We consider the classical integrable 1+1 trigonometric ${rm gl}_N$ Landau-Lifshitz models constructed by means of quantum $R$-matrices satisfying also the associative Yang-Baxter equation. It is shown that 1+1 field analogue of the trigonometric Calogero-Moser-Sutherland model is gauge equivalent to the Landau-Lifshitz model, which arises from the Antonov-Hasegawa-Zabrodin trigonometric non-standard $R$-matrix. The latter generalizes the Cherednik's 7-vertex $R$-matrix in ${rm GL}_2$ case to the case of ${rm GL}_N$. Explicit change of variables between the 1+1 models is obtained.
{"title":"Gauge equivalence of 1+1 Calogero-Moser-Sutherland field theory and higher rank trigonometric Landau-Lifshitz model","authors":"K. Atalikov, A. Zotov","doi":"arxiv-2403.00428","DOIUrl":"https://doi.org/arxiv-2403.00428","url":null,"abstract":"We consider the classical integrable 1+1 trigonometric ${rm gl}_N$\u0000Landau-Lifshitz models constructed by means of quantum $R$-matrices satisfying\u0000also the associative Yang-Baxter equation. It is shown that 1+1 field analogue\u0000of the trigonometric Calogero-Moser-Sutherland model is gauge equivalent to the\u0000Landau-Lifshitz model, which arises from the Antonov-Hasegawa-Zabrodin\u0000trigonometric non-standard $R$-matrix. The latter generalizes the Cherednik's\u00007-vertex $R$-matrix in ${rm GL}_2$ case to the case of ${rm GL}_N$. Explicit\u0000change of variables between the 1+1 models is obtained.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the following simple root systems: $A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to addition of a point in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that mutation equivalent to the $q$-Painlev'e III quiver found by Okubo. The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces.
{"title":"New cluster algebras from old: integrability beyond Zamolodchikov periodicity","authors":"Andrew N. W. Hone, Wookyung Kim, Takafumi Mase","doi":"arxiv-2403.00721","DOIUrl":"https://doi.org/arxiv-2403.00721","url":null,"abstract":"We consider discrete dynamical systems obtained as deformations of mutations\u0000in cluster algebras associated with finite-dimensional simple Lie algebras. The\u0000original (undeformed) dynamical systems provide the simplest examples of\u0000Zamolodchikov periodicity: they are affine birational maps for which every\u0000orbit is periodic with the same period. Following on from preliminary work by\u0000one of us with Kouloukas, here we present integrable maps obtained from\u0000deformations of cluster mutations related to the following simple root systems:\u0000$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise,\u0000by considering Laurentification, that is, a lifting to a higher-dimensional map\u0000expressed in a set of new variables (tau functions), for which the dynamics\u0000exhibits the Laurent property. For the integrable map obtained by deformation\u0000of type $A_3$, which already appeared in our previous work, we show that there\u0000is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a\u0000composition of mutations and a permutation applied to the same cluster algebra\u0000of rank 6, with an additional 2 frozen variables. Furthermore, both the\u0000deformed $A_3$ map and the QRT map correspond to addition of a point in the\u0000Mordell-Weil group of a rational elliptic surface of rank two, and the\u0000underlying cluster algebra comes from a quiver that mutation equivalent to the\u0000$q$-Painlev'e III quiver found by Okubo. The deformed integrable maps of types\u0000$B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerisation process is suggested and exact solutions are found in this case as well.
{"title":"The Shigesada-Kawasaki-Teramoto model: conditional symmetries, exact solutions and their properties","authors":"Roman Cherniha, Vasyl' Davydovych, John R. King","doi":"arxiv-2402.19050","DOIUrl":"https://doi.org/arxiv-2402.19050","url":null,"abstract":"We study a simplification of the well-known Shigesada-Kawasaki-Teramoto\u0000model, which consists of two nonlinear reaction-diffusion equations with\u0000cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is\u0000derived using an algorithm adopted for the construction of conditional\u0000symmetries. The symmetries obtained are applied for finding a wide range of\u0000exact solutions, possible biological interpretation of some of which being\u0000presented. Moreover, an alternative application of the simplified model related\u0000to the polymerisation process is suggested and exact solutions are found in\u0000this case as well.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Transport processes in crowded periodic structures are often mediated by cooperative movements of particles forming clusters. Recent theoretical and experimental studies of driven Brownian motion of hard spheres showed that cluster-mediated transport in one-dimensional periodic potentials can proceed in form of solitary waves. We here give a comprehensive description of these solitons. Fundamental for our analysis is a static presoliton state, which is formed by a periodic arrangements of basic stable clusters. Their size follows from a geometric principle of minimum free space. Adding one particle to the presoliton state gives rise to solitons. We derive the minimal number of particles needed for soliton formation, number of solitons at larger particle numbers, soliton velocities and soliton-mediated particle currents. Incomplete relaxations of the basic clusters are responsible for an effective repulsive soliton-soliton interaction seen in measurements. Our results provide a theoretical basis for describing experiments on cluster-mediated particle transport in periodic potentials.
