocks, Phys status solidi B 153, 49-57 (1989)
[4]Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, , 223-276 (1997)
[5]Afanasiev, V. V.; Sagdeev, R. Z.; Zaslavsky, G. M.: Chaotic jets with multifractal space – time random walk, Chaos 1, 143-159 (1991) · Zbl 0902.60071 · doi:10.1063/1.165824
[6]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys rep 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[7]Barkai, E.; Metzler, R.; Klafter, J.: From continuous time random walks to the fractional Fokker – Planck equation, Phys rev E 61, 132-138 (2000)
[8]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
[9], Applications of fractional calculus in physics (2000)
[10]West, B.; Bologna, M.; Grigolini, P.: Physics of fractal operators, (2003)
[11]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives theory and applications, (1993) · Zbl 0818.26003
[12]Podlubny, I.: Fractional differential equations, (1999)
[13]Miller, K.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[14]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and application of fractional differential equations, (2006)
[15]Meerschaert, M. M.; Benson, D. A.; Baeumer, B.: Operator Lévy motion and multiscaling anomalous diffusion, Phys rev E 63, 1112-1117 (2001)
[16]Zaslavsky, G. M.: Multifractional kinetics, Physica A 288, 431-443 (2000)
[17]Meerschaert, M. M.; Mortensen, J.; Wheatcraft, S. W.: Fractional vector calculus for fractional advection dispersion, Physica A 367, 181-190 (2006)
[18]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
[19]Schumer, R.; Benson, D. A.; Meerschaert, M. M.; Wheatcraft, S. W.: Eulerian derivation of the fractional advection-dispersion equation, J contamin hydrol 48, 6988 (2001)
[20]Laskin, N.; Zaslavsky, G. M.: Nonlinear fractional dynamics on a lattice with long-range interactions, Physica A 368, 38-54 (2006)
[21]Tarasov, V. E.; Zaslavsky, G. M.: Fractional dynamics of coupled oscillators with long-range interaction, Chaos 16, 023110 (2006) · Zbl 1152.37345 · doi:10.1063/1.2197167
[22]Tarasov, V. E.; Zaslavsky, G. M.: Fractional dynamics of systems with long-range space interaction and temporal memory, Physica A (2007)
[23]Cresson J. Fractional embedding of differential operators and Lagrangian systems. Preprint math.DS/0605752.
[24]Frederico GSF, Torres DFM. A formulation of Noether’s theorem for fractional problems of the calculus of variations. Preprint math.OC/0701187.
[25]Frohlich, J.; Israel, R.; Lieb, E. H.; Simon, B.: Phase transitions and reflection positivity. I. general theory and long range lattice models, Commun math phys 62, 1-34 (1978)
[26]Barre, J.; Bouchet, F.; Dauxois, T.; Ruffo, S.: Large deviation techniques applied to systems with long-range interactions, J stat phys 119, 677-713 (2005) · Zbl 1170.82302 · doi:10.1007/s10955-005-3768-8
[27]Noether E. Invariant variation problems physics/0503066 M.A. Tavel’s English translation of Invariante Variationsprobleme, Nachr. d. Konig. Gesellsch. d. Wiss. zu Gottingen, Math-phys. Klasse 1918;235 – 57; Transport Theory and Statistical Physics 1971;1:183 – 207.
[28]Bogoliubov, N. N.; Shirkov, D. V.: Introduction to the theory of quantized fields, (1980)
[29]Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics, Phys rev E 53, 1890-1899 (1996)
[30]Baleanu, D.; Muslih, S. I.: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys scr 72, 119-121 (2006) · Zbl 1122.70360 · doi:10.1238/Physica.Regular.072a00119
[31]Stanislavsky, A. A.: Hamiltonian formalism of fractional systems, Eur phys J B 49, 93-101 (2006)
[32]Tarasov, V. E.: Fractional variations for dynamical systems: Hamilton and Lagrange approaches, J phys A 39, 8409-8425 (2006) · Zbl 1122.70013 · doi:10.1088/0305-4470/39/26/009
[33]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
[34]Cottrill-Shepherd K, Naber M. Fractional differential forms II” Preprint math-ph/0301016.
[35]Tarasov, V. E.: Fractional generalization of gradient and Hamiltonian systems, J phys A 38, 5929-5943 (2005) · Zbl 1074.70012 · doi:10.1088/0305-4470/38/26/007
[36]De Donder Th. Theorie invariantive du calcul des variations. Nuov ed. Gauthier-Villars; Paris: 1935. · Zbl 0013.16901
[37]Kastrup, H. A.: Canonical theories of Lagrangian dynamical systems in physics, Phys rep 101, 1-167 (1983)
[38]Born, M.: On the quantum theory of the electromagnetic field, Proc roy soc (London) A 143, 410-437 (1934) · Zbl 0008.13803 · doi:10.1098/rspa.1934.0010
[39]Giachetta, G.; Mangiarotti, L.; Sardanashvily, G.: New Lagrangian and Hamiltonian methods in field theory, (1997)
[40]Kanatchikov, I. V.: Canonical structure of classical field theory in the polymomentum phase space, Rep math phys 41, 49-90 (1998) · Zbl 0947.70020 · doi:10.1016/S0034-4877(98)80182-1
[41]Pierantozzi, T.; Vazquez, L.: An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like, J math phys 46, 113512 (2005) · Zbl 1111.35049 · doi:10.1063/1.2121167
[42]Bollini, C. G.; Giambiagi, J. J.: Arbitrary powers of d’alembertians and the Huygens principle, J math phys 34, 610-621 (1993) · Zbl 0808.35155 · doi:10.1063/1.530263
[43]Kempfle, S.: Causality criteria for solutions of linear fractional differential equations, Fract calc appl anal 1, No. 4, 351-364 (1998) · Zbl 1042.34017
[44]Caldeira, A. O.; Leggett, A. J.: Quantum tunneling in a dissipative system, Ann phys (NY) 149, 374-456 (1983)
[45]Ao, P.: Potential in stochastic differential equations: novel construction, J phys A 37, L25-L30 (2004) · Zbl 1050.60056 · doi:10.1088/0305-4470/37/3/L01