The theory behind analytical algorithms for computer-algebra systems is discussed. A computer system is described which is capable of optimizing input data, constructing equations of equilibrium and motion, formulating and solving the basic mechanics problems for a broad class of holonomic systems with elastic and dissipative constraints on the basis of the Lagrange-D'Alembert principle Keywords: computer-algebra system, equation of motion, holonomic system, Lagrange-D'Alembert principle Problem Formulation. The advent of computer-algebra systems (CASs) stimulated the development of problem-oriented systems (POSs) for PCs with the purpose of research automation and solution of the basic mechanics problems. The major results obtained in this filed are reported in [2][3][4][5][6][7][8][9][10][11] and pertain to discrete mechanical systems modeled by rigid bodies with elastic, dissipative, kinematic, and geometrical constraints. However, no unified approach to the problem has been worked out yet. Developments in the field are based on different principles of analytical mechanics [5]. Because of the specific nature of analytic transformations on PCs, methods having proved themselves well for manual use (Lagrange equations of the second kind, for one) may appear poorly suited for computer use. The classical and original methods may produce better results [3,6].The present paper discusses algorithms for computer-aided analytic representation and analysis of holonomic rigid-body systems with elastic and dissipative constraints based on the D'Alembert-Lagrange principle, proposes a theory and analytic algorithms for CASs and POSs, and describes a computer system that can optimize input data, construct analytic equations of motion and equilibrium, and formulate and solve the basic mechanics problems.1. Analytic Description of a Mechanical System. Let us consider a holonomic mechanical system consisting of n bodies, each having mass m i and principal central moments of inertia J ix , J iy , and J iz . The velocities of the centers of mass are defined in projections onto the axes of an absolute coordinate system (CS); and the angular velocities of the bodies, in projections, ω ix , ω iy , and ω iz , onto the principal central axes of the body-fixed CSs. The kinetic energy of the system is

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D'Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities Keywords: computer algebra system, equations of motion for systems of rigid bodies, holonomic and nonholonomic systems, D'Alembert-Lagrange principleIntroduction. Problems of nonholonomic mechanics are complicated regarding the formalization of description. They play a special role in studying mechanical systems and are important for engineering applications [2,[5][6][7][8][13][14][15]. There are different methods for describing, deriving, and integrating the equations of motion for nonholonomic systems [3,4,9,10]. These methods employ different representations of kinetic energy (or the Lagrangian function): as a function of generalized coordinates and velocities (equation with Lagrange multipliers); as a function of generalized coordinates and independent velocities (Chaplygin's and Voronets' equations); and as a function of generalized coordinates and pseudovelocities (Lagrange-Euler equations derived by Hamel). The acceleration energy function introduced by Appell [3] and the modified Hamiltonian function [10] are used with the same purpose. The listed methods are not universal-resulting expressions are rather awkward and do not guarantee correct results; and if the mechanical system changes its structure, then the equations should be derived anew. Therefore, the methods are of little use in the development of universal problem-oriented systems (POSs) for PCs with the purpose of research automation and solution of nonholonomic mechanics problems. The paper [12] proposed a theory and analytic algorithms for computer-algebra systems (CASs) used to develop a POS for PCs with the purpose of research automation and solution of basic problems in the mechanics of holonomic rigid-body systems with elastic and dissipative constraints based on the D'Alembert-Lagrange principle.In what follows, we will first calculate the number of operations to demonstrate the numerical efficiency of the equations of motion for holonomic systems and the fundamental possibility of deriving the equations of motion for nonholonomic systems using the method proposed in [12]. Then we will discuss algorithms (based on the D'Alembert-Lagrange principle) for computer-assisted derivation of the equations of motion for systems with nonholonomic constraints linear with respect to generalized velocities. We will also outline modifications of the algorithms from [12] that make it possible to use one CAS to find analytic and...