{"title":"Solitary cluster waves in periodic potentials: Formation, propagation, and soliton-mediated particle transport","authors":"Alexander P. Antonov, Artem Ryabov, Philipp Maass","doi":"arxiv-2402.17469","DOIUrl":"https://doi.org/arxiv-2402.17469","url":null,"abstract":"Transport processes in crowded periodic structures are often mediated by\u0000cooperative movements of particles forming clusters. Recent theoretical and\u0000experimental studies of driven Brownian motion of hard spheres showed that\u0000cluster-mediated transport in one-dimensional periodic potentials can proceed\u0000in form of solitary waves. We here give a comprehensive description of these\u0000solitons. Fundamental for our analysis is a static presoliton state, which is\u0000formed by a periodic arrangements of basic stable clusters. Their size follows\u0000from a geometric principle of minimum free space. Adding one particle to the\u0000presoliton state gives rise to solitons. We derive the minimal number of\u0000particles needed for soliton formation, number of solitons at larger particle\u0000numbers, soliton velocities and soliton-mediated particle currents. Incomplete\u0000relaxations of the basic clusters are responsible for an effective repulsive\u0000soliton-soliton interaction seen in measurements. Our results provide a\u0000theoretical basis for describing experiments on cluster-mediated particle\u0000transport in periodic potentials.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a 2-component Camassa-Holm equation, as well as a 2-component generalization of the modified Camassa-Holm equation, nonlocal infinitesimal symmetries quadratically depending on eigenfunctions of linear spectral problems are constructed from functional gradients of spectral parameters. With appropriate pseudo-potentials, these nonlocal infinitesimal symmetries are prolonged to enlarged systems, and then explicitly integrated to generate symmetry transformations in finite form for enlarged systems. As implementations of these finite symmetry transformations, some kinds of nontrivial solutions and B"{a}cklund transformations are derived for both equations.
{"title":"Nonlocal Symmetries of Two 2-component Equations of Camassa-Holm Type","authors":"Ziqi Li, Kai Tian","doi":"arxiv-2402.12618","DOIUrl":"https://doi.org/arxiv-2402.12618","url":null,"abstract":"For a 2-component Camassa-Holm equation, as well as a 2-component\u0000generalization of the modified Camassa-Holm equation, nonlocal infinitesimal\u0000symmetries quadratically depending on eigenfunctions of linear spectral\u0000problems are constructed from functional gradients of spectral parameters. With\u0000appropriate pseudo-potentials, these nonlocal infinitesimal symmetries are\u0000prolonged to enlarged systems, and then explicitly integrated to generate\u0000symmetry transformations in finite form for enlarged systems. As\u0000implementations of these finite symmetry transformations, some kinds of\u0000nontrivial solutions and B\"{a}cklund transformations are derived for both\u0000equations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"276 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider an integro-differential equation model for traffic flow which is an extension of the Burgers equation model. To discuss the model, we first examine general settings for integrable integro-differential equations and find that they are obtained through a simple residue formula from integrable eqations in a complex domain. As demonstration of the efficiency of this approach, we list several integrable equations including a difference equation with double singular integral and an equation with elliptic singular integral. Then, we discuss the traffic model with singular integral and show that the model exhibits interaction between free flow region and congested region depending on the parameter of non-locality.
{"title":"Non-local time evolution equation with singular integral and its application to traffic flow model","authors":"Kohei Higashi","doi":"arxiv-2402.13128","DOIUrl":"https://doi.org/arxiv-2402.13128","url":null,"abstract":"We consider an integro-differential equation model for traffic flow which is\u0000an extension of the Burgers equation model. To discuss the model, we first\u0000examine general settings for integrable integro-differential equations and find\u0000that they are obtained through a simple residue formula from integrable\u0000eqations in a complex domain. As demonstration of the efficiency of this\u0000approach, we list several integrable equations including a difference equation\u0000with double singular integral and an equation with elliptic singular integral.\u0000Then, we discuss the traffic model with singular integral and show that the\u0000model exhibits interaction between free flow region and congested region\u0000depending on the parameter of non-locality.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Volterra lattice, when imposing non-zero constant boundary values, admits the structure of a completely integrable Hamiltonian system if the system size is sufficiently small. Such a Volterra lattice can be regarded as an epidemic model known as the SIR model with vaccination, which extends the celebrated SIR model to account for vaccination. Upon the introduction of an appropriate variable transformation, the SIR model with vaccination reduces to an Abel equation of the first kind, which corresponds to an exact differential equation. The equipotential curve of the exact differential equation is the Lambert curve. Thus, the general solution to the initial value problem of the SIR model with vaccination, or the Volterra lattice with constant boundary values, is implicitly provided by using the Lambert W function.
