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This letter focuses on studying the theory of Hojman conserved quantity of the discrete non-conservative dynamical systems. The operators of discrete translation and discrete differentiation to the right and left are introduced in discrete non-conservative dynamical systems. The Hojman theorems, the determining equations and Hojman conserved quantities of the Lie symmetry are obtained for discrete non-conservative dynamical systems. Finally, an example is discussed to illustrate the application of the results.

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature.

Multisymplectic variational integrators are structure-preserving numerical schemes especially designed for PDEs derived from covariant spacetime Hamilton principles. The goal of this paper is to study the properties of the temporal and spatial discrete evolution maps obtained from a multisymplectic numerical scheme. Our study focuses on a (1+1)-dimensional spacetime discretized by triangles, but our approach carries over naturally to more general cases. In the case of Lie group symmetries, we explore the links between the discrete Noether theorems associated to the multisymplectic spacetime discretization and to the temporal and spatial discrete evolution maps, and emphasize the role of boundary conditions. We also consider in detail the case of multisymplectic integrators on Lie groups. Our results are illustrated with the numerical example of a geometrically exact beam model.

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

In this paper we develop the construction of geometric integrators for higher-order mechanics. These integrators are naturally symplectic and preserve momentum since they have a discrete variational origin.

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler–Lagrange equations. Interesting applications as, for instance, a discrete derivation of the Euler–Poincaré equations for second-order Lagrangians and its application to optimal control of a rigid body, and of a Cosserat rod are shown at the end of the paper.

After the recall of the theory of incursive discrete harmonic oscillator, it is shown the hyperincursive discrete harmonic oscillator is separable into two incursive oscillators. It is shown that any differential continuous derivative bifurcates into two difference discrete derivatives. For second order differential equations, a generalized discrete derivative is presented, that can become complex, defining so a complex velocity. The hyperincursive discrete time equation of the Schrödinger quantum oscillator is recalled, to show that the hyperincursive discrete equations contribute to a unification of classical and quantum mechanics. Finally, we develop the theoretical presentation of the hyperincursive discrete equation of the Klein-Gordon differential second order equation which bifurcates to 4 first order discrete equations that gives the original Dirac quantum relativist equation. This is a remarkable result because the Dirac equation is rediscovered from this new method based on the hyperincursive discrete second order equation which bifurcates to first order equations.