The aim of this chapter is to present some very basic principles of classical mechanics. They can help us understand the novelty of quantum theory. Classical mechanics was founded by Newton and contemporary 17th-century natural philosophers, and further developed in the 18th and 19th centuries by many physicists and mathematicians, as a result of the search for a rational, causal, and scientific explanation of the universe. In classical mechanics, real-world objects are modeled as a collection of point particles, i.e., objects with negligible size. The motion of a point particle is characterized by a small number of parameters such as its position, mass, velocity, and the forces applied to it. The position of a point particle is defined with respect to an arbitrary fixed reference point in the three- dimensional Euclidean space, which is usually chosen as the origin of some coordinate system. The velocity of a point particle is the rate of change of its position with time. The force applied to a point particle (called the external force) is the cause for the point particle x y z A to undergo a certain change in its velocity, i.e., an acceleration (Newton’s second law of motion). For the sake of simplicity, here we shall refer to a point particle or a collection of point particles simply as an object. Newton’s first law of motion states that if an object experiences no net external force, then its velocity is constant, i.e., the object is either at rest or it moves in a straight line with a constant speed. The first law of motion postulates the existence of a certain set of frames of reference called inertial reference frames, relative to which the motion of an object not subject to external forces is a straight line at a constant speed. For this reason, Newton’s first law is also referred to as the law of inertia.
The aim of this chapter is to present some very basic principles of classical mechanics. They can help us understand the novelty of quantum theory. Classical mechanics was founded by Newton and contemporary 17th-century natural philosophers, and further developed in the 18th and 19th centuries by many physicists and mathematicians, as a result of the search for a rational, causal, and scientific explanation of the universe. In classical mechanics, real-world objects are modeled as a collection of point particles, i.e., objects with negligible size. The motion of a point particle is characterized by a small number of parameters such as its position, mass, velocity, and the forces applied to it. The position of a point particle is defined with respect to an arbitrary fixed reference point in the three- dimensional Euclidean space, which is usually chosen as the origin of some coordinate system. The velocity of a point particle is the rate of change of its position with time. The force applied to a point particle (called the external force) is the cause for the point particle x y z A to undergo a certain change in its velocity, i.e., an acceleration (Newton’s second law of motion). For the sake of simplicity, here we shall refer to a point particle or a collection of point particles simply as an object. Newton’s first law of motion states that if an object experiences no net external force, then its velocity is constant, i.e., the object is either at rest or it moves in a straight line with a constant speed. The first law of motion postulates the existence of a certain set of frames of reference called inertial reference frames, relative to which the motion of an object not subject to external forces is a straight line at a constant speed. For this reason, Newton’s first law is also referred to as the law of inertia.
The branch of physics referred to as classical mechanics describes and predicts with great accuracy the motion of macroscopic objects moving much slower than the speed of light when subject to one or more forces. At the center of classical mechanics is Newton’s famous second law of motion: force = mass × acceleration. Classical mechanics is deterministic in that it exactly describes (within its limitations) the future motion of an object, provided that the object’s mass, initial position, and velocity, as well as the forces acting on the object, are sufficiently well known. The objects in classical mechanics are often approximated as fictional particles of a certain mass, their sizes being negligible or irrelevant. In cases where the physical extent of an object needs to come into play, the object often takes the form of a sphere or some other calculation-friendly shape (recall, for example, our spherical asteroid from the preface to this book).