当施加非零常数边界值时，如果系统规模足够小，Volterra 网格就会具有完全可积分哈密顿系统的结构。这种 Volterra 网格可被视为一种流行病模型，即带疫苗接种的 SIR 模型，它扩展了著名的 SIR 模型，以考虑疫苗接种。在引入适当的变量变换后，带疫苗接种的 SIR 模型就简化为第一类阿贝勒方程，相当于精确微分方程。精确微分方程的等势线就是兰伯特曲线。因此，有疫苗接种的 SIR 模型或具有恒定边界值的 Volterra 晶格的初值问题的一般解，是通过使用兰伯特 W 函数隐含提供的。
{"title":"The Volterra lattice, Abel's equation of the first kind, and the SIR epidemic models","authors":"Atsushi Nobe","doi":"arxiv-2402.11888","DOIUrl":"https://doi.org/arxiv-2402.11888","url":null,"abstract":"The Volterra lattice, when imposing non-zero constant boundary values, admits\u0000the structure of a completely integrable Hamiltonian system if the system size\u0000is sufficiently small. Such a Volterra lattice can be regarded as an epidemic\u0000model known as the SIR model with vaccination, which extends the celebrated SIR\u0000model to account for vaccination. Upon the introduction of an appropriate\u0000variable transformation, the SIR model with vaccination reduces to an Abel\u0000equation of the first kind, which corresponds to an exact differential\u0000equation. The equipotential curve of the exact differential equation is the\u0000Lambert curve. Thus, the general solution to the initial value problem of the\u0000SIR model with vaccination, or the Volterra lattice with constant boundary\u0000values, is implicitly provided by using the Lambert W function.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Perturbation theory of vacuum spherically-symmetric spacetimes (including the cosmological constant) has greatly contributed to the understanding of black holes, relativistic compact stars and even inhomogeneous cosmological models. The perturbative equations can be decoupled in terms of (gauge-invariant) master functions satisfying $1+1$ wave equations. In this work, building on previous work on the structure of the space of master functions and equations, we study the reconstruction of the metric perturbations in terms of the master functions. To that end, we consider the general situation in which the perturbations are driven by an arbitrary energy-momentum tensor. Then, we perform the metric reconstruction in a completely general perturbative gauge. In doing so, we investigate the role of Darboux transformations and Darboux covariance, responsible for the isospectrality between odd and even parity in the absence of matter sources and also of the physical equivalence between the descriptions based on all the possible master equations. We also show that the metric reconstruction can be carried out in terms of any of the possible master functions and that the expressions admit an explicitly covariant form.
{"title":"Gauge-Independent Metric Reconstruction of Perturbations of Vacuum Spherically-Symmetric Spacetimes","authors":"Michele Lenzi, Carlos F. Sopuerta","doi":"arxiv-2402.10004","DOIUrl":"https://doi.org/arxiv-2402.10004","url":null,"abstract":"Perturbation theory of vacuum spherically-symmetric spacetimes (including the\u0000cosmological constant) has greatly contributed to the understanding of black\u0000holes, relativistic compact stars and even inhomogeneous cosmological models.\u0000The perturbative equations can be decoupled in terms of (gauge-invariant)\u0000master functions satisfying $1+1$ wave equations. In this work, building on\u0000previous work on the structure of the space of master functions and equations,\u0000we study the reconstruction of the metric perturbations in terms of the master\u0000functions. To that end, we consider the general situation in which the\u0000perturbations are driven by an arbitrary energy-momentum tensor. Then, we\u0000perform the metric reconstruction in a completely general perturbative gauge.\u0000In doing so, we investigate the role of Darboux transformations and Darboux\u0000covariance, responsible for the isospectrality between odd and even parity in\u0000the absence of matter sources and also of the physical equivalence between the\u0000descriptions based on all the possible master equations. We also show that the\u0000metric reconstruction can be carried out in terms of any of the possible master\u0000functions and that the expressions admit an explicitly covariant form.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The influence of an external oscillating in time magnetic field on the dynamics of the Davydov's soliton is investigated. It is shown that it essentially depends not only on the amplitude and frequency of the magnetic field, but also on the field orientation with respect to the molecular chain axis. The soliton velocity and phase are calculated. They are oscillating in time functions with the frequency of the main harmonic, given by the external field frequency, and higher multiple harmonics. It is concluded that such complex effects of external time-depending magnetic fields on the dynamics of solitons modify the charge transport in low-dimensional molecular systems, which can affect functioning of the devices based on such systems. These results suggest also the physical mechanism of therapeutic effects of oscillating magnetic fields, based on the field influence on the dynamics of solitons which provide charge transport through biological macro-molecules in the redox processes.
