Physics
Classical Mechanics
Classical mechanics is a branch of physics that deals with the motion of objects and the forces acting on them. It encompasses the study of Newton's laws of motion, the conservation of energy and momentum, and the behavior of particles and rigid bodies. Classical mechanics provides a framework for understanding and predicting the behavior of macroscopic objects at everyday speeds and scales.
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10 Key excerpts on "Classical Mechanics"
eBook - PDFQuantum Dynamics
Applications in Biological and Materials Systems
- Eric R. Bittner(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
1 Survey of Classical Mechanics Quantum mechanics is in many ways the cumulation of many hundreds of years of work and thought about how mechanical things move and behave. Since ancient times, scientists have wondered about the structure of matter and have tried to develop a generalized and underlying theory that governs how matter moves at all length scales. For ordinary objects, the rules of motion are very simple. By ordinary, I mean objects that are more or less on the same length and mass scale as you and I, say (conservatively) 10 − 7 m to 10 6 m and 10 − 25 g to 10 8 g moving at less than 20% of the speed of light. On other words, almost everything you can see and touch and hold obeys what are called classical laws of motion. The term classical means that that the basic principles of this class of motion have their foundation in antiquity. Classical Mechanics is an extremely well-developed area of physics. While you may think that because Classical Mechanics has been studied extensively for hundreds of years there really is little new development in this field, it remains a vital and extremely active area of research. Why? Because the majority of universe “lives” in a dimensional realm where Classical Mechanics is extremely valid. Classical Mechanics is the workhorse for atomistic simulations of fluids, proteins, and polymers. It provides the basis for understanding chaotic systems. It also provides a useful foundation of many of the concepts in quantum mechanics. Quantum mechanics provides a description of how matter behaves at very small length and mass scales, that is, the realm of atoms, molecules, and below. It has been developed over the past century to explain a series of experiments on atomic systems that could not be explained using purely classical treatments. The advent of quantum mechanics forced us to look beyond the classical theories. However, it was not a drastic and complete departure.
eBook - PDFPhases of Matter and their Transitions
Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists
- Gijsbertus de With(Author)
- 2023(Publication Date)
- Wiley-VCH(Publisher)
15 2 Classical Mechanics The microscopic description of the motion of particles can be done by either clas- sical mechanics or quantum mechanics. In this chapter, a brief review of the first “tool” – classical particle mechanics (PM) based on Hamilton’s principle – is provided. The presentation is abstract but efficient and useful in quantum mechanics as well as in statistical thermodynamics [1]. We first introduce the notion of a particle, generalized coordinates, and the basic mechanics principles. Thereafter, we deal with Hamilton’s prin- ciple, the Lagrange equations, and their consequences. Finally, we introduce Hamilton’s equations and an important consequence of all of these, the virial theorem. 2.1 Frames, Particles, and Coordinates In mechanics we distinguish between kinematics – the description of motion and change without considering the cause – and kinetics – why things move and change. Here, we dis- cuss kinematics, and in the remaining sections kinetics. To describe phenomena, we need a frame of reference. A basic assumption of nonrelativistic mechanics is that space–time is isotropic and homogeneous with zero curvature, that is, it is a three-dimensional (3D) Euclidean space with a universal time, independent of relative motion. It follows that a so-called inertial frame can be found in which the laws of mechanics do not depend explicitly on the position and orientation of the system in space and are independent of the time origin. A freely moving particle (one that experiences no external force) moves with constant velocity in such a frame. This is the law of inertia. Thus, any frame I ′ moving with a constant velocity V with respect to the inertial frame I is also an inertial frame. Galilei’s Principle of Relativity 1 (PoR) states that all laws of nature are identical in all inertial frames, that is, the laws are form invariant or covariant with respect to transformations of coordinates and time from one inertial frame to another.
