Discrete Calculus Of Variations Research Articles

Brachistochrone of off-centered cylinders

We consider the problem of finding paths of shortest transit time between two points (popularly known as brachistochrone) for cylinders with off-centered center of mass, rolling down without slip, subject solely to the force of gravity. This problem is set up using principles of classical rigid body dynamics and the desired path function is solved for numerically using the method of discrete calculus of variations. We discover a distinct array of brachistochrone trajectories for off-centered cylinders, demonstrate a critical dependence of such paths on the initial location and orientation of cylinders’ centers of mass and bring new insights into the family of brachistochrone problems and solutions.

A structure-preserving algorithm for time-scale non-shifted Hamiltonian systems

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus. Not only can the combination of Δ and ∇ derivatives be beneficial to obtaining higher convergence order in numerical analysis, but also it prompts the time-scale numerical computational scheme to have good properties, for instance, structure-preserving. In this letter, a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed. By using the time-scale discrete variational method and calculus theory, and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems, the corresponding discrete Hamiltonian principle can be obtained. Furthermore, the time-scale discrete Hamilton difference equations, Noether theorem, and the symplectic scheme of discrete Hamiltonian systems are obtained. Finally, taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples, they show that the time-scale discrete variational method is a structure-preserving algorithm. The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.

Asynchronous finite differences in most probable distribution with finite numbers of particles

For a discrete function fx on a discrete set, the finite difference can be either forward and backward. If fx is a sum of two such functions fx=f1x+f2x, the first order difference of Δfx can be grouped into four possible combinations, in which two are the usual synchronous ones Δff1x+Δff2x and Δbf1x+Δbf2x, and other two are asynchronous ones Δff1x+Δbf2x and Δbf1x+Δff2x, where Δf and Δb denote the forward and backward difference respectively. Thus, the first order variation equation δfx=0 for this function fx gives at most four different solutions which contain both true and false one. A formalism of the discrete calculus of variations is developed to single out the true one by means of comparison of the second order variations, in which the largest value in magnitude indicates the true solution, yielding the exact form of the distributions for Boltzmann, Bose and Fermi system without requiring the numbers of particle to be infinitely large. When there is only one particle in the system, all distributions reduce to be the Boltzmann one.

On the inverse problem of the discrete calculus of variations

In this paper we present an algorithm to find the discrete Lagrangian for an autonomous recurrence relation of arbitrary even order 2k with k > 1. The method is based on the existence of a set of differential operators called annihilation operators which can be used to convert a functional equation into a system of linear partial differential equations. This completely solves the inverse problem of the calculus of variations in this setting.

Optimal control of a slot car racer

AbstractIn this contribution, we present a simulation method for the optimal control of a mechatronic system that is based on discrete variational calculus and apply it to compute the time‐minimal path of a slot car racer. Here, the DMOC (Discrete Mechanics and Optimal Control [4]) method is used to generate offline optimal trajectories for the electro‐mechanically coupled system, i.e. sequences of discrete configurations and sequences of driving voltages. The time‐minimal path is achieved by the choice of different cost functions, the sum of time steps, the negative sum of the quadratic momenta and the negative sum of the quadratic velocities. Simulation results show that the momentum formulation yields the lowest number of iterations. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discrete Moving Frames on Lattice Varieties and Lattice-Based Multispaces

In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver’s one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.

Helmholtz's inverse problem of the discrete calculus of variations

We derive the discrete version of the classical Helmholtz's condition. Precisely, we state a theorem characterizing second-order finite difference equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.

Discrete Variational Calculus for B-Spline Curves

In this work, we study discrete variational problems, for B-spline curves, which are invariant under translation and rotation. We show this approach has advantages over studying smooth variational problems whose solutions are approximated by B-spline curves. The latter method has been well studied in the literature but leads to high order approximation problems. We are particularly interested in Lagrangians that are invariant under the special Euclidean group for which B-spline approximated curves are well suited. The main application we present here is the curve completion problem in 2D and 3D. Here, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions of these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the curve completion problem. The novel methods we develop for the discrete curve completion problem are general, and can be used to solve other discrete variational problems for B-spline curves. Our method completely avoids the difficulties of high order smooth differential invariants.

