Elias Maciel, Inocencio Ortiz, Christian E. Schaerer
We propose a numerical scheme for the time-integration of nonholonomic mechanical systems, both conservative and nonconservative. The scheme is obtained by simultaneously discretizing the constraint equations and the Herglotz variational principle. We validate the method using numerical simulations and contrast them against the results of standard methods from the literature.
{"title":"A Herglotz-based integrator for nonholonomic mechanical systems","authors":"Elias Maciel, Inocencio Ortiz, Christian E. Schaerer","doi":"10.3934/jgm.2023012","DOIUrl":"https://doi.org/10.3934/jgm.2023012","url":null,"abstract":"<abstract><p>We propose a numerical scheme for the time-integration of nonholonomic mechanical systems, both conservative and nonconservative. The scheme is obtained by simultaneously discretizing the constraint equations and the Herglotz variational principle. We validate the method using numerical simulations and contrast them against the results of standard methods from the literature.</p></abstract>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135534446","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}
Recently, T. Masson, J. Francois, S. Lazzarini, C. Fournel and J. Attard have introduced a new method of the reduction of gauge symmetries called the dressing field method. In this paper we analyse this method from the fiber bundle point of view and we show the geometric implications for a principal bundle underlying a given gauge theory.We show how the existence of a dressing field satisfying certain conditions naturally leads to the reduction of the principal bundle and, as a consequence, to the reduction of the configuration and phase bundle of the system.
最近，T. Masson, J. Francois, S. Lazzarini, C. Fournel和J. Attard提出了一种新的减少规范对称性的方法，称为修整场法。本文从纤维束的角度分析了这种方法，并给出了给定规范理论下的主束的几何含义。我们证明了满足一定条件的修整场的存在如何自然地导致主束的缩减，从而导致系统的构型和相束的缩减。
{"title":"The dressing field method in gauge theories - geometric approach","authors":"Marcin Zając","doi":"10.3934/jgm.2023007","DOIUrl":"https://doi.org/10.3934/jgm.2023007","url":null,"abstract":"Recently, T. Masson, J. Francois, S. Lazzarini, C. Fournel and J. Attard have introduced a new method of the reduction of gauge symmetries called the dressing field method. In this paper we analyse this method from the fiber bundle point of view and we show the geometric implications for a principal bundle underlying a given gauge theory.We show how the existence of a dressing field satisfying certain conditions naturally leads to the reduction of the principal bundle and, as a consequence, to the reduction of the configuration and phase bundle of the system.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86727009","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}
In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.
{"title":"Lagrangian–Hamiltonian formalism for cocontact systems","authors":"X. Rivas, Daniel Torres","doi":"10.3934/jgm.2023001","DOIUrl":"https://doi.org/10.3934/jgm.2023001","url":null,"abstract":"In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88760731","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}
A. Pigazzini, C. Özel, Saeid Jafari, R. Pinčák, A. DeBenedictis
We derive the general formulas for a special configuration of the sequential warped-product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a generic diagonal conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $ n_1 $-dimensional and $ n_2 $-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $ (n_1-1) $-dimensional group of transformations to the case of positive constant Ricci curvature ($ lambda > 0 $).
{"title":"A family of special case sequential warped-product manifolds","authors":"A. Pigazzini, C. Özel, Saeid Jafari, R. Pinčák, A. DeBenedictis","doi":"10.3934/jgm.2023006","DOIUrl":"https://doi.org/10.3934/jgm.2023006","url":null,"abstract":"We derive the general formulas for a special configuration of the sequential warped-product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a generic diagonal conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $ n_1 $-dimensional and $ n_2 $-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $ (n_1-1) $-dimensional group of transformations to the case of positive constant Ricci curvature ($ lambda > 0 $).","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91070293","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}
The real (compact) Stiefel manifold realized as set of orthonormal frames is considered as a pseudo-Riemannian submanifold of an open subset of a vector space equipped with a multi-parameter family of pseudo-Riemannian metrics. This family contains several well-known metrics from the literature. Explicit matrix-type formulas for various differential geometric quantities are derived. The orthogonal projections onto tangent spaces are determined. Moreover, by computing the metric spray, the geodesic equation as an explicit second order matrix valued ODE is obtained. In addition, for a multi-parameter subfamily, explicit matrix-type formulas for pseudo-Riemannian gradients and pseudo-Riemannian Hessians are derived. Furthermore, an explicit expression for the second fundamental form and an explicit formula for the Levi-Civita covariant derivative are obtained. Detailed proofs are included.
