Hamiltonian formalism for nonlocal gravity models

Nonlocal gravity models are constructed to explain the current acceleration of the universe. These models are inspired by the infrared correction appearing in Einstein–Hilbert action. Here, we develop the Hamiltonian formalism of a nonlocal model by considering only terms to quadratic order in Riemann tensor, Ricci tensor and Ricci scalar. We show how to count degrees of freedom using Hamiltonian formalism including Ricci tensor and Ricci scalar terms. In this model, we have also worked out with a choice of a nonlocal action which has only two degrees of freedom equivalent to GR. Finally, we find the existence of additional constraints in Hamiltonian required to remove the ghosts in our full action. We also compare our results with that of obtained using Lagrangian formalism.

Modeling of centrifugal deployment of three-section minisatellite boom

The aim of the article is to model the processes of centrifugal deployment of a three-section boom and preliminary analyze the feasibility of this deployment method for an Earth remote sensing (ERS) minisatellite (MS). During the research, methods of theoretical mechanics, multibody dynamics, control theory, and computer modeling were used. Centrifugal deployment of multi-section booms have been successfully used on spin stabilized satellites, but not on ERS satellites, which have other features of operation and require additional studies. The main part of the MS is a platform to which a transformable antenna is attached by means of a transformable boom. Before deployment, the stowed boom and antenna are attached to the MS platform. The boom sections are connected by joints with one rotational degree of freedom and deployed sequentially due to centrifugal forces when the MS rotates in the required direction. Each of the boom joints has a locking mechanism that latches when a predetermined deploy angle is reached. To model the processes of the boom deployment, the MS is presented as a system of connected bodies, where the platform and the stowed antenna are absolutely rigid bodies, and the boom consists of three flexible beams of a tubular cross-section. The differential equations of the MS dynamics during the deployment are obtained using the Lagrangian formalism, which are supplemented by algebraic equations describing the constraints from the joints. The scenarios of the boom deployment with a constant control torque and a constant angular velocity of the MS are considered. These scenarios are simulated, and estimates of the control actions needed to ensure full deployment of the boom and the stabilization of the MS after latching of the joints are calculated. Dependences of variations of the loads on the boom structure during deployment are obtained. The simulation results allow us to conclude that it is feasible to implement the method of the boom centrifugal deployment for the MS, which can perform fast rotations about the three axes of the body reference frame. Implementation of this method allows designers to reduce mass of the MS because it does not require any servo drives in the boom deployment system.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.

Wilberforce pendulum: modelling linearly damped coupled oscillations of a spring-mass system

The Wilberforce pendulum is a coupled spring-mass system, where a mass with adjustable moment of inertia is suspended from a helical spring. Energy is converted between the translational and torsional modes, and this energy conversion is most clearly observed at resonance, which occurs when the damped natural frequencies of the two oscillation modes are equal. A theoretical model—with energy losses due to viscous damping accounted for—was formulated using the Lagrangian formalism to predict the pendulum mass’ trajectory. Theoretical predictions were compared with experimental data, showing good agreement. Fourier analysis of both theoretical predictions and experimental data further corroborate the validity of our quantitative model. The dependence of oscillation features like beat frequency and maximum conversion amplitude on relevant parameters such as the initial vertical displacement, initial angular displacement and moment of inertia was also investigated and experimentally verified.

Aplicación del formalismo de Lagrangianos Controlados en la regulación de velocidad del motor de inducción trifásico

In this work the Controlled Lagrangian Formalism applied to electrical machines is explored for the first time. It begins with an analysis of the purely mechanical systems, once understood, the study is carried out on a two-phase induction motor, this implying a greater degree of complexity because there is no reference that has done it before. Finally, this study is expanded to the three-phase motor, this being the main research object of the project. The main guide used was the Bloch article cite bloch2000controlled on the analysis of mechanical systems. Regarding the procedure, the first thing that is done is the selection of the generalized coordinates, the Lagrangian is proposed and the model is obtained from it through the Euler-Lagrange equations, followed by that the symmetries are identified (which in the case of MI is especially interesting because these symmetries are obvious from the choice of coordinates) and the configuration space is divided into vertical and horizontal directions, the horizontal directions are redefined and the Controlled Lagrangian is proposed. Finally, generalized forces are sought, using Noether's Theorem as support and thus establishing the control law. The development to obtain the Controlled Lagrangian and the control law is done in detail, explaining each step of the procedure and using specific algebraic methods of this formalism that are strongly based on the geometric structure of the variety of configuration. The results obtained are an approach in the direction of Controlled Lagrangians applied electrical machines.

