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We present a numerical framework for modeling extended hyperelastic bodies based on a Lagrangian formulation of general relativistic elasticity theory. We use finite element methods to discretize the body, then use the semi--discrete action to derive ordinary differential equations of motion for the discrete nodes. The nodes are evolved in time using fourth--order Runge--Kutta. We validate our code against the normal modes of oscillation of a hyperelastic sphere, which are known analytically in the limit of small (linear), slow (Newtonian) oscillations. The algorithm displays second order convergence. This numerical framework can be used to obtain the orbital motion and internal dynamics of a hyperelastic body of any shape, for any spacetime metric, and for varying hyperelastic energy models., Comment: Corrected typos

Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular. That is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption insures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definition of the conjugate momenta can be inverted for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive -- there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950's. After a brief review of the Dirac--Bergmann algorithm, several physical examples are constructed from familiar components: point masses connected by massless springs, rods, cords and pulleys. The algorithm is also used to develop an initial value formulation of systems with holonomic constraints., Comment: 28 pages, 7 figures

The Dirac-Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories -- general relativity, electromagnetism, Yang Mills, string theory, etc. The Dirac-Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac-Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this paper I analyze a finite-dimensional system that exhibits all of the major steps in the algorithm. The system includes primary and secondary constraints, first and second class constraints, restrictions on Lagrange multipliers, and both physical and gauge degrees of freedom. This relatively simple system provides a platform for discussing the Dirac conjecture, constructing Dirac brackets, and applying gauge conditions., Comment: Final Version

The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally treated as the same order in smallness as the separation itself. This assumption is discussed in various papers, but is typically not recognized in textbooks. Relaxing this assumption leads to the generalized geodesic deviation equation (GGDE). We demonstrate the distinction between the GDE and the GGDE by computing the relative acceleration between timelike geodesics in two-dimensional de Sitter spacetime. We do this by considering a fiducial geodesic and a secondary geodesic (both timelike) that cross with nonzero speed. These geodesics are spanned by a spacelike geodesic, whose tangent evaluated at the fiducial geodesic defines the separation. The second derivative of the separation describes the relative acceleration between the fiducial and secondary geodesics. Near the crossing point, where the separation between the timelike geodesics is small but their rates of separation can be large, we show that the GGDE holds but the GDE fails to apply., Comment: 9 pages, 6 figures. Final version published in Classical and Quantum Gravity