refined Classical_mechanics Information, explanation, recent texts, monographs, and related patents.
Information & explanations, latest texts & monographs on Classical_mechanics (including recent related patents.)


Classical mechanics

Classical mechanics is the physics of forces, acting upon bodies. It is often referred to as "Newtonian mechanics" after Newton and his laws of motion. Classical mechanics is subdivided into statics (which deals with objects in equilibrium) and dynamics (which deals with objects in motion). See also mechanics. Classical mechanics produces very accurate results within the domain of everyday experience. It is superseded by relativistic mechanics for systems moving at large velocities near the speed of light, quantum mechanics for systems at small distance scales, and relativistic quantum field theory for systems with both properties. Nevertheless, classical mechanics is still very useful, because (i) it is much simpler and easier to apply than these other theories, and (ii) it has a very large range of approximate validity. Classical mechanics can be used to describe the motion of human-sized objects (such as tops and baseballs), many astronomical objects (such as planets and galaxies), and even certain microscopic objects (such as organic molecules.) Although classical mechanics is roughly compatible with other "classical" theories such as classical electrodynamics and thermodynamics, there are inconsistencies that were discovered in the late 19th century that can only be resolved by more modern physics. In particular, classical nonrelativistic electrodynamics predicts that the speed of light is a constant relative to an aether medium, a prediction that is difficult to reconcile with classical mechanics and which led to the development of special relativity. When combined with classical thermodynamics, classical mechanics leads to the Gibbs paradox in which entropy is not a well-defined quantity and to the ultraviolet catastrophe in which a blackbody is predicted to emit infinite amounts of energy. The effort at resolving these problems led to the development of quantum mechanics. Table of contents showTocToggle("show","hide") 1 Description of the theory 1.1 Position and its derivatives 1.1.1 Velocity 1.1.2 Acceleration 1.1.3 Frames of Reference 1.2 Forces; Newton's Second Law 1.3 Energy 1.4 Further results 1.5 Example 2 History 3 See also 4 Further Reading 5 External links Description of the theory We will now introduce the basic concepts of classical mechanics. For simplicity, we only deal with a point particle, which is an object with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied on it. We will discuss each of these parameters in turn. In reality, the kind of objects which classical mechanics can describe always have a non-zero size. True point particles, such as the electron, are properly described by quantum mechanics. Objects with non-zero size have more complicated behavior than our hypothetical point particles, because their internal configuration can change - for example, a baseball can spin while it is moving. However, we will be able to use our results for point particles to study such objects by treating them as composite objects, made up of a large number of interacting point particles. We can then show that such composite objects behave like point particles, provided they are small compared to the distance scales of the problem, which indicates that our use of point particles is self-consistent. Position and its derivatives The position of a point particle is defined with respect to an arbitrary fixed point in space, which is sometimes called the origin, O. It is defined as the vector r from O to the particle. In general, the point particle need not be stationary, so r is a function of t, the time elapsed since an arbitrary initial time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute in all reference frames. Velocity The velocity, or the rate of change of position with time, is defined as . In pre-Einstein relativity velocities are directly additive and subtractive. For example, if one car traveling at 60 km/h passes another car traveling at 50 km/h, from the perspective of the car it passes it is traveling at 60-50 = 10 km/h. Mathematically, if we define the velocity of the second reference frame in our previous discussion above as the vector u = ux (x being the x-dimensional unit vector), following the above formulas gives us: v' = v - u as we would expect. Acceleration The acceleration, or rate of change of velocity, is . The acceleration vector can be changed by changing its magnitude, changing its direction, or both. If the magnitude of v decreases, this is sometimes referred to as deceleration; but generally any change in the velocity, including deceleration, is simply referred to as acceleration. Frames of Reference The following consequences can be derived about the perspective of an event in two reference frames, S and S', where S' is traveling at a relative speed of u to S.
  • v' = v - u (the velocity of a particle from the perspective of S' is slower by u than from the perspective of S)
  • a' = a (the acceleration of a particle remains the same regardless of reference frame)
  • F' = F (since F = ma) (the force on a particle remains the same regardless of reference frame; see Newton's law)
  • the speed of light is not a constant
  • the form of Maxwell's equations is not preserved in different reference frames