Trajectories of a ball moving inside a spherical cavity using first integrals of the governing nonlinear system

Analytical study of ball vibration absorber behavior is presented in the paper. The dynamics of trajectories of a heavy ball moving without slipping inside a spherical cavity are analyzed. Following our previous work, where a similar system was investigated through various numerical simulations, research of the dynamic properties of a sphere moving in a spherical cavity was carried out by methods of analytical dynamics. The strategy of analytical investigation enabled definition of a set of special and limit cases which designate individual domains of regular trajectories. In order to avoid any mutual interaction between the domains along a particular trajectory movement, energy dissipation at the contact of the ball and the cavity has been ignored, as has any kinematic excitation due to cavity movement. A governing system was derived using the Lagrangian formalism and complemented by appropriate non-holonomic constraints of the Pfaff type. The three first integrals are defined, enabling the evaluation of trajectory types with respect to system parameters, the initial amount of total energy, the angular momentum of the ball and its initial spin velocity. The neighborhoods of the limit trajectories and their dynamic stability are assessed. Limit and transition special cases are investigated along with their individual elements. The analytical means of investigation enabled the performance of broad parametric studies. Good agreement was found when comparing the results achieved by the analytical procedures in this paper with those obtained by means of numerical simulations, as they followed from the Lagrangian approach and the Appell–Gibbs function presented in previous papers.

Chiral Dirac Equation and Its Spacetime and CPT Symmetries

The Dirac equation with chiral symmetry is derived using the irreducible representations of the Poincaré group, the Lagrangian formalism, and a novel method of projection operators that takes as its starting point the minimal assumption of four linearly independent physical states. We thereby demonstrate the fundamental nature of this form of the Dirac equation. The resulting equation is then examined within the context of spacetime and CPT symmetries with a discussion of the implications for the general formulation of physical theories.

Generation and collective interaction of giant magnetic dipoles in laser cluster plasma

Interaction of circularly polarized laser pulses with spherical nano-droplets generates nanometer-size magnets with lifetime on the order of hundreds of femtoseconds. Such magnetic dipoles are close enough in a cluster target and magnetic interaction takes place. We investigate such system of several magnetic dipoles and describe their rotation in the framework of Lagrangian formalism. The semi-analytical results are compared to particle-in-cell simulations, which confirm the theoretically obtained terrahertz frequency of the dipole oscillation.

A straightforward and Lagrangian proof of the Einsteinian equivalence between the mass and the inernal energy (i.e. rest energy)

I propose a Lagrangian proof of Einstein's well-known law that the mass of any system is its internal energy. The interest of this proof is to show how the distinction between internal degrees of freedom and the center of mass appears in the Lagrangian formalism. Considering that the Lagrangian depends on a particular set of variables for the internal degree of freedom, I show in a standard Lagrangian way how one can naturally find the desired law. This proof does not use the tensors of energy-momentum and can be easily used by students familiar with Lagrangian mechanics and the basis of Special Relativity. I apply the method for the particles and for the field, using the scalar field for simplification but it is easy to generalize for other fields (containing only the first derivative in Lagrangian). I give the example for the gravitation field. The method permits us to observe a strong relation between the Einstein’s E=mc² law and his other famous law of the time dilation. I carefully analyze the meaning of the particular choice of the variable and showing a sort of a modified speed addition formula without contradicting, of course, the one of Einstein (& Poincaré). I also try to untangle, with this point of view, the relation between the mass and the origin of the energy scale. Finally I analyze the reason why in Newtonian mechanic we don’t have a such law. In the annex I give some elements of the Hamiltonian analysis checking again the coherence of the particular set of variables and I apply this way of thinking to the old Lorentz-Poincaré model of the electron (useful for an explicit classical renormalization of the mass).