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

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\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash 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.Comment: 45 pages. Published paper is open access, and can be found either at the Journal of Physics Communications website or at the DOI belo

Generalized Lie Symmetries and Almost Regular Lagrangians: A Link Between Symmetry and Dynamics

The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on the evolution of dynamical systems is determined. It is found that if the action has a generalized Lie symmetry, then the Lagrangian is necessarily singular; the converse is not true, as we show with a specific example. It is also found that the generalized Lie symmetry of the action is a Lie subgroup of the generalized Lie symmetry of the Euler-Lagrange equations of motion. The converse is once again not true, and there are systems for which the Euler-Lagrange equations of motion have a generalized Lie symmetry while the action does not, as we once again show through a specific example. Most importantly, it is shown that each generalized Lie symmetry of the action contributes one arbitrary function to the evolution of the dynamical system. The number of such symmetries gives a lower bound to the dimensionality of the family of curves emanating from any set of allowed initial data in the Lagrangian phase space. Moreover, if second- or higher-order Lagrangian constraints are introduced during the application of the Lagrangian constraint algorithm, these additional constraints could not have been due to the generalized Lie symmetry of the action.Comment: 34 pages with one table. Published paper is open access, and can be found either at the Journal of Physics Communications website or at the DOI below. This is a follow-up paper to "Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics", DOI: 10.1088/2399-6528/ab923c. arXiv admin note: text overlap with arXiv:2006.0261

Entanglement Generation by a Three-Dimensional Qubit Scattering: Concurrence vs. Path (In)Distinguishability

A scheme for generating an entangled state in a two spin-1/2 system by means of a spin-dependent potential scattering of another qubit is presented and analyzed in three dimensions. The entanglement is evaluated in terms of the concurrence both at the lowest and in full order in perturbation with an appropriate renormalization for the latter, and its characteristics are discussed in the context of (in)distinguishability of alternative paths for a quantum particle.Comment: 4 pages, 2 figures; minor corrections in the text, references updated, typos in a couple of equations correcte

Dark Energy and Extending the Geodesic Equations of Motion: Its Construction and Experimental Constraints

With the discovery of Dark Energy, $\Lambda_{DE}$, there is now a universal length scale, $\ell_{DE}=c/(\Lambda_{DE} G)^{1/2}$, associated with the universe that allows for an extension of the geodesic equations of motion. In this paper, we will study a specific class of such extensions, and show that contrary to expectations, they are not automatically ruled out by either theoretical considerations or experimental constraints. In particular, we show that while these extensions affect the motion of massive particles, the motion of massless particles are not changed; such phenomena as gravitational lensing remain unchanged. We also show that these extensions do not violate the equivalence principal, and that because $\ell_{DE}=14010^{800}_{820}$ Mpc, a specific choice of this extension can be made so that effects of this extension are not be measurable either from terrestrial experiments, or through observations of the motion of solar system bodies. A lower bound for the only parameter used in this extension is set.Comment: 19 pages. This is the published version of the first half of arXiv:0711.3124v2 with corrections include

Dark energy and extending the geodesic equations of motion: its construction and experimental constraints

With the discovery of Dark Energy, ΛDE, there is now a universal length scale, $${\ell_{\rm DE}=c/(\Lambda_{\rm DE} G)^{1/2}}$$ , associated with the universe that allows for an extension of the geodesic equations of motion. In this paper, we will study a specific class of such extensions, and show that contrary to expectations, they are not automatically ruled out by either theoretical considerations or experimental constraints. In particular, we show that while these extensions affect the motion of massive particles, the motion of massless particles are not changed; such phenomena as gravitational lensing remain unchanged. We also show that these extensions do not violate the equivalence principal, and that because $${\ell_{\rm DE}=14010^{800}_{820}}$$ Mpc, a specific choice of this extension can be made so that effects of this extension are not be measurable either from terrestrial experiments, or through observations of the motion of solar system bodies. A lower bound for the only parameter used in this extension is set

Dark energy and extending the geodesic equations of motion: A spectrum of galactic rotation curv

A recently proposed extension of the geodesic equations of motion, where the worldlinetraced by a test particle now depends on the scalar curvature, is used to study theformation of galaxies and galactic rotation curves. This extension is applied to themotion of a fluid in a spherical geometry, resulting in a set of evolution equations forthe fluid in the nonrelativistic and weak gravity limits. Focusing on the stationary solutionsof these equations and choosing a specific class of angular momenta for the fluidin this limit, we show that dynamics under this extension can result in the formation ofgalaxies with rotational velocity curves (RVC) that are consistent with the UniversalRotation Curve (URC), and through previous work on the URC, the observed rotationalvelocity profiles of 1100 spiral galaxies. In particular, a spectrum of RVCs canform under this extension, and we find that the two extreme velocity curves predictedby it brackets the ensemble of URCs constructed from these 1100 velocity profiles.We also find that the asymptotic behavior of the URC is consistent with that of themost probable asymptotic behavior of the RVCs predicted by the extension. A stabilityanalysis of these stationary solutions is also done, and we find them to be stable in thegalactic disk, while in the galactic hub they are stable if the period of oscillations ofperturbations is longer than 0.91+/- 0.31 to 1.58+/- 0.46 billion years