DGDSA - Geometria Diferencial, Sistemes Dinàmics i Aplicacions
http://hdl.handle.net/2117/3202
Fri, 23 Jul 2021 17:03:38 GMT2021-07-23T17:03:38ZContinuous singularities in hamiltonian relative equilibria with abelian momentum isotropy
http://hdl.handle.net/2117/343814
Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy
Rodríguez Olmos, Miguel Andrés
We survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.
Fri, 16 Apr 2021 09:33:56 GMThttp://hdl.handle.net/2117/3438142021-04-16T09:33:56ZRodríguez Olmos, Miguel AndrésWe survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.Multisymplectic unified formalism for Einstein-Hilbert gravity
http://hdl.handle.net/2117/117223
Multisymplectic unified formalism for Einstein-Hilbert gravity
Gaset Rifà, Jordi; Román Roy, Narciso
We present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interest- ing when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the co- variant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomen- tum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geo- metrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.
Tue, 15 May 2018 09:54:12 GMThttp://hdl.handle.net/2117/1172232018-05-15T09:54:12ZGaset Rifà, JordiRomán Roy, NarcisoWe present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interest- ing when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the co- variant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomen- tum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geo- metrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.Hamilton-Jacobi theory in multisymplectic classical field theories
http://hdl.handle.net/2117/115384
Hamilton-Jacobi theory in multisymplectic classical field theories
De León, Manuel; Prieto Martínez, Pedro Daniel; Román Roy, Narciso; Vilariño Fernández, Silvia
The geometric framework for the Hamilton-Jacobi theory developed in the studies of Carinena et al. [Int. J. Geom. Methods Mod. Phys. 3(7), 1417-1458 (2006)], Carinena et al. [Int. J. Geom. Methods Mod. Phys. 13(2), 1650017 (2015)], and de Léon et al. [Variations, Geometry and Physics (Nova Science Publishers, New York, 2009)] is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.
Mon, 19 Mar 2018 10:04:33 GMThttp://hdl.handle.net/2117/1153842018-03-19T10:04:33ZDe León, ManuelPrieto Martínez, Pedro DanielRomán Roy, NarcisoVilariño Fernández, SilviaThe geometric framework for the Hamilton-Jacobi theory developed in the studies of Carinena et al. [Int. J. Geom. Methods Mod. Phys. 3(7), 1417-1458 (2006)], Carinena et al. [Int. J. Geom. Methods Mod. Phys. 13(2), 1650017 (2015)], and de Léon et al. [Variations, Geometry and Physics (Nova Science Publishers, New York, 2009)] is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.The Hamiltonian tube of a cotangent-lifted action
http://hdl.handle.net/2117/108284
The Hamiltonian tube of a cotangent-lifted action
Rodríguez Olmos, Miguel Andrés; Teixidó Román, Miguel
The Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawback of the MGS form is that it does not have an explicit expression. We will obtain a MGS form for cotangent-lifted actions on cotangent bundles that, in addition to its defining features, respects the additional fibered structure present. This model generalizes previous results obtained by T. Schmah for orbits with fully-isotropic momentum. In addition, our construction is explicit up to the integration of a differential equation on G. This equation can be easily solved for the groups SO(3) or SL(2), thus giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle.
Mon, 02 Oct 2017 13:47:26 GMThttp://hdl.handle.net/2117/1082842017-10-02T13:47:26ZRodríguez Olmos, Miguel AndrésTeixidó Román, MiguelThe Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawback of the MGS form is that it does not have an explicit expression. We will obtain a MGS form for cotangent-lifted actions on cotangent bundles that, in addition to its defining features, respects the additional fibered structure present. This model generalizes previous results obtained by T. Schmah for orbits with fully-isotropic momentum. In addition, our construction is explicit up to the integration of a differential equation on G. This equation can be easily solved for the groups SO(3) or SL(2), thus giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle.Regularity properties of fiber derivatives associated with higher-order mechanical systems
http://hdl.handle.net/2117/104055
Regularity properties of fiber derivatives associated with higher-order mechanical systems
Colombo, Leonardo; Prieto Martínez, Pedro Daniel
The aim of this work is to study fiber derivatives associated to Lagrangian and Hamiltonian functions describing the dynamics of a higher-order autonomous dynamical system. More precisely, given a function in T*T(k-1)Q, we find necessary and sufficient conditions for such a function to describe the dynamics of a kth-order autonomous dynamical system, thus being a kth-order Hamiltonian function. Then, we give a suitable definition of (hyper)regularity for these higher-order Hamiltonian functions in terms of their fiber derivative. In addition, we also study an alternative characterization of the dynamics in Lagrangian submanifolds in terms of the solutions of the higher-order Euler-Lagrange equations.