Classical mechanics and electromagnetismThe mechanics of motion formulated by Newton and contemporaries in the 17th century is the foundation of contemporary physics.It is immensely successful, explaining and quantifying a vast array of phenomena, ranging from planetary motion to the flow of fluids.By the latter part of the 19th century the study of electric and magnetic fields and their inclusion in the framework of Newtonian physics appeared near completion.Since motion and electromagnetism are central to the failures of Newtonian physics in the atomic realm, it is useful to review the essential ideas, concepts, and the implicit assumptions of what is now referred to as classical physics.Many of the results presented in this chapter are assumed to be familiar to the reader, but some concepts are more thoroughly presented because of their relevance to quantum mechanics. Newtonian mechanicsFor simplicity initial consideration is limited to a single point particle of inertial mass m.A point particle is treated as infinitesimally small and indivisible.It is possible to formulate Newtonian mechanics in terms of continuous matter distributions, referred to as continuum mechanics, but these will not be considered in this text.In that regard, point particles are often used as an approximation to the constituents of real matter in Newtonian mechanics.A point particle's position therefore coincides with a point in space at every given moment of time, parameterized with t, and can be described mathematically by a vector x from the choice of origin to that point.As the point particle moves its position is given by the time-dependent trajectory x t ( ).An important assumption of Newtonian mechanics is that all observers are equipped with identical meter sticks and synchronized clocks, and they all agree upon the current position at the time t, although the choice of each observer's origin is arbitrary.The Cartesian components of the vector trajectory are denoted x t ( ) i , where the subscript i runs over the dimensions of the space under consideration.In one dimension the subscript is
The mathematician plays a game in which he himself invents the rules, while the physicist plays a game in which the rules are provided by nature, but as time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen. (Paul A. M. Dirac) As described in Chapter 5, classical ( Newtonian ) mechanics is centered around Hamilton's principle and the associated Newton's second law, particularly as applied to statics and dynamics of nondeformable (discrete) and deformable (continuous) bodies. Such systems are typically of O (1) size operating at O (1) speeds. Modern physics generally refers to branches of physics that have arisen since Einstein's relativity theory in the early part of the twentieth century, including special relativity, general relativity, and quantum mechanics, for example. Modern physics addresses physical phenomena that occur at very small sizes, such as atoms, or very large sizes, such as galaxies, and/or very high speeds close to the speed of light. We focus on the theory of relativity and quantum mechanics, both of which reduce to the classical mechanics limit that has occupied our considerations thus far. Relativity reduces to classical mechanics when υ ≪ c , where c is the speed of light in a vacuum, and υ is a characteristic velocity of the system; quantum mechanics reduces to classical mechanics when length scales dominant within the system are much larger than Planck's universal constant ħ.
In Newtonian mechanics, a particle is described as an object that is characterized by certain properties. The most important characteristics of a particle are its mass, position, velocity, and acceleration. In this chapter, it is shown how a particle follows a well defined classical trajectory. The main characteristics of the dynamics of particles such as linear and angular momentum, force, energy, moment of inertia, and torque are presented. An understanding of these effects is essential in understanding and appreciating the laws of quantum mechanics. As an example of the Newtonian mechanics, the motion of an electron in electric and magnetic fields experiencing Lorentz force is discussed. This example explains how Thomson discovered the electron in the late nineteenth century.
In the 17th century, it was thought that motion of macroscopic and microscopic objects could be well-expressed by using the laws of classical mechanics. But towards the end of 19th century, experimental studies showed that classical mechanics failed when applied to microscopic objects. This was mainly because classical mechanics did not take into account the concept of de-Broglie’s dual nature of matter and Heisenberg’s uncertainty principle. Also, classical mechanics does not put any restrictions on the values of dynamical properties (likes position, momentum, energy etc.) calculated for microscopic object. According to classical mechanics, microscopic objects can have any value of dynamical properties, ( exact, large or small is permissible). Also any number of properties can be calculated simultaneously for a system.
In the 17th century, it was thought that motion of macroscopic and microscopic objects could be well-expressed by using the laws of classical mechanics. But towards the end of 19th century, experimental studies showed that classical mechanics failed when applied to microscopic objects. This was mainly because classical mechanics did not take into account the concept of de-Broglie’s dual nature of matter and Heisenberg’s uncertainty principle. Also, classical mechanics does not put any restrictions on the values of dynamical properties (likes position, momentum, energy etc.) calculated for microscopic object. According to classical mechanics, microscopic objects can have any value of dynamical properties, ( exact , large or small is permissible). Also any number of properties can be calculated simultaneously for a system.