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# Classical Mechanics Point Particles And Relativity Pdf

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Classical [note 1] mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery , and astronomical objects , such as spacecraft , planets , stars , and galaxies.

For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future determinism , and how it has moved in the past reversibility.

The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton , and the mathematical methods invented by Gottfried Wilhelm Leibniz , Joseph-Louis Lagrange , Leonard Euler , and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces.

Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of analytical mechanics. They are, with some modification, also used in all areas of modern physics. Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the speed of light.

When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics : quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity becomes applicable.

However, a number of modern sources do include relativistic mechanics in classical physics, which in their view represents classical mechanics in its most developed and accurate form. The following introduces the basic concepts of classical mechanics.

For simplicity, it often models real-world objects as point particles objects with negligible size. The motion of a point particle is