Thu, 04 May 2017 10:50:10 GMThttp://hdl.handle.net/2117/1040552017-05-04T10:50:10ZColombo, LeonardoPrieto Martínez, Pedro DanielThe aim of this work is to study fiber derivatives associated to Lagrangian and Hamiltonian functions describing the dynamics of a higher-order autonomous dynamical system. More precisely, given a function in T*T(k-1)Q, we find necessary and sufficient conditions for such a function to describe the dynamics of a kth-order autonomous dynamical system, thus being a kth-order Hamiltonian function. Then, we give a suitable definition of (hyper)regularity for these higher-order Hamiltonian functions in terms of their fiber derivative. In addition, we also study an alternative characterization of the dynamics in Lagrangian submanifolds in terms of the solutions of the higher-order Euler-Lagrange equations.Equivalence between the Hamiltonian and Lagrangian formalisms for constrained systems
http://hdl.handle.net/2117/103145
Equivalence between the Hamiltonian and Lagrangian formalisms for constrained systems
Batlle Arnau, Carles; Gomis Torné, Joaquin; Pons Ràfols, Josep Maria; Román Roy, Narciso
The equivalence between the Lagrangian and Hamiltonian formalism is studied for constraint systems. A procedure to construct the Lagrangian constraints from the Hamiltonian constraints is given. Those Hamiltonian constraints that are first class with respect to the Hamiltonian constraints produce Lagrangian constraints that are FL-projectable.
Fri, 31 Mar 2017 10:40:29 GMThttp://hdl.handle.net/2117/1031452017-03-31T10:40:29ZBatlle Arnau, CarlesGomis Torné, JoaquinPons Ràfols, Josep MariaRomán Roy, NarcisoThe equivalence between the Lagrangian and Hamiltonian formalism is studied for constraint systems. A procedure to construct the Lagrangian constraints from the Hamiltonian constraints is given. Those Hamiltonian constraints that are first class with respect to the Hamiltonian constraints produce Lagrangian constraints that are FL-projectable.Structural aspects of Hamilton–Jacobi theory
http://hdl.handle.net/2117/102965
Structural aspects of Hamilton–Jacobi theory
Cariñena Marzo, José F.; Gràcia Sabaté, Francesc Xavier; Marmo, Giuseppe; Martínez Fernandez, Eduardo; Muñoz Lecanda, Miguel Carlos; Román Roy, Narciso
In our previous papers [11, 13] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (‘slicing vector fields’) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton– Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.
The final publication is available at Springer via http://dx.doi.org/10.1142/S0219887816500171
Tue, 28 Mar 2017 13:13:36 GMThttp://hdl.handle.net/2117/1029652017-03-28T13:13:36ZCariñena Marzo, José F.Gràcia Sabaté, Francesc XavierMarmo, GiuseppeMartínez Fernandez, EduardoMuñoz Lecanda, Miguel CarlosRomán Roy, NarcisoIn our previous papers [11, 13] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (‘slicing vector fields’) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton– Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.Variational principles and symmetries on fibered multisymplectic manifolds
http://hdl.handle.net/2117/101999
Variational principles and symmetries on fibered multisymplectic manifolds
Gaset Rifà, Jordi; Prieto Martínez, Pedro Daniel; Román Roy, Narciso
The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.
Tue, 07 Mar 2017 08:22:02 GMThttp://hdl.handle.net/2117/1019992017-03-07T08:22:02ZGaset Rifà, JordiPrieto Martínez, Pedro DanielRomán Roy, NarcisoThe standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.The wave equation for stiff strings and piano tuning
http://hdl.handle.net/2117/101752
The wave equation for stiff strings and piano tuning
Gràcia Sabaté, Francesc Xavier; Sanz Perela, Tomás
We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.
Wed, 01 Mar 2017 06:51:11 GMThttp://hdl.handle.net/2117/1017522017-03-01T06:51:11ZGràcia Sabaté, Francesc XavierSanz Perela, TomásWe study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.Order reduction, projectability and constrainsts of second-order field theories and higuer-order mechanics
http://hdl.handle.net/2117/100664
Order reduction, projectability and constrainsts of second-order field theories and higuer-order mechanics
Gaset Rifà, Jordi; Román Roy, Narciso
The consequences of the projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ to a lower-order jet bundle are analyzed using the constraint algorithm for the Euler-Lagrange equations in $J^3\pi$. The results are applied to the Hilbert Lagrangian for the Einstein equations. Furthermore, the case of higher-order mechanics is also studied as a particular situation.
Wed, 08 Feb 2017 10:31:46 GMThttp://hdl.handle.net/2117/1006642017-02-08T10:31:46ZGaset Rifà, JordiRomán Roy, NarcisoThe consequences of the projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ to a lower-order jet bundle are analyzed using the constraint algorithm for the Euler-Lagrange equations in $J^3\pi$. The results are applied to the Hilbert Lagrangian for the Einstein equations. Furthermore, the case of higher-order mechanics is also studied as a particular situation.