Stress tensors, Riemannian metrics and the alternative descriptions in elasticity

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Stability Analysis of a Rigid Body with Attached Geometrically Nonlinear Rod by the Energy-Momentum Method

This paper applies the energy-momentum method to the problem of nonlinear stability of relative equilibria of a rigid body with attached flexible appendage in a uniformly rotating state. The appendage is modeled as a geometrically exact rod which allows for finite bending, shearing and twist in three dimensions. Application of the energy-momentum method to this example depends crucially on a special choice of variables in terms of which the second variation block diagonalizes into blocks associated with rigid body modes and internal vibration modes respectively. The analysis yields a nonlinear stability result which states that relative equilibria are nonlinearly stable provided that; (i) the angular velocity is bounded above by the square root of the minimum eigenvalue of an associated linear operator and, (ii) the whole assemblage is rotating about the minimum axis of inertia

Normalizing connections and the energy-momentum method

The block diagonalization method for determining the stability of relative equilibria is discussed from the point of view of connections. We construct connections whose horizontal and vertical decompositions simultaneosly put the second variation of the augmented Hamiltonian and the symplectic structure into normal form. The cotangent bundle reduction theorem provides the setting in which the results are obtained

A block diagonalization theorem in the energy-momentum method

We prove a geometric generalization of a block diagonalization theorem first found by the authors for rotating elastic rods. The result here is given in the general context of simple mechanical systems with a symmetry group acting by isometries on a configuration manifold. The result provides a choice of variables for linearized dynamics at a relative equilibrium which block diagonalizes the second variation of an augmented energy these variables effectively separate the rotational and internal vibrational modes. The second variation of the effective Hamiltonian is block diagonal. separating the modes completely. while the symplectic form has an off diagonal term which represents the dynamic interaction between these modes. Otherwise, the symplectic form is in a type of normal form. The result sets the stage for the development of useful criteria for bifurcation as well as the stability criteria found here. In addition, the techniques should apply to other systems as well, such as rotating fluid masses

Earth-Moon Lagrangian points as a testbed for general relativity and effective field theories of gravity

We first analyse the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid. This scheme allows us to take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. A vehicle initially placed at L4 or L5 will not remain near the respective points. In particular, in the classical case the vehicle moves on a trajectory about the libration points for at least 700 days before escaping away. We show that this is true also if the modified long-distance Newtonian potential of effective gravity is employed. We also evaluate the impulse required to cancel out the perturbing force due to the Sun in order to force the spacecraft to stay precisely at L4 or L5. It turns out that this value is slightly modified with respect to the corresponding Newtonian one. In the second part of the paper, we first evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters and describe a tiny departure from the equilateral triangle. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. In other words, we deal with a theory involving quantum corrections to Einstein gravity, rather than to Newtonian gravity. By virtue of the effective-gravity correction to the long-distance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. In the latter case, for the point L1, general relativity corrects Newtonian theory by 7.61 meters, comparable, as an order of magnitude, with the lunar geodesic precession of about 3 meters per orbit. The latter is a cumulative effect accurately measured at the centimeter level through the lunar laser ranging positioning technique. Thus, it is possible to study a new laser ranging test of general relativity to measure the 7.61-meter correction to the L1 Lagrangian point, an observable never used before in the Sun-Earth-Moon system. Performing such an experiment requires controlling the propulsion to precisely reach L1, an instrumental accuracy comparable to the measurement of the lunar geodesic precession, understanding systematic effects resulting from thermal radiation and multi-body gravitational perturbations. This will then be the basis to consider a second-generation experiment to study deviations of effective field theories of gravity from general relativity in the Sun-Earth-Moon system

Trajectory and spacecraft design for a pole-sitter mission

This paper provides a detailed mission analysis and systems design of a pole-sitter mission. It considers a spacecraft that is continuously above either the North or South Pole and, as such, can provide real-time, continuous and hemispherical coverage of the polar regions. Two different propulsion strategies are proposed, which result in a near-term pole-sitter mission using solar electric propulsion and a far-term pole-sitter mission where the electric thruster is hybridized with a solar sail. For both propulsion strategies, minimum propellant pole-sitter orbits are designed. Optimal transfers from Earth to the pole-sitter are designed assuming Soyuz and Ariane 5 launch options, and a controller is shown to be able to maintain the trajectory under unexpected conditions such as injection errors. A detailed mass budget analysis allows for a trade-off between mission lifetime and payload mass capacity, and candidate payloads for a range of applications are investigated. It results that a payload of about 100 kg can operate for approximately 4 years with the solar-electric spacecraft, while the hybrid propulsion technology enables extending the missions up to 7 years. Transfers between north and south pole-sitter orbits are also considered to observe either pole when illuminated by the Sun

Relativistic effects in two-particle emission for electron and neutrino reactions

Two-particle two-hole contributions to electroweak response functions are computed in a fully relativistic Fermi gas, assuming that the electroweak current matrix elements are independent of the kinematics. We analyze the genuine kinematical and relativistic effects before including a realistic meson-exchange current (MEC) operator. This allows one to study the mathematical properties of the non-trivial seven-dimensional integrals appearing in the calculation and to design an optimal numerical procedure to reduce the computation time. This is required for practical applications to CC neutrino scattering experiments, where an additional integral over the neutrino flux is performed. Finally we examine the viability of this model to compute the electroweak 2p-2h response functions.Comment: Major revision (shortened). 22 pages, 18 figure

2p-2h excitations in neutrino scattering: angular distribution and frozen approximation

We study the phase-space dependence of 2p-2h excitations in neutrino scattering using the relativistic Fermi gas model. We follow a similar approach to other authors, but focusing in the phase-space properties, comparing with the non-relativistic model. A careful mathematical analysis of the angular distribution function for the outgoing nucleons is performed. Our goals are to optimize the CPU time of the 7D integral to compute the hadron tensor in neutrino scattering, and to conciliate the different relativistic and non relativistic models by describing general properties independently of the two-body current. For some emission angles the angular distribution becomes infinite in the Lab system, and we derive a method to integrate analytically around the divergence. Our results show that the frozen approximation, obtained by neglecting the momenta of the two initial nucleons inside the integral of the hadron tensor, reproduces fairly the exact response functions for constant current matrix elements.Comment: 8 pages, 4 figures. Contribution to 16th International Workshop on Neutrino Factories and Future Neutrino Beam Facilities, 25-30 August, 2014. Held at University of Glasgow, United Kingdo