Forces; Newton's Second Law Newton's second law relates the mass and velocity of a particle to a vector quantity known as the force. Suppose m is the mass of a particle and F is the vector sum of all applied forces (i.e. the net applied force.) Then Newton's second law states that . The quantity mv is called the momentum. Typically, the mass m is constant in time, and Newton's law can be written in the simplified form where a is the acceleration, as defined above. It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used. Newton's second law is insufficient to describe the motion of a particle. In addition, we require a description of F, which is to be obtained by considering the particular physical entities with which our particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, say with λ a positive constant. Once we have independent relations for each force acting on a particle, we can substitute it into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion. Continuing our example, suppose that friction is the only force acting on the particle. Then the equation of motion is . This can be integrated to obtain where v0 is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. This expression can be further integrated to obtain the position r of the particle as a function of time. Important forces include the gravitational force and the Lorentz force for electromagnetism. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if we know that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, -F, on A. Energy If a force F is applied to a particle that achieves a displacement δr, the work done by the force is the scalar quantity . Suppose the mass of the particle is constant, and δWtotal is the total work done on the particle, which we obtain by summing the work done by each applied force. From Newton's second law, we can show that δWtotal = δT-, where T is called the kinetic energy. For a point particle, it is defined as . For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the individual particles' kinetic energies. A particular class of forces, known as conservative forces, can be expressed as the gradient of a scalar function, known as the potential energy and denoted V: . Suppose all the forces acting on a particle are conservative, and V is the total potential energy, obtained by summing the potential energies corresponding to each force. Then . This result is known as the conservation of energy, and states that the total energy, E = T + V, is constant in time. It is often useful, because most commonly encountered forces are conservative. Further results Newton's laws provide many important results for composite bodies. See angular momentum. There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are often more useful for solving problems. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems. Example Consider two reference frames, one of which is traveling at a relative speed of u to the other. For example, for a car passing another car at a relative speed of 10 km/h, u is 10 km/h. Two reference frames S and S', with S' traveling at a relative speed of u to S; an event has space-time coordinates of (x,y,z,t) in S and (x',y',z',t') in S'. The space-time coordinates of an event in Galilean-Newtonian relativity are governed by the set of formulas which defines a group transformation known as the Galilean transformation: Assuming time is considered an absolute in all reference frames, the relationship between space-time coordinates in reference frames differing by a relative speed of u in the x direction (let x = ut when x' = 0) is: x' = x - ut y' = y z' = z t' = t The set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). History The Greeks and Aristotle in particular were the first to propose that there are abstract principles governing nature. One of the first scientists who suggested abstract laws was Galileo Galilei who also performed the famous experiment of dropping two canon balls from the tower of Pisa (The theory, and the practice showed that they both hit the ground at the same time). Sir Isaac Newton was the first to propose the three laws of motion (the law of inertia, the second law mentioned above, and the law of action and reaction), and to prove that these laws govern both everyday objects and celestial objects. Newton also developed the calculus which is necessary to perform the mathematical calculations involved in classical mechanics. After Newton the field became more mathematical and more abstract. See also Edmund Halley -- List of equations in classical mechanics Further Reading External links General subfields within physics Classical mechanics | Condensed matter physics | Electromagnetism | Field theory (physics) | Continuum mechanics | General relativity | Particle physics Quantum mechanics | Quantum field theory | Solid state physics | Electronic Structure of Materials | Special relativity | Standard Model | Statistical mechanics | Thermodynamics
This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License

Classical Mechanics (3rd Edition) by Herbert Goldstein

Classical Dynamics of Particles and Systems by Stephen T. Thornton

Classical Mechanics by John R. Taylor

Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, No 60) by V. I. Arnold

Classical Mechanics: Point Particles and Relativity (Classical Theoretical Physics) by Walter Greiner

Classical Mechanics : 2nd Edition by H.C. Corben

Classical Dynamics : A Contemporary Approach by Jorge V. José

The Classical Electromagnetic Field by Leonard Eyges

Thermodynamics and Statistical Mechanics (Classical Theoretical Physics) by Walter Greiner

New Foundations for Classical Mechanics: Fundamental Theories of Physics by David Hestenes

Classical Dynamics by Donald T. Greenwood

Classical Dynamics of Particles and Systems by Jerry B. Marion

Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman

Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing by Harvey S. Leff

Concepts of Mass in Classical and Modern Physics by Max Jammer


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