Regular and Chaotic Dynamics最新文献

{"title":"Normal Form of the Equations of Perturbed Motion near Triangular Libration Points\u0000at Third-Order Resonances","authors":"Anatoly P. Markeev","doi":"10.1134/S1560354725050053","DOIUrl":"10.1134/S1560354725050053","url":null,"abstract":"<div><p>A treatment is given of the spatial restricted elliptic problem of three\u0000bodies interacting under Newtonian gravity. The problem depends on two parameters:\u0000the ratio between the masses of the main attracting bodies and the eccentricity of their\u0000elliptic orbits. The eccentricity is assumed to be small. Nonlinear equations of motion of\u0000the test mass near a triangular libration point are analyzed. It is assumed that the\u0000parameters of the problem lie on the curves of third-order resonances corresponding to\u0000the planar restricted problem.\u0000In addition to these resonances (their number is equal to five), the spatial problem\u0000has a resonance that takes place at any parameter values since the\u0000the frequency of small linear oscillations of the test mass along the axis perpendicular to\u0000the plane of the orbit of the main bodies is equal to the frequency of Keplerian motion of\u0000these bodies.\u0000In this paper, the normal form of the Hamiltonian function of perturbed motion\u0000through fourth-degree terms relative to deviations from the libration point is obtained.\u0000Explicit expressions for the coefficients of normal form up to and including the second\u0000degree of eccentricity are found.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"837 - 846"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Pendulum-Type Motions and Permanent Rotations in an Approximate Problem of the Dynamics of a Rigid Body with a Vibrating Suspension","authors":"Mikhail V. Belichenko,&nbsp;Olga V. Kholostova","doi":"10.1134/S1560354725050065","DOIUrl":"10.1134/S1560354725050065","url":null,"abstract":"<div><p>The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced.\u0000It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines the\u0000results of the problem of gravitation (without vibration) and that of vibration\u0000(ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"847 - 865"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"When Knowledge of a Single Integral of Motion is Sufficient for Integration of Newton Equations (ddot{boldsymbol{q}}=boldsymbol{M}(boldsymbol{q}))","authors":"Stefan Rauch-Wojciechowski,&nbsp;Maria Przybylska","doi":"10.1134/S1560354725050077","DOIUrl":"10.1134/S1560354725050077","url":null,"abstract":"<div><p>For an autonomous dynamical system of <span>(n)</span> differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of <span>((n-1))</span> integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in <span>(2n)</span>-dimensional phase space only <span>(n)</span> integrals are sufficient as equations are generated by 1 function — the Hamiltonian.</p><p>There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations <span>(ddot{boldsymbol{q}}=boldsymbol{M}(boldsymbol{q}))</span>, <span>(boldsymbol{q}inmathbb{R}^{n})</span>, which allows for integration by quadratures.</p><p>In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"866 - 885"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Contact Magnetic Geodesic and Sub-Riemannian Flows on (V_{n,2}) and Integrable Cases of a Heavy Rigid Body with a Gyrostat","authors":"Božidar Jovanović","doi":"10.1134/S156035472505003X","DOIUrl":"10.1134/S156035472505003X","url":null,"abstract":"<div><p>We prove the integrability of magnetic geodesic flows of <span>(SO(n))</span>-invariant Riemannian metrics on the rank two Stefel variety <span>(V_{n,2})</span> with respect to the magnetic field <span>(eta dalpha)</span>, where <span>(alpha)</span> is the standard contact form on <span>(V_{n,2})</span> and <span>(eta)</span> is a real parameter.\u0000Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for <span>(SO(n))</span>-invariant sub-Riemannian structures on <span>(V_{n,2})</span>. All statements in the limit <span>(eta=0)</span> imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by <span>(SO(n)times SO(2))</span>-invariant Riemannian metrics. For <span>(n=3)</span>, using the isomorphism <span>(V_{3,2}cong SO(3))</span>, the obtained integrable magnetic models reduce to\u0000integrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point:\u0000the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski\u0000top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange\u0000gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"799 - 818"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the 175th Anniversary of S. V. Kovalevskaya","authors":"","doi":"10.1134/S1560354725050016","DOIUrl":"10.1134/S1560354725050016","url":null,"abstract":"","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"765 - 766"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sonya Kowalewski’s Legacy to Mechanics and Complex Lie Algebras","authors":"Velimir Jurdjevic","doi":"10.1134/S1560354725050028","DOIUrl":"10.1134/S1560354725050028","url":null,"abstract":"<div><p>This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski’s seminal work on the motions of a rigid body around a fixed point under the influence of gravity.\u0000The point of departure for understanding Kowalewski’s work\u0000begins with Kirchhoff’s model for the equilibrium configurations of an elastic rod in <span>({mathbb{R}}^{3})</span> subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I. V. Komarov and V. B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere <span>(S^{3})</span> and the hyperboloid <span>(H^{3})</span> [17] and, secondly, it shows\u0000that these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of <span>(so(4,mathbb{C}))</span> generated by\u0000an affine quadratic Hamiltonian <span>(H)</span> (Kirchhoff – Kowalewski type).