{"title":"A multi-parameter family of metrics on stiefel manifolds and applications","authors":"Markus Schlarb","doi":"10.3934/jgm.2023008","DOIUrl":"https://doi.org/10.3934/jgm.2023008","url":null,"abstract":"The real (compact) Stiefel manifold realized as set of orthonormal frames is considered as a pseudo-Riemannian submanifold of an open subset of a vector space equipped with a multi-parameter family of pseudo-Riemannian metrics. This family contains several well-known metrics from the literature. Explicit matrix-type formulas for various differential geometric quantities are derived. The orthogonal projections onto tangent spaces are determined. Moreover, by computing the metric spray, the geodesic equation as an explicit second order matrix valued ODE is obtained. In addition, for a multi-parameter subfamily, explicit matrix-type formulas for pseudo-Riemannian gradients and pseudo-Riemannian Hessians are derived. Furthermore, an explicit expression for the second fundamental form and an explicit formula for the Levi-Civita covariant derivative are obtained. Detailed proofs are included.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79861677","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}
This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmuller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra.
{"title":"On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras","authors":"J. Huebschmann","doi":"10.3934/jgm.2021009","DOIUrl":"https://doi.org/10.3934/jgm.2021009","url":null,"abstract":"This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmuller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76339193","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}
We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms, we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.
{"title":"A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian","authors":"Jordi Gaset Rifà, Arnau Mas","doi":"10.3934/jgm.2023014","DOIUrl":"https://doi.org/10.3934/jgm.2023014","url":null,"abstract":"We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms, we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85935358","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}
We study the modular class of begin{document}$ Q $end{document}-manifolds, and in particular of negatively graded Lie begin{document}$ infty $end{document}-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie begin{document}$ infty $end{document}-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie begin{document}$ infty $end{document}-algebroids and their begin{document}$ Q $end{document}-manifold equivalent, which we hope to be of use for future reference.
We study the modular class of begin{document}$ Q $end{document}-manifolds, and in particular of negatively graded Lie begin{document}$ infty $end{document}-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie begin{document}$ infty $end{document}-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie begin{document}$ infty $end{document}-algebroids and their begin{document}$ Q $end{document}-manifold equivalent, which we hope to be of use for future reference.
{"title":"Modular class of Lie $ infty $-algebroids and adjoint representations","authors":"R. Caseiro, C. Laurent-Gengoux","doi":"10.3934/jgm.2022008","DOIUrl":"https://doi.org/10.3934/jgm.2022008","url":null,"abstract":"<p style='text-indent:20px;'>We study the modular class of <inline-formula><tex-math id=\"M2\">begin{document}$ Q $end{document}</tex-math></inline-formula>-manifolds, and in particular of negatively graded Lie <inline-formula><tex-math id=\"M3\">begin{document}$ infty $end{document}</tex-math></inline-formula>-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie <inline-formula><tex-math id=\"M4\">begin{document}$ infty $end{document}</tex-math></inline-formula>-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie <inline-formula><tex-math id=\"M5\">begin{document}$ infty $end{document}</tex-math></inline-formula>-algebroids and their <inline-formula><tex-math id=\"M6\">begin{document}$ Q $end{document}</tex-math></inline-formula>-manifold equivalent, which we hope to be of use for future reference.</p>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82496223","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}
This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories—called $ Q $-local minimizers and $ Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.
{"title":"Local minimizers for variational obstacle avoidance on Riemannian manifolds","authors":"Jacob R. Goodman","doi":"10.3934/jgm.2023003","DOIUrl":"https://doi.org/10.3934/jgm.2023003","url":null,"abstract":"This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories—called $ Q $-local minimizers and $ Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80781625","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}
A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in [1] and [2]. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in [3] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.
{"title":"Time-adaptive Lagrangian variational integrators for accelerated optimization","authors":"Valentin Duruisseaux, M. Leok","doi":"10.3934/jgm.2023010","DOIUrl":"https://doi.org/10.3934/jgm.2023010","url":null,"abstract":"<abstract><p>A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in <sup>[<xref ref-type=\"bibr\" rid=\"b1\">1</xref>]</sup> and <sup>[<xref ref-type=\"bibr\" rid=\"b2\">2</xref>]</sup>. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in <sup>[<xref ref-type=\"bibr\" rid=\"b3\">3</xref>]</sup> to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.</p></abstract>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79031673","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}
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