{"title":"Davydov's soliton in an external alternating magnetic field","authors":"Larissa Brizhik","doi":"arxiv-2402.09172","DOIUrl":"https://doi.org/arxiv-2402.09172","url":null,"abstract":"The influence of an external oscillating in time magnetic field on the\u0000dynamics of the Davydov's soliton is investigated. It is shown that it\u0000essentially depends not only on the amplitude and frequency of the magnetic\u0000field, but also on the field orientation with respect to the molecular chain\u0000axis. The soliton velocity and phase are calculated. They are oscillating in\u0000time functions with the frequency of the main harmonic, given by the external\u0000field frequency, and higher multiple harmonics. It is concluded that such\u0000complex effects of external time-depending magnetic fields on the dynamics of\u0000solitons modify the charge transport in low-dimensional molecular systems,\u0000which can affect functioning of the devices based on such systems. These\u0000results suggest also the physical mechanism of therapeutic effects of\u0000oscillating magnetic fields, based on the field influence on the dynamics of\u0000solitons which provide charge transport through biological macro-molecules in\u0000the redox processes.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139760311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is the second part in a series aiming at exploiting tools from Hamiltonian mechanics to study the motion of an extended body in general relativity. In the first part of this work, we constructed a 10-dimensional, covariant Hamiltonian framework that encodes all the linear-in-spin corrections to the geodesic motion. This formulation, although non-canonical, revealed that, at this linear-in-spin order, the integrability of Schwarzschild and Kerr geodesics remain. Building on this formalism, in the present work, we translate this abstract integrability result into tangible applications for linear-in-spin dynamics of a compact object into a Schwarzschild background spacetime. In particular, we construct a canonical system of coordinates which exploits the spherical symmetry of the Schwarzschild spacetime. They are based on a relativistic generalization of the classical Andoyer variables of Newtonian rigid body motion. This canonical setup, then, allows us to derive ready-to-use formulae for action-angle coordinates and gauge-invariant Hamiltonian frequencies, which automatically include all linear-in-spin effects. No external parameters or ad hoc choices are necessary, and the framework can be used to find complete solutions by quadrature of generic, bound, linear-in-spin orbits, including orbital inclination, precession and eccentricity, as well as spin precession. We demonstrate the strength of the formalism in the simple setting of circular orbits with arbitrary spin and orbital precession, and validate them against known results in the literature.
{"title":"Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime","authors":"Paul Ramond, Soichiro Isoyama","doi":"arxiv-2402.05049","DOIUrl":"https://doi.org/arxiv-2402.05049","url":null,"abstract":"This work is the second part in a series aiming at exploiting tools from\u0000Hamiltonian mechanics to study the motion of an extended body in general\u0000relativity. In the first part of this work, we constructed a 10-dimensional,\u0000covariant Hamiltonian framework that encodes all the linear-in-spin corrections\u0000to the geodesic motion. This formulation, although non-canonical, revealed\u0000that, at this linear-in-spin order, the integrability of Schwarzschild and Kerr\u0000geodesics remain. Building on this formalism, in the present work, we translate\u0000this abstract integrability result into tangible applications for\u0000linear-in-spin dynamics of a compact object into a Schwarzschild background\u0000spacetime. In particular, we construct a canonical system of coordinates which\u0000exploits the spherical symmetry of the Schwarzschild spacetime. They are based\u0000on a relativistic generalization of the classical Andoyer variables of\u0000Newtonian rigid body motion. This canonical setup, then, allows us to derive\u0000ready-to-use formulae for action-angle coordinates and gauge-invariant\u0000Hamiltonian frequencies, which automatically include all linear-in-spin\u0000effects. No external parameters or ad hoc choices are necessary, and the\u0000framework can be used to find complete solutions by quadrature of generic,\u0000bound, linear-in-spin orbits, including orbital inclination, precession and\u0000eccentricity, as well as spin precession. We demonstrate the strength of the\u0000formalism in the simple setting of circular orbits with arbitrary spin and\u0000orbital precession, and validate them against known results in the literature.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139760314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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