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C Sardon - One of the best experts on this subject based on the ideXlab platform.
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a geometric hamilton jacobi Theory on a nambu jacobi manifold
International Journal of Geometric Methods in Modern Physics, 2019Co-Authors: M De Leon, C SardonAbstract:In this paper, we propose a geometric Hamilton–Jacobi (HJ) Theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ Theory is that if a Hamiltonian vector field XH can be projected i...
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a hamilton jacobi Theory for implicit differential systems
Journal of Mathematical Physics, 2018Co-Authors: Ogul Esen, Manuel De Leon, C SardonAbstract:In this paper, we propose a geometric Hamilton-Jacobi Theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of TT*Q generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton–Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.
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geometry of the discrete hamilton jacobi equation applications in optimal control
Reports on Mathematical Physics, 2018Co-Authors: Manuel De Leon, C SardonAbstract:In this paper, we review the discrete Hamilton—Jacobi equation from a geometric point of view. In similarity with the continuous geometric Hamilton—Jacobi Theory, we propose two different discrete geometric interpretations for the equation. The first approach is based on the construction of a discrete Hamilton—Jacobi equation using discrete projective flows. For it, we develop some former results on discrete Hamiltonian systems and provide a discrete equation explicitly, which matches some previous results depicted in the literature. The interest of our method is that it retrieves some already known results, but starting from a new outlook. The second approach is formulated in terms of discrete vector fields, whose definition is not straightfoward. For this, we revisit the discrete Theory of mechanics by relying on the construction of discrete vector fields taken from optimal control backgrounds. From here, we reconstruct a discrete Hamilton—Jacobi equation in a novel way, and which has not been devised in the literature before. As a last result, both interpretations are proven to be equivalent theoretically, but the numerical results differ slightly. The discrete vector field approach seems fairly more accurate concerning numerical values in the specific example that we show, that is an optimal control problem for a nonlinear system.
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a geometric hamilton jacobi Theory on a nambu jacobi manifold
arXiv: Mathematical Physics, 2017Co-Authors: M De Leon, C SardonAbstract:In this paper we propose a geometric Hamilton--Jacobi Theory on a Nambu--Jacobi manifold. The advantange of a geometric Hamilton--Jacobi Theory is that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a one-form $dW$, then the integral curves of the projected vector field $X_H^{dW}$ can be transformed into integral curves of the vector field $X_H$ provided that $W$ is a solution of the Hamilton--Jacobi equation. This procedure allows us to reduce the dynamics to a lower dimensional manifold in which we integrate the motion. On the other hand, the interest of a Nambu--Jacobi structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric Hamilton--Jacobi equation on a Nambu--Jacobi manifold and apply it to the third-order Riccati differential equation as an example.
Edith Padron - One of the best experts on this subject based on the ideXlab platform.
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Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures
'MDPI AG', 2021Co-Authors: Sergio Grillo, Juan Carlos Marrero, Edith PadronAbstract:In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g
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extended hamilton jacobi Theory contact manifolds and integrability by quadratures
Journal of Mathematical Physics, 2020Co-Authors: Sergio Grillo, Edith PadronAbstract:A Hamilton–Jacobi Theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper, we shall apply such a Theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton–Jacobi equation related to these systems. Then, we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.A Hamilton–Jacobi Theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper, we shall apply such a Theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton–Jacobi equation related to these systems. Then, we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.
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extended hamilton jacobi Theory contact manifolds and integrability by quadratures
arXiv: Differential Geometry, 2019Co-Authors: Sergio Grillo, Edith PadronAbstract:A Hamilton-Jacobi Theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a Theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton-Jacobi Equation (HJE) related to these systems. Then we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.
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a hamilton jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and poisson manifolds
Journal of Geometry and Physics, 2016Co-Authors: Sergio Grillo, Edith PadronAbstract:Abstract In this paper we develop, in a geometric framework, a Hamilton–Jacobi Theory for general dynamical systems. Such a Theory contains the classical Theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton–Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton–Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.
Richard Kerner - One of the best experts on this subject based on the ideXlab platform.
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time reparametrization invariance and hamilton jacobi approach to the cosmological σ model
Protein Science, 2014Co-Authors: J W Van Holten, Richard KernerAbstract:The construction of physical models with local time-reparametrization invariance is reviewed. Negative-energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton-Jacobi Theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker type. After a discussion of empty universes, the FLRW Theory is extended with homogeneous scalar elds generically described by a -model on some scalar manifold. An explicit gauge-invariant solution is constructed for the non-linear O(N)-models.
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time reparametrization invariance and hamilton jacobi approach to the cosmological sigma model
arXiv: High Energy Physics - Theory, 2013Co-Authors: J W Van Holten, Richard KernerAbstract:The construction of physical models with local time-reparametrization invariance is reviewed. Negative-energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton-Jacobi Theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker type. After a discussion of empty universes, the FLRW Theory is extended with homogeneous scalar fields generically described by a $\sg$-model on some scalar manifold. An explicit gauge-invariant solution is constructed for the non-linear O(N)-models.
Gregory A Fiete - One of the best experts on this subject based on the ideXlab platform.
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analog of hamilton jacobi Theory for the time evolution operator
Physical Review A, 2019Co-Authors: Michael Vogl, Pontus Laurell, Aaron Barr, Gregory A FieteAbstract:In this paper we develop an analogue of Hamilton-Jacobi Theory for the time-evolution operator of a quantum many-particle system. The Theory offers a useful approach to develop approximations to the time-evolution operator, and also provides a unified framework and starting point for many well-known approximations to the time-evolution operator. In the important special case of periodically driven systems at stroboscopic times, we find relatively simple equations for the coupling constants of the Floquet Hamiltonian, where a straightforward truncation of the couplings leads to a powerful class of approximations. Using our Theory, we construct a flow chart that illustrates the connection between various common approximations, which also highlights some missing connections and associated approximation schemes. These missing connections turn out to imply an analytically accessible approximation that is the "inverse" of a rotating frame approximation and thus has a range of validity complementary to it. We numerically test the various methods on the one-dimensional Ising model to confirm the ranges of validity that one would expect from the approximations used. The Theory provides a map of the relations between the growing number of approximations for the time-evolution operator. We describe these relations in a table showing the limitations and advantages of many common approximations, as well as the new approximations introduced in this paper.
Sergio Grillo - One of the best experts on this subject based on the ideXlab platform.
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Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures
'MDPI AG', 2021Co-Authors: Sergio Grillo, Juan Carlos Marrero, Edith PadronAbstract:In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g
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extended hamilton jacobi Theory contact manifolds and integrability by quadratures
Journal of Mathematical Physics, 2020Co-Authors: Sergio Grillo, Edith PadronAbstract:A Hamilton–Jacobi Theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper, we shall apply such a Theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton–Jacobi equation related to these systems. Then, we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.A Hamilton–Jacobi Theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper, we shall apply such a Theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton–Jacobi equation related to these systems. Then, we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.
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extended hamilton jacobi Theory contact manifolds and integrability by quadratures
arXiv: Differential Geometry, 2019Co-Authors: Sergio Grillo, Edith PadronAbstract:A Hamilton-Jacobi Theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a Theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton-Jacobi Equation (HJE) related to these systems. Then we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.
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a hamilton jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and poisson manifolds
Journal of Geometry and Physics, 2016Co-Authors: Sergio Grillo, Edith PadronAbstract:Abstract In this paper we develop, in a geometric framework, a Hamilton–Jacobi Theory for general dynamical systems. Such a Theory contains the classical Theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton–Jacobi Theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton–Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is established. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.
