On Some Aspects of the Geometry of Differential Equations in Physics

Gràcia, Xavier; Muñoz-Lecanda, Miguel C.; Román‐Roy, Narciso

doi:10.1142/s0219887804000150

Cited by 9 publications

(7 citation statements)

References 77 publications

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“…There exists C such that φ (2) a := φ(1) a = 0, (12) where the derivative is taken in the direction of Eq. ( 4) and u = C. The subset obtained, that we will denote by M 2 , is again a linear submanifold of M 1 (this is also true in the time-dependent case) and we shall denote the functions defining M 2 in M 1 by φ…”

Section: Constraint Algorithms For Singular Lq Systemsmentioning

confidence: 99%

A numerical algorithm for singular optimal LQ control systems

Delgado-Téllez

Ibort

2009

Numer Algor

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Abstract. A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear-quadratic optimal control problems is presented. The algorithm is based on the presymplectic constraint algorithm (PCA) by GotayNester [11,26] that allows to solve presymplectic hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems [9]. The numerical implementation of the algorithm is based on the singular value decomposition that, on each step allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 3 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour.

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Section: Constraint Algorithms For Singular Lq Systemsmentioning

confidence: 99%

A numerical algorithm for singular optimal LQ control systems

Delgado-Téllez

Ibort

2009

Numer Algor

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“…The Lagrangian phase space formulation of mechanics [1][2][3][4], with its roots in differential geometry, provides an especially fruitful framework with which to analyze dynamical systems of singular Lagrangians L. Instead of trajectories ( ) ( ( ) ( )) = q t q t q t ,..., D 1 on a D-dimensional configuration space  that are solutions of the Euler-Lagrange equations of motion, trajectories in the Lagrangian phase space formulation with E being the energy of the system. For regular Lagrangians, W L is symplectic.…”

Section: Introductionmentioning

confidence: 99%

Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics

Speliotopoulos¹

2020

J. Phys. Commun.

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Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation-called second-order, Euler-Lagrange vector fields (SOELVFs)-with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.

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Section: Introductionmentioning

confidence: 99%

Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics

Speliotopoulos

2020

Preprint

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Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation-called second-order, Euler-Lagrange vector fields (SOELVFs)-with integral flows that have this symmetry are determined.Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.

show abstract

On Some Aspects of the Geometry of Differential Equations in Physics

Cited by 9 publications

References 77 publications

A numerical algorithm for singular optimal LQ control systems

A numerical algorithm for singular optimal LQ control systems

Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics

Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics

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