eBook - PDF- Mackillo Kira, Stephan W. Koch(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
These investigations can always be traced back to fundamental equations that define the axioms in a mathematical form. In particular, these equations can be applied to develop the systematic theory for semiconductor quantum-optical phenomena. As background for our quantum- mechanical studies, we summarize the basic equations needed to formulate the theory of Classical Mechanics in this first chapter. 1.1 Classical description The motion of a classical particle can be analyzed when its position x is known as func- tion of time t , yielding a well-defined trajectory x (t ). For example, the velocity of the particle, v(t ) = d dt x (t ) = ˙ x (t ), follows from the first-order time derivative of the trajectory. 1 2 Central concepts in Classical Mechanics To define x (t ) and v(t ) under arbitrary conditions, one also needs to know the mass m of the particle which defines the particle’s inertia resisting changes of v(t ) resulting from exter- nal forces F (x , t ) that can depend on both position and time. For simplicity, and since we want to focus on the underlying concepts, we use the notation for simple one-dimensional motion. More generally, position, velocity, force etc. are all three-dimensional vectorial quantities. The fundamental description of the classical motion can always be expressed using Hamilton’s formulation of mechanics. As a first step, we identify the system Hamiltonian H = p 2 2m + U (x ), (1.1) where p is the momentum of the particle and U (x ) is the potential of the external forces. In the Hamilton formalism, one describes x and p as independent variables such that the Hamiltonian is actually a function H (x , p) of x and p. In general, (x , p) define the canonical phase-space coordinates and (x (t ), p(t )) describes a phase-space trajec- tory. Figure 1.1 (solid and dashed lines) presents several trajectories connecting the same initial (x i , p i ) and final (x f , p f ) phase-space points.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
C H A P T E R 3 Fundamentals of Newtonian Mechanics 3.1 Introduction The science of mechanics seeks to provide a precise and consistent description of the dynamics of particles and systems of particles. That is, we attempt to discover a set of physical laws which provide us with a method for mathematically describing the motions of bodies and aggre-gates of bodies. In order to do this, we need to introduce certain funda-mental concepts. It is implicit in Newtonian theory that the concept of distance is intuitively understandable from a geometrical viewpoint. Further-more, time is considered to be an absolute quantity, capable of precise definition by an arbitrary observer. In relativity theory, however, we must modify these Newtonian ideas (see Chapter 4). The combination of the concepts of distance and time allows us to define the velocity and accelera-tion of a particle. The third fundamental concept, mass, requires some ela-boration which we shall give in connection with the discussion of Newton's laws. The physical laws which we introduce must be based on experimental fact. A physical law may be characterized by the statement that it might have been otherwise. Thus, there is no a priori reason to expect that the 56 3.2 NEWTON'S LAWS 57 gravitational attraction between two bodies must vary exactly as the inverse square of the distance between them. But experiment indicates that this is so. Once a set of experimental data is correlated and a postulate is formula-ted regarding the phenomena to which the data refer, then various impli-cations can be worked out. If these implications are all verified by experi-ment, there is reason to believe that the postulate is generally true. The postulate then assumes the status of a physical law. If some experiments are found to be in disagreement with the predictions of the law, then the theory must be modified in order to be consistent with all known facts. Newton has provided us with the fundamental laws of mechanics.
