Classical mechanics

In this paper, we investigate the treatment of constraints in rate equations describing the temporal evolution of biological populations or chemical reactions. We present a formulation for arbitrary holonomic and linear nonholonomic constraints which ensures the positivity of the dynamical variables and which is an analog of Gauss' principle of least constraint in classical mechanics. The approach is illustrated for the replication of molecular species in the Schuster-Eigen hypercycle model, imposing the conservation of the total number of molecules and the entropy production as constraints. The latter is used to model the behavior of an isolated system tending toward equilibrium and, for comparison, a stationary nonequilibrium state of an open system, which is characterized by undamped oscillations.

We derive the equations of motion for linked rigid bodies from Lagrange mechanics and from Gauss's principle of least constraint. The rotational motion of the subunits is described in terms of quaternion parameters and angular velocities. Different types of joints can be incorporated via axis constraints for the angular velocities. The resulting equations of motion are generalizations of the Euler equations of motion for a single rotor.

The moving frame method for multi-body dynamics, established by Murakami in [10] and [11], embodies a consistent notation and mathematical framework that simplifies the derivation of equations of motion of complex systems. The derivation of the equations of motion follows Hamilton’s principle and requires the calculation of virtual angular velocities and the corresponding virtual rotational displacements. The goal of this paper is to present a projection-based approach, which only requires knowledge of Euler’s first and second law, that results in the same equation of motion. The constraints need not satisfy d’Alembert’s principle and the projection is based on a generalization of Gauss’ principle of least constraint [14]. One advantage of the proposed method is that it avoids variational principles and therefore is more accessible to undergraduate students. In addition, the final form of the equation of motion is more easily understood. We motivate our approach using the example of the simple pendulum, derive the main result, and apply the methodology for derivation of the equations of motion for a modified Chaplygin sleigh and a rotary pendulum.

We show that the Gaussian isokinetic thermostat used in nonequilibrium molecular-dynamics calculations can be derived from a Hamiltonian. An alternative variational approach involves geodesic motion in a curved space. We consider the implications of this result for the nonequilibrium Lorentz gas and thermostated systems in general. \textcopyright{} 1996 The American Physical Society.

Dynamics and Control of Machines

We present another way to teach the least-action principle in classical mechanics. Beginning with Newton’s second law of motion, we keep finding equivalent propositions based mainly on the method of undetermined coefficients in a multivariable-dependent identity to arrive at the Euler–Lagrange equations and the conventional principle of stationary action $$\delta S=0$$ δ S = 0 with the boundary condition for the variation of the physical path. Pedagogically, by following the logical steps that we show, students can easily learn that Lagrangian mechanics naturally arises from Newtonian mechanics. This approach makes it further clear that Newton’s laws do not require the minimum action—true for a free particle, just as the shortest path between two points is a straight line. However, the action is not always the minimum once the potential energy is turned on. The derivation of the least-action principle from Newton’s second law may resemble Euclid’s approach to geometric axioms.

In this paper, the derivation of the equations of dynamics of a four-wheeled mobile robot is carried out using the variational principle of least constraint, known as the Gauss principle. Equations of nonholonomic constraints are obtained. The function of the measure of coercion of the four-wheeled mobile robot is composed. Dynamic equations based on the Gauss principle are obtained taking into account the dynamic characteristics of two DC motors. Methods for taking into account the friction forces on the wheels and random perturbations due to the unevenness of the canvas are proposed. On the Maple platform, an algorithm and a program for modeling the dynamics of a mobile robot based on the Gauss principle were developed the correctness of the obtained equations of robot motion were proved.

Introduction In this chapter we focus on a number of techniques used in MD simulation, primarily the methods for computing the interactions and integrating the equations of motion. The goal is to generate the atomic trajectories; subsequent chapters will deal with the all-important question of analyzing this raw ‘experimental’ data. We continue to work with the simplest atomic systems, in other words, with monatomic fluids based on the LJ potential, not only because we want to introduce the methodology gradually, but also because a lot of the actual qualitative (and even quantitative) behavior of many-body systems is already present in this simplest of models. Models of this kind are widely used in MD studies of basic many-body behavior, examples of which will be encountered in later chapters. Equations of motion While Newton's second law suffices for the dynamics of the simple atomic fluid discussed in this chapter, later chapters will require more complex forms of the equations of motion. The Lagrangian formulation of classical mechanics provides a general basis for dealing with these more advanced problems, and we begin with a brief summary of the relevant results. There are, of course, other ways of approaching the subject, and we will also make passing reference to Hamilton's equations. A full treatment of the subject can be found in textbooks on classical mechanics, for example [go180].

The calculus of variations is concerned with finding extremal paths of functionals in analogy with the way that classical calculus seeks to find critical points of functions. Variational calculus plays a central role in classical mechanics, connecting the “Principle of Least Action” and Lagrange’s equations of motion (also called the Euler– Lagrange equations). In that setting, generalized coordinates are introduced to describe the geometric configuration of a mechanical system. In this chapter, classical variational calculus is reviewed and extended to describe systems on Lie groups. Of course, the introduction of coordinates such as Euler angles to describe the orientation of a rigid body can be used to formulate classical variational problems at the expense of introducing singularities. However, it is possible to formulate variational problems on Lie groups without coordinates. This results in the Euler’Poincaré equations.