A variational approach to define robustness for parametric multiobjective optimization problems

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter $${\lambda \in \mathbb{R}}$$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the ?-dependent Pareto set. In this work we give a new definition that allows to characterize ?-robust Pareto points, meaning points which hardly vary under the variation of the parameter ?. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe ?-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples.

DISCRETE SECOND-ORDER EULER–POINCARÉ EQUATIONS: APPLICATIONS TO OPTIMAL CONTROL

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.

Time scale differential, integral, and variational embeddings of Lagrangian systems

We introduce differential, integral, and variational delta embeddings. We prove that the integral delta embedding of the Euler–Lagrange equations and the variational delta embedding coincide on an arbitrary time scale. In particular, a new coherent embedding for the discrete calculus of variations that is compatible with the least-action principle is obtained.

Continuous and discrete approaches to vakonomic mechanics

In this paper we summarize the main features of vakonomic mechanics (or constrained variational calculus), both from continuous and discrete points of views. In the continuous case, we focus ourselves on Lagrangian systems defined by the following data: a Riemannian metric (kinetic term) and constraints linear on the velocities. We show that, for such kind of systems, it is possible to find an explicit Hamiltonian description. For the numerical setting, we describe two methods to design geometric integrators, first, applying discrete variational calculus to a discretization of the continuous vakonomic problem or, second, applying standard symplectic integration to the Hamiltonian description of the initial vakonomic system. We show that, in a particular case, both constructions match exactly.

Optimal control of underactuated mechanical systems: A geometric approach

In this paper, we consider a geometric formalism for optimal control of underactuated mechanical systems. Our techniques are an adaptation of the classical Skinner and Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics of the optimal control problem. These developments will allow us to develop a new class of geometric integrators based on discrete variational calculus.

Symmetric Three-Term Recurrence Equations and Their Symplectic Structure

We revive the study of the symmetric three-term recurrence equations. Our main result shows that these equations have a natural symplectic structure, that is, every symmetric three-term recurrence equation is a special discrete symplectic system. The assumptions on the coefficients in this paper are weaker and more natural than those in the current literature. In addition, our result implies that symmetric three-term recurrence equations are completely equivalent with Jacobi difference equations arising in the discrete calculus of variations. Presented applications of this study include the Riccati equation and inequality, detailed Sturmian separation and comparison theorems, and the eigenvalue theory for these three-term recurrence and Jacobi equations.

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.

Equivalent conditions to the nonnegativity of a quadratic functional in discrete optimal control

AbstractIn this paper we provide a characterization of the nonnegativity of a discrete quadratic functional ℐ with fixed right endpoint in the optimal control setting. This characterization is closely related to the kernel condition earlier introduced by M. Bohner as a part of a focal points definition for conjoined bases of the associated linear Hamiltonian difference system. When this kernel condition is satisfied only up to a certain critical index m, the traditional conditions, which are the focal points, conjugate intervals, implicit Riccati equation, and partial quadratic functionals, must be replaced by a new condition. This new condition is determined to be the nonnegativity of a block tridiagonal matrix, representing the remainder of ℐ after the index m, on a suitable subspace. Applications of our result include the discrete Jacobi condition, a unification of the nonnegativity and positivity of ℐ into one statement, and an improved result for the special case of the discrete calculus of variations. Even when both endpoints of ℐ are fixed, this paper provides a new result. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ℐ with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ℐ in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ℐ in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ℐ. Finally, we define partial quadratic functionals associated with ℐ and a (strong) regularity of ℐ, which we relate to the positivity and nonnegativity of ℐ.

Nonnegativity of a discrete quadratic functional in terms of the (strengthened) legendre and jacobi conditions

This paper contains a complete characterization of the nonnegativity of a discrete quadratic functional with one endpoint allowed to vary. In particular, we derive the exact form and explain the role of the (strengthened) Legendre condition in the discrete calculus of variations. Under this condition, the nonnegativity of the quadratic functional is equivalent to each of the following conditions: the nonexistence of intervals conjugate to 0, the existence of a certain conjoined basis of the associated Jacobi difference equation, the nonnegativity of certain recurrence matrices, and, under a natural additional assumption, the existence of a symmetric solution to the Riccati matrix difference equation. Moreover, an extension of the discrete Legendre condition is derived for the given discrete variational problem.

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.

First integrals in the discrete variational calculus

The intent of this paper is to show that first integrals of the discrete Euler equation can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analog of the classical theorem of E. Noether in the Calculus of Variations.