- M Rasetti(Author)
- 1986(Publication Date)
- WSPC(Publisher)
1 Chapter 1 Review of Basic Classical Mechanics 1. Newtonian Mechanics of a System of Particles The dynamics of a single particle of mass m (assumed to be constant, no relativistic effects being considered) whose position in space (unless otherwise specified ]R or a finite part Q thereof) is defined by the position vector q = q(t) at time t, is completely described by the equation of motion (Newton's second law) £(4) -£(«nv) = F , (1.1) in which F is the total force acting on the particle and v its velocity. A dot is used when no ambiguity arises to denote a time derivative. It is interesting to note that the formulation (1.1) — which Newton himself used —remains valid even in the case of variable m, which is not the case for the more familiar form of the equation of motion (1.1), F = ma = mq , (1.2) where a is the acceleration of the particle. If the particle belongs to an isolated system, consisting of N mass points, Eq. (1.1) (or (1.2)) still holds for each particle and the dynamics of the system is described by the system of N equations 2 j Here m., q. (i = 1,...,N) denote, respectively, the mass and position of the i-th particle in the system. , Since the system has been assumed to be isolated, namely the only forces acting on the particles in the system are mutual inter-particle forces, the force F.. acting on parti-cle i has been written in (1.3) as the vector sum of the (N-l) forces F.. exerted by all other particles j (j7i) on particle i. * j Particularly interesting is the case when the forces F.. are conservative, i.e. can be derived from a two-body potential U...: V- u i.j ' (1,4) where V-* is the gradient operator with respect to the coordinates of ^i particle i. In most cases, U..., which describes the potential energy of the pair, is function only of the distance between the two particles i,j U 1J-U 1J
eBook - PDF- Yoni Kahn, Adam Anderson(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
1 Classical Mechanics Classical Mechanics is the cornerstone of the GRE, making up 20% of the exam, and at the same time has the dubious distinc- tion of being the subject that turns so many people away from physics. Your first physics class was undoubtedly a mechan- ics class, at which point you probably wondered what balls, springs, ramps, rods, and merry-go-rounds had to do in the slightest with the physics of the real world. So rather than (a) attempt the impossible task of covering your 1000-page freshman-year textbook in this much shorter reference, or (b) risk turning you away from physics before you’ve even taken the exam, we’ll structure this chapter a little differently than the rest of the book. We’re not going to review such things as Newton’s laws, force balancing, or the definition of momen- tum; you should know these things in your sleep, or the rest of the exam will seem impossibly hard. Rather than review basic topics, we’ll review standard problem types you’re likely to encounter on the GRE. The more advanced topics will get their own brief treatment as well. After finishing this chapter, you will have reviewed nearly all the material you’ll need for the Classical Mechanics section of the test, but in a format that is much more useful for the way the problems will likely be presented on the test. If you need a more detailed review of any of these topics, just open up any undergraduate physics text. 1.1 Blocks One of the first things you learned in the first semester of freshman year physics was probably how to balance forces using free-body diagrams. Rather than rehash that discus- sion, which you can find in absolutely any textbook, we’ll review it through a series of example problems that are GRE favorites. They involve objects, usually called “blocks,” with certain masses, doing silly things like sitting on ramps, being pushed against springs, and traveling on carts.
eBook - PDF- P R Wallace(Author)
- 1991(Publication Date)
- World Scientific(Publisher)
Symmetry remains, and conservation laws, and causality. Each con-cept has been broadened and refined, and new relations found be-tween them, but the essential features of the concept have remained. It is important, therefore, that we talk about these concepts in the context of Newtonian mechanics, for only then will we be able to understand their subtler significance in modern physics. 2.1. M a s s Let us start with the concept of mass. A less terse definition for the mass of a body (or a particle) is coefficient of inertia. Consider, for example, two objects made of the same material, one having twice the volume of the other (twice as big). Suppose that each is acted upon by an identical force for an identical length of time. It is found that the smaller body is accelerated to twice the speed of the larger one. The larger one has more resistance to acceleration, more inertia than the smaller. We say it has twice the mass. Of course, inertia is only proportional to size for objects made of the same material. A ball of lead of 20 cm diameter has more inertia, and hence more mass than one of aluminum. How do we test their relative masses? In the same way as before, by comparing the relative accelerations which they are given by identical forces. The above statements, which define mass (in relative terms at least) seem almost precisely equivalent to Newton's Law of Motion. Galileo had already formulated the rather strange law that every body, left to itself (i.e. not acted upon by any force) would continue indefinitely in its state of rest or uniform motion in a straight line. What is so difficult about this law is that it is hard to imagine something not acted upon by any force! It certainly has never existed in the real universe. It is an abstraction, a mental construct. We can, however, imagine it better today than in the time of Newton Fundamental* of Newtonian Mechanic » 23 or Galileo, due to our increasing familiarity with space travel.