</p><p>The paper shows that the passage to complex variables\u0000is synonymous with the representation of <span>(so(4,mathbb{C}))</span> as <span>(sl(2,mathbb{C})times sl(2,mathbb{C}))</span> and the embedding of <span>(H)</span> into <span>(sp(4,mathbb{C}))</span>, an important intermediate step towards uncovering the origins of Kowalewski’s integral. There is a quintessential Kowalewski type integral of motion on <span>(sp(4,mathbb{C}))</span> that appears as a spectral invariant for the Poisson system associated with a Hamiltonian <span>(mathcal{H})</span> (a natural extension of <span>(H)</span>) that satisfies Kowalewski’s conditions.</p><p>The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski’s ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"767 - 798"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Painlevé Test, First Integrals and Exact Solutions of Nonlinear Dissipative Differential Equations","authors":"Nikolay A. Kudryashov","doi":"10.1134/S1560354725050041","DOIUrl":"10.1134/S1560354725050041","url":null,"abstract":"<div><p>The Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid are considered. The Cauchy problems for all these partial differential equations are not solved by the\u0000inverse scattering transform. Reductions of these equations to nonlinear ordinary differential\u0000equations do not pass the Painlevé test. However, there are local expansions of the general\u0000solutions in the Laurent series near movable singular points.\u0000We demonstrate that the obtained information from the Painlevé test for reductions of\u0000nonlinear evolution dissipative differential equations can be used to construct the\u0000nonautonomous first integrals of nonlinear ordinary differential equations. Taking into\u0000account the found first integrals, we also obtain analytical solutions of nonlinear evolution\u0000dissipative differential equations. Our approach is illustrated to obtain the\u0000nonautonomous first integrals for reduction of the Korteweg – de Vries – Burgers equation,\u0000the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and\u0000the nonlinear differential equation for description surface waves in a convecting fluid.\u0000The obtained first integrals are used to construct exact solutions of the above-mentioned\u0000nonlinear evolution equations with as many arbitrary constants as possible. It is shown that\u0000some exact solutions of the equation for description of nonlinear waves in a convecting\u0000liquid are expressed via the Painlevé transcendents.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"819 - 836"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145256268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Integrability of Homogeneous Exact Magnetic Flows on Spheres","authors":"Vladimir Dragović,&nbsp;Borislav Gajić,&nbsp;Božidar Jovanović","doi":"10.1134/S1560354725040082","DOIUrl":"10.1134/S1560354725040082","url":null,"abstract":"<div><p>We consider motion of a material point placed in a constant homogeneous magnetic field in <span>(mathbb{R}^{n})</span> and also motion restricted to the sphere <span>(S^{n-1})</span>.\u0000While there is an obvious integrability of the magnetic system in <span>(mathbb{R}^{n})</span>, the integrability of the system restricted to the sphere <span>(S^{n-1})</span> is highly nontrivial. We prove\u0000complete integrability of the obtained restricted magnetic systems for <span>(nleqslant 6)</span>. The first integrals of motion of the magnetic flows on the spheres <span>(S^{n-1})</span>, for <span>(n=5)</span> and <span>(n=6)</span>, are polynomials of degree\u0000<span>(1)</span>, <span>(2)</span>, and <span>(3)</span> in momenta.\u0000We prove noncommutative integrability of the obtained magnetic flows for any <span>(ngeqslant 7)</span> when the systems allow a reduction to the cases with <span>(nleqslant 6)</span>. We conjecture that the restricted magnetic systems on <span>(S^{n-1})</span> are integrable for all <span>(n)</span>.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"582 - 597"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Oscillations in a Neighborhood of Lagrangian Libration Points in One\u0000Resonance Case","authors":"Anatoly P. Markeev","doi":"10.1134/S1560354725040136","DOIUrl":"10.1134/S1560354725040136","url":null,"abstract":"<div><p>This paper addresses the spatial restricted elliptic problem\u0000of three bodies (material points) gravitating toward each other under Newton’s law of\u0000gravitation. The eccentricity of the orbit of the main attracting bodies is assumed to be\u0000small, and nonlinear oscillations of\u0000a passively gravitating body near a Lagrangian triangular libration point are studied.\u0000It is assumed that in the limiting case of the circular problem the ratio\u0000of the frequency of rotation of the main bodies about their common center of mass\u0000to the value of one of the frequencies of small linear oscillations of the passive body\u0000is exactly equal to three. A detailed analysis is made of two different particular cases of\u0000influence of the three-dimensionality of the\u0000problem on the characteristics of nonlinear oscillations of the passive body.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"666 - 676"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spinning Top in Quadratic Potential and Matrix Dressing Chain","authors":"Vsevolod E. Adler,&nbsp;Alexander P. Veselov","doi":"10.1134/S1560354725040021","DOIUrl":"10.1134/S1560354725040021","url":null,"abstract":"<div><p>We show that the equations of motion of a rigid body about a fixed point in the Newtonian\u0000field with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schrödinger operators are discussed.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"30 Editors:","pages":"464 - 480"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}