scholarly journals Toward a classification of dynamical symmetries in classical mechanics

1983 ◽  
Vol 27 (1) ◽  
pp. 53-71 ◽  
Author(s):  
Geoff Prince

A one-parameter group on evolution space which permutes the classical trajectories of a Lagrangian system is called a dynamical symmetry. Following a review of the modern approach to the “symmetry-conservation law” duality an attempt is made to classify such invariance groups according to the induced transformation of the Cartan form. This attempt is fairly successful inasmuch as the important cases of Lie, Noether and Cartan symmetries can be distinguished. The theory is illustrated with a presentation of results for the classical Kepler problem.

2020 ◽  
Vol 17 (11) ◽  
pp. 2050172
Author(s):  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman ◽  
B. B. I. Gadjagboui

The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.


1973 ◽  
Vol 28 (3-4) ◽  
pp. 538-540 ◽  
Author(s):  
D. J. Simms

AbstractThis is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.


2003 ◽  
Vol 56 (6) ◽  
pp. B80-B81
Author(s):  
H Iro, ◽  
K Anderson,

In classical mechanics (c.m.), and near the semi-classical limit h →0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ 0 /ρ ∞ are found for scattering by attractive central potentials U(r) ; here ρ 0,∞ (and v 0,∞ ) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials ( U (0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v ∞ ), α 1 = v ∞ / v 0 , α 2 = 1, α 3 = v 0 / v ∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U ( r →0) ~ ─ C/r q , with 0 < q < 1 in ID (where r ≡ | x | ), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α 1 (c.m.) = 0, α 3 (c.m.) = ∞, and α 2 (c.m.) = (1 —½ q ) N , where N = [integer part of (1 ─½ q ) -1 ]is the number of trajectories through any point ( r , θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v ∞ ) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r ; for small v ∞ / u it yields the leading terms α 1 (s.c.l.) = Λ 1 (q) v ∞ / u , α 2 (s.c.I) = (1 ─½ q ) -1 , α 3 (s.c.l.)= Λ 3 ( q ) u/v ∞ , where u ≡ (C/h q m 1-q ) 1/(2-q) is a generalized Bohr velocity. Here Λ 1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α 2 (s.c.l.) and α 2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ C exp (─r/a)/r q ) , and if, near the s.c.l., v ∞ / u is regarded as a function not of h but more realistically of v ∞ , then the expressions α (s.c.l.) above apply only in an intermediate range 1/ a ≪ mv ∞ / h ≪ ( mC/h 2 ) 1/(2- q ) which exists only if a ≫ ( h 2 / mC ) 1/(2- q ) ).


Author(s):  
Uspenskaya Inna

The article is devoted to the systematization of the criteria of the classification of concert music for violin, in which, along with the traditional genre criteria, stylistic and textured ones are highlighted It is noted that such a comprehensive consideration allows solving a number of tasks of both research and performance profile. Based on the modern approach to the genre system, the article extrapolates it to concert violin music, which covers the range from solo miniature pieces to concerts for violin and symphony orchestra. It is emphasized that the least researched is the question of the stylistics of concert violin genres, constituted according to the same parameters as the musical texture – horizontal, vertical and depth (E. Nazaikinsky). The article proposes an original classification of the genre-stylistic complex of concert violin music, that is based on the following factors: the style of the highest levels (epoch-making, national, specific), genre (the complex of existing genres of violin music), texture in the aspect of stylistics (the main “identification mark” of the genre) and the style of concretized levels (author’s individual level and separate work). Considering the first classification criterion – the genre one, its universal nature it should be noted, covering two levels of the concert violin music system: functional – performers, the way of performance – and semantic-compositional – genre content and style (I. Tukova). The style criterion acts as a parallel to the genre criterion and means the differentiation of the genre system according to the signs of introversion (style as an introvert category, according to V. Kholopova). Here the phenomena and concepts are formed that cover all levels of the style hierarchy in its distribution to concert music for violin – from the historical to the author’s individual and even the style of a separate piece. It is emphasized that the least explored area of violin concert is its stylistics, which is closely related to its texture – the “external form” of the genre manifestation (L. Shapovalova). The stylistic aspect in violin music-making is reviewed in the article according to the same parameters as the texture aspect, since they largely coincide (E. Nazaikinsky). We are talking about the factors of horizontal (the types of texture that form the stylistic relief of the text of the work), vertical (the combination of textures in their different stylistic meanings), depth (based on the author’s handwriting of his connections with the texture and style sources – historical, national ones, characteristic of certain violin schools and directions). It is noted that this refers to both sides of the genre-stylistic system of concert music for violin (with the participation of a violin) – functional and semanticcompositional – and is realized in the following variants of textured style: solo orchestra (violin or several violins with an orchestra); solo ensemble (the same accompanied by a chamber ensemble); solo piano (violin and piano duet); solo violin (violin without accompaniment). It is proved that all these textured and stylistic varieties of concert violin music are combined on the basis of the idea of a concert style – “competition-agreement” (B. Asafiev) of the participants in the act of playing music. The measure of the correlation of performing forces in a concert dialogue ultimately determines the choice of criteria for classifying its varieties in their extrapolation to a concert violin. The article reveals the features of all four above-named options for this dialogue, taking into account their possible combination. It is noted that this combination is most fully reflected in a violin concert with an orchestra, where other forms of concert appear occasionally – solo without accompaniment (solo cadenzas), ensemble (microdialogues of the violin and other orchestral instruments). The classification criteria highlighted in the article, first of all texture-stylistic ones, together form the following system of genres of concert music for violin (with the participation of a violin), considered from the standpoint of: 1) concert dialogue in its textured manifestations (gradation in the dominance of the soloist instrument over accompaniment or, conversely, accompaniment over a solo part); 2) the principle of intimacy, bordering on concertness, but meaning the parity of the performing parts (a distinctive feature of chamber ensembles, in which it stands out as the leading violin part); 3) the self-sufficiency of the violin as a universal instrument suitable for the implementation of concert dialogue in the solo form of music-making (a wide range of genre forms of violin music – from miniatures and their cycles to suites, partitas and solo sonatas). It is noted that, in the future, the classification patterns identified in this article can be considered using the example of specific samples belonging to a particular genre group. The author of this article plans to do this on the basis of concert genres of violin music created by the composers of the Kharkiv school. Focusing on classical and modern samples, as well as the traditions of the Kharkiv stringbow performing school represented by A. Leshchinsky, A. Yuriev, S. Kocharyan, G. Averyanov, E. Shchelkanovtseva, L. Kholodenko, E. Kupriyanenko and other string players, Kharkiv authors interpret the concert-violin style in various ways, revealing in it both the general (the “image” of the violin in the system of specific instrumental styles), and the special (the styles of the national and regional schools), as well as the unique, individual (the representations of the latter are their best works).