KEY FEATURES The key features of this chapter are Newton's laws of motion, the definitions of mass and force , the law of gravitation , the principle of equivalence , and gravitation by spheres . This chapter is concerned with the foundations of dynamics and gravitation . Kinematics is concerned purely with geometry of motion, but dynamics seeks to answer the question as to what motion will actually occur when specified forces act on a body. The rules that allow one to make this connection are Newton's laws of motion . These are laws of physics that are founded upon experimental evidence and stand or fall according to the accuracy of their predictions. In fact, Newton's formulation of mechanics has been astonishingly successful in its accuracy and breadth of application, and has survived, essentially intact, for more than three centuries. The same is true for Newton's universal law of gravitation which specifies the forces that all masses exert upon each other. Taken together, these laws represent virtually the entire foundation of classical mechanics and provide an accurate explanation for a vast range of motions from large molecules to entire galaxies. NEWTON'S LAWS OF MOTION Isaac Newton's three famous laws of motion were laid down in Principia , written in Latin and published in 1687. These laws set out the founding principles of mechanics and have survived, essentially unchanged, to the present day. Even when translated into English, Newton's original words are hard to understand, mainly because the terminology of the seventeenth century is now archaic.

Obtaining models of mechanical processes with dissipation based on the Euler-Lagrange theory has undoubted advantages over Newton's theory due to the smaller size of the considered vector of variables included in the equations. However, the Euler-Lagrange variation theory is not applicable to the description of the motion of systems with dissipation. The aim of the work is to demonstrate the possibility of using the Euler-Lagrange theory in relation to dissipative systems with different types of friction. Mathematical models of systems with dissipation are based on the superposition of mechanical and thermodynamic Lagrangians. To obtain a mathematical description of dissipative systems it is proposed to use the field theory as applied to the thermodynamics of dissipative processes within the framework of the Lagrange formalism. The Euler-Lagrange equations are obtained for the Stokes and Coulomb friction models. As it was referred to the research results obtained there is possibility of accounting the energy dissipation in the Lagrange formalism. The mathematical models proposed describe dynamic processes in heterogeneous structures with friction based on the Euler-Lagrange theory. There are presented mathematical transformations that allow transition from models based on the Lagrange formalism to models based on Newtonian mechanics

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce the thermodynamic and structural properties. The objective is to develop a means for exploiting strong coupling classical methods (e.g., MD, integral equations, DFT) to describe quantum systems. The classical system has an effective temperature, local chemical potential, and pair interaction that are defined by requiring equivalence of the grand potential and its functional derivatives with respect to the external and pair potentials for the classical and quantum systems. Practical inversion of this mapping for the classical properties is effected via the hypernetted chain approximation, leading to representations as functionals of the quantum pair correlation function. As an illustration, the parameters of the classical system are determined approximately such that ideal gas and weak coupling RPA limits are preserved (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Evolution of the concept of chemical equilibrium has been strongly influenced by Newtonian mechanics. Even in the late nineteenth century scientists viewed chemical equilibrium as resulting from, an equality of the contending forces, that is, the forward and the reverse reactions. The main purpose of this article is to show that freshman students conceptualize the rates of the forward and reverse reactions in chemical equilibrium as forces, perhaps in the same sense as used in the evolution of the concept of chemical equilibrium and student misconceptions about Newton's third law of motion. Results obtained show that at least some students consider the forward and reverse reactions as a sort of chemical analogue of Newton's third law of motion. Based on this interpretation, it is plausible to hypothesize that student conceptualization of the forward and reverse reactions as forces is ontogenetically (Piaget & Garcia, 1983) a step towards, the deeper understanding of a dynamic chemical equilibrium, leading to a progressive ‘problemshift’ (Lakatos, 1970). In spite of the commonalities between psychogenesis and the history of science, it is important to point out that ontogenesis is not an exact and detailed recapitulation of phylogenesis. Results obtained in this study and their interpretation are important, as they help us to anticipate student utilization of Newton's third law in understanding chemical equilibrium, before the dynamic nature of equilibrium is understood.

There is a great deal of justified concern about continuity through scientific theory change. Our thesis is that, particularly in physics, such continuity can be appropriately captured at the level of conceptual frameworks (the level above the theories themselves) using conceptual space models. Indeed, we contend that the conceptual spaces of three of our most important physical theories—Classical Mechanics (CM), Special Relativity Theory (SRT), and Quantum Mechanics (QM)—have already been so modelled as phase-spaces. Working with their phase-space formulations, one can trace the conceptual changes and continuities in transitioning from CM to QM, and from CM to SRT. By offering a revised severity-ordering of changes that conceptual frameworks can undergo, we provide reasons to doubt the commonly held view that CM is conceptually closer to SRT than QM.

A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman–von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum–classical hybrid systems.