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[en] The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is established, and the fractional Lagrange equations are obtained by virtue of the d'Alembert—Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal transformations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results. (general)
Journal
Chinese Physics. B; ISSN 1674-1056; 
; v. 23(11); [7 p.]
; v. 23(11); [7 p.]
[en] For a wide class of Lagrangian systems it is shown rigorously that the conventional formulation of Noether's theorem provides a bijective map from the set of equivalence classes of Noether's symmetries onto the set of equivalence classes of conserved currents. The author further discusses if Noether's theorem is generalized in a significant way by several formulations proposed in this decade. (Auth.)
Journal
Letters in Mathematical Physics; ISSN 0377-9017; ; v. 3(5); p. 419-424
; v. 3(5); p. 419-424
[en] The energy functional of Thomas-Fermi-Dirac-von Weizsäcker model with external potential is studied. The minimizer for the functional is investigated. Furthermore, the value and some properties of the minimizer are estimated from the functional without solving the associate Euler-Lagrange equation
Journal
Journal of Physics. Conference Series (Online); ISSN 1742-6596; ; v. 539(1); [4 p.]
; v. 539(1); [4 p.]
[en] In this paper, we present a new method to construct the conservation laws for relativistic mechanical systems by finding corresponding integrating factors. First, the Lagrange equations of relativistic mechanical systems are established, and the definition of integrating factors of the systems is given; second, the necessary conditions for the existence of conserved quantities of the relativistic mechanical systems are studied in detail, and the relation between the conservation laws and the integrating factors of the systems is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the systems are established, and an example is given to illustrate the application of the results.
Journal
Communications in Theoretical Physics; ISSN 0253-6102; ; v. 44(2); p. 231-234
; v. 44(2); p. 231-234
[en] I prove that the necessary and sufficient condition for two Lagrangian densities L1(psi/sup A/;psi/sup A//sub ,alpha/) and L2(psi/sup A/;psi/sup A//sub ,alpha/) to have exactly the same Euler-Lagrange derivatives is that their difference Δ(psi/sup A/;psi/sup A//sub ,alpha/) be the divergence of ω/sup μ/(psi/sup A/;psi/sup A//sub ,alpha/;x/sup μ/) with a given dependence on psi/sup A//sub ,alpha/. The main point is that ω/sup μ/ depends on psi/sup A//sub ,alpha/ but Δ does not depend on second derivatives of the field psi/sup A/. Therefore, the function Δ need not be linear in psi/sup A//sub ,alpha/
Journal
Physical Review. D, Particles Fields; ISSN 0556-2821; ; v. 27(2); p. 451-453
; v. 27(2); p. 451-453
[en] The French electricity generation system comprises several dozens nuclear units (mostly Pressurized Water Reactors), coal and oil-fired units, and several hundreds smaller hydro units, with a total capacity of about 90 000 MW. The various optimal operation problems of that system, whether they cover a few hours (unit scheduling), or several years (unit refuelling), have many characteristics in common: - unit dynamics are decoupled, - technical constraints of one unit are complex, and their formulation sometimes requires integer variables, - there are only a little number of coupling constraints (load-demand equilibrium...). Decomposition-coordination methods (sometimes called Lagrangian relaxation) appear particularly adequate to solve such problems: they make it possible to handle very precisely the technical constraints of each unit, at the decomposition level (each local problem being solved by dynamic programming or linear programming). Global optimization is achieved at the coordination level
[fr]
Le parc de production d'electricite d'EDF comprend plusieurs dizaines de tranches nucleaires (essentiellement des reacteurs a eau pressurisee), des tranches au charbon et au fioul, plusieurs centaines de centrales hydroelectriques, de taille plus modeste. La puissance totale du systeme est de l'ordre de 90 000 MW. Les differents problemes d'exploitation de ce systeme, qu'ils couvrent quelques heures (etablissement des planning de marche), ou plusieurs annees (rechargement des unites en combustible) ont plusieurs caracteristiques en commun: - les dynamiques des unites sont decouplees, - les contraintes techniques d'une unite sont complexes, et leur formulation fait parfois intervenir des variables entieres, - il n'y a qu'un petit nombre de contraintes couplantes (equilibre production-consommation...). Les methodes de decomposition-coordination (parfois appelees relaxation lagrangienne) sont particulierement adaptees a la resolution de tels problemes: elles permettent de prendre en compte tres precisement les contraintes techniques de chaque unite au niveau de decomposition (chaque probleme local etant resolu par programmation dynamique ou programmation lineaire). Et l'optimisation globale est assuree par le niveau de coordinationJournal
Bulletin de la Direction des Etudes et Recherches, Serie C; ISSN 0013-4511; ; CODEN EDBCA; (no.4); p. 65-78
; CODEN EDBCA; (no.4); p. 65-78
[en] We deal with the question of what it means to define a minimal coupling prescription in presence of torsion and/or non-metricity, carefully explaining while the naive substitution ∂→∇ introduces extra couplings between the matter fields and the connection that can be regarded as non-minimal in presence of torsion and/or non-metricity. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCPF) lead to different dynamics. To that end, we will first write the Euler–Lagrange equations for matter fields in terms of the covariant derivatives of a general non-Riemannian space, and derivate the form of the associated Noether currents and charges. Then we will see that if the minimal coupling prescriptions is applied as we discuss, for spin 0 and 1 fields the results of MCPL and MCPF are equivalent, while for spin 1/2 fields there is a difference if one applies the MCPF or the MCPL, since the former leads to charge violation.
Journal
European Physical Journal. C, Particles and Fields (Online); ISSN 1434-6052; ; CODEN EPCFFB; v. 80(8); p. 1-17
; CODEN EPCFFB; v. 80(8); p. 1-17
[en] We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Euler-Lagrange equations for an action principle. Two ways of constructing the action principle are presented. From simple consideration, we derive the necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order differential equations with the structure of the Euler-Lagrange equations. An explicit form of the action is constructed if such a multiplier exists. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples
Journal
Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 40(33); p. 10071-10081
; v. 40(33); p. 10071-10081
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