2020 ◽  
Vol 22 (1) ◽  
pp. 131-138
Author(s):  
N. V. Kozlova ◽  
T. V. Rogacheva ◽  
T. E. Levickaya ◽  
E. A. Tsehmeistruk ◽  
D. G. Nazmetdinova

The research substantiates the role of clinical psychologists in the system of comprehensive rehabilitation within the conditions of bio-psychosocial model of rehabilitation. Practical experience and the analysis of scientific literature made it possible to describe peculiarities of modern complex rehabilitation and the problems experienced by clinical psychologists. The authors determined difficulties and characteristics of professional work of rehabilitation psychologist within modern legal terrain. One problem is the lack of professional standard for the profession of clinical psychologist. The other problem is the inconsistency in the use of terms denoting professional specifics. Тhe paper focuses on the specificity of psychodiagnostic and rehabilitative work of a psychologist in the light of changing theoretical basis of rehabilitation and the transition from the International Classification of Disability Disorders and Social Insufficiency to the International Classification of Functioning, Disability, and Health. The authors showed the discrepancy between the theoretical approaches of domestic and foreign psychology to the assessment of mental disorders. New professional tasks of the psychologist are defined taking into account multidisciplinary orientation of rehabilitation process. The paper describes the problems of professional training of clinical psychologists and specifies new educational tasks are specified proceeding from the modern approach to rehabilitation. Positive dynamics in the solution of the declared problem can be found in new legislative initiatives, development of General clinical recommendations for rehabilitation psychologists, scientific publications on psychological rehabilitation, update of educational programs for training clinical psychologists.


2006 ◽  
Vol 18 (03) ◽  
pp. 329-347 ◽  
Author(s):  
P. A. HORVÁTHY

The dynamical (super)symmetries for various monopole systems are reviewed. For a Dirac monopole, non-smooth Runge–Lenz vector can exist; there is, however, a spectrum-generating conformal o(2,1) dynamical symmetry that extends into osp(1/1) or osp(1/2) for spin 1/2 particles. Self-dual 't Hooft–Polyakov-type monopoles admit an su(2/2) dynamical supersymmetry algebra, which allows us to reduce the fluctuation equation to the spin 0 case. For large r, the system reduces to a Dirac monopole plus a suitable inverse-square potential considered before by McIntosh and Cisneros, and by Zwanziger in the spin 0 case, and to the "dyon" of D'Hoker and Vinet for spin 1/2. The asymptotic system admits a Kepler-type dynamical symmetry as well as a "helicity-supersymmetry" analogous to the one Biedenharn found in the relativistic Kepler problem. Similar results hold for the Kaluza–Klein monopole of Gross–Perry–Sorkin. For the magnetic vortex, the N = 2 supersymmetry of the Pauli Hamiltonian in a static magnetic field in the plane combines with the o(2) × o(2,1) bosonic symmetry into an o(2) × osp(1/2) dynamical superalgebra.


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