eBook - PDFMechanics
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
The study of this excellent critical history is recom-mended to all students of mechanics, especially since in our book we must restrict ourselves to the concepts of mechanics in a form ready for use and cannot delve into the origin and gradual clarification of these concepts. This should not be interpreted to mean, however, that we agree with Mach's positivistic philosophy as it is developed in Chapter IV, 4, of his book, with its attendant overemphasis of the Economy Principle, the denial of atomic theory and the preference for formal continuity theories. 2 Newton's Principia, translated by Andrew Motte.—TRANSLATOR. s Evident, that is, once a reference system has been chosen in which the velocity is to be measured. 3 4 Mechanics of a Particle LI matter, which is to be explained physically. Newton attempts the latter in his definition 1, in which he says that the quantity of matter is measured by its density and volume conjunctly. This is obviously only a mock definition, since density itself cannot be defined in any other way than by the amount of matter in unit volume. In the same definition 1, Newton also states that instead of quantity of matter he will use the word mass. We shall follow him in this, but shall postpone the physical definition of mass (as well as that of force) until later. The quantity of motion accordingly becomes the product of mass and velocity. Like the latter it is a directed magnitude, a vector. We write 4 (1) p=rav and formulate the first law of motion in its final form : (2) p=constant in the absence of forces. We shall put the law of inertia thus formulated at the head of our mechanics. It is the result of an evolution extending over many centuries, and is by no means as self-evident as it appears to us today.
eBook - PDFThe Road to Maxwell's Demon
Conceptual Foundations of Statistical Mechanics
- Meir Hemmo, Orly R. Shenker(Authors)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
In this book we show that this is not the case. 3.1 The fundamental theory of the world 41 thermodynamics by mechanics, and which play a role in the Demon argument. This is the aim of the present chapter. 3.2 Introducing Classical Mechanics It is a fundamental premise in Classical Mechanics 3 that the world is made up of particles. 4 Richard Feynman expressed this idea as follows. If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms. 5 The laws governing the behavior of atoms, or more generally of particles, are – by our working hypothesis – the laws of Classical Mechanics. 6 Since thermodynamic systems, such as the systems in Joule’s experiment and in Carnot’s cycle, are nothing but collections of particles, they must obey the laws that particles obey – namely, the laws of mechanics. In other words, the fact that thermodynamic systems are made of particles that obey the laws of mechanics ought to underwrite the laws of thermodynamics, to the extent that these laws are true. 7 3 What exactly is the content of the theory of Classical Mechanics, and what are its auxiliary hypotheses, is an open question, with which we do not deal here; for us, in this book, the theory includes the ontology as we present it here. For a discussion of the scope of the theory see, for example, Hutchison (1993) and Callender (1995). 4 Classical physics introduces fields, in addition to particles. These include the electric and magnetic fields. Fields are extended in space and in this sense differ significantly from particles. The microscopic state of the world at a given time actually consists of the microstates of the particles and the microstates of the fields.
eBook - PDF- Delo E. Mook, Thomas Vargish(Authors)
- 2018(Publication Date)
- Princeton University Press(Publisher)
Much of the power and influence of Newtonian mechanics reside in the second law. It provides a recipe for making quantitative predictions about the physical world. If we know the position, speed, and mass of an object at any given time (the so-called initial conditions of an object), and if 40 THE CLASSICAL BACKGROUND we further know the values of all forces acting on that object, then we can use the second law to predict the consequences of the actions of those forces in the following way. We first use the second law to calculate the acceler-ation (the change in speed and direction) that the object will undergo as a result of the forces acting. Given that calculated acceleration, we can pre-dict at any subsequent instant what the new values of the object's speed and position will be. The job is most easily done with calculus—a mathematical tool Newton himself invented for the purpose. Because Newton's laws are expressed in terms of the most general prop-erties of objects (their motions and masses) and not in terms of the exact nature of the object itself (such as the material from which it is made, its color or age) the domain of validity of those laws was, until the advent of relativity theory and quantum thec-y in the early twentieth century, thought to be universal. Given the initial conditions of any object, one could apply Newton's model to find its subsequent conditions of motion and position at any time in the future, whether the object studied is a planet, a cannon ball, a billiard ball, an arrow, a stone, or even a human body. Newton's laws can also be used to investigate the motion of any object at any time in the past as well as in the future. One needs only a specifica-tion of the initial conditions of speed, position, and force at a stated time, and then the laws can be run in reverse to predict what the values of speed, direction of travel, and position had to have been in the past in order for the present set of observed conditions to come about.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
