DGDSA - Geometria Diferencial, Sistemes Dinàmics i Aplicacions http://hdl.handle.net/2117/3202 2020-03-30T00:31:59Z 2020-03-30T00:31:59Z Multisymplectic unified formalism for Einstein-Hilbert gravity Gaset Rifà, Jordi Román Roy, Narciso http://hdl.handle.net/2117/117223 2020-02-12T22:41:51Z 2018-05-15T09:54:12Z 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. 2018-05-15T09:54:12Z 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. Hamilton-Jacobi theory in multisymplectic classical field theories De León, Manuel Prieto Martínez, Pedro Daniel Román Roy, Narciso Vilariño Fernández, Silvia http://hdl.handle.net/2117/115384 2020-02-12T22:26:44Z 2018-03-19T10:04:33Z 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. 2018-03-19T10:04:33Z 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. The Hamiltonian tube of a cotangent-lifted action Rodríguez Olmos, Miguel Andrés Teixidó Román, Miguel http://hdl.handle.net/2117/108284 2019-12-19T09:42:20Z 2017-10-02T13:47:26Z 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. 2017-10-02T13:47:26Z 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. Regularity properties of fiber derivatives associated with higher-order mechanical systems Colombo, Leonardo Prieto Martínez, Pedro Daniel http://hdl.handle.net/2117/104055 2019-12-19T09:42:11Z 2017-05-04T10:50:10Z 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. 2017-05-04T10:50:10Z 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. Equivalence between the Hamiltonian and Lagrangian formalisms for constrained systems Batlle Arnau, Carles Gomis Torné, Joaquin Pons Ràfols, Josep Maria Román Roy, Narciso http://hdl.handle.net/2117/103145 2020-02-12T21:57:41Z 2017-03-31T10:40:29Z 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. 2017-03-31T10:40:29Z 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. 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 http://hdl.handle.net/2117/102965 2020-02-12T22:49:04Z 2017-03-28T13:13:36Z 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 2017-03-28T13:13:36Z 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. Variational principles and symmetries on fibered multisymplectic manifolds Gaset Rifà, Jordi Prieto Martínez, Pedro Daniel Román Roy, Narciso http://hdl.handle.net/2117/101999 2020-02-12T21:57:14Z 2017-03-07T08:22:02Z 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. 2017-03-07T08:22:02Z 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. The wave equation for stiff strings and piano tuning Gràcia Sabaté, Francesc Xavier Sanz Perela, Tomás http://hdl.handle.net/2117/101752 2020-02-12T21:26:29Z 2017-03-01T06:51:11Z 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. 2017-03-01T06:51:11Z 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. Order reduction, projectability and constrainsts of second-order field theories and higuer-order mechanics Gaset Rifà, Jordi Román Roy, Narciso http://hdl.handle.net/2117/100664 2020-02-12T21:57:08Z 2017-02-08T10:31:46Z 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. 2017-02-08T10:31:46Z 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. A Hamiltonian study of the stability and bifurcations for the satellite problem Muñoz Lecanda, Miguel Carlos Rodríguez Olmos, Miguel Andrés Teixidó Román, Miguel http://hdl.handle.net/2117/85174 2020-02-12T05:51:54Z 2016-04-05T09:34:40Z A Hamiltonian study of the stability and bifurcations for the satellite problem Muñoz Lecanda, Miguel Carlos; Rodríguez Olmos, Miguel Andrés; Teixidó Román, Miguel We study the dynamics of a rigid body in a central gravitational field modeled as a Hamiltonian system with continuous rotational symmetries following the geometric framework of Wang et al. Novelties of our work are the use the reduced energy-momentum for the stability analysis and the treatment of axisymmetric bodies. We explicitly show the existence of new relative equilibria and study their stability and bifurcation patterns. The final publication is available at Springer via http://dx.doi.org/10.1007/s00332-015-9257-6 2016-04-05T09:34:40Z Muñoz Lecanda, Miguel Carlos Rodríguez Olmos, Miguel Andrés Teixidó Román, Miguel We study the dynamics of a rigid body in a central gravitational field modeled as a Hamiltonian system with continuous rotational symmetries following the geometric framework of Wang et al. Novelties of our work are the use the reduced energy-momentum for the stability analysis and the treatment of axisymmetric bodies. We explicitly show the existence of new relative equilibria and study their stability and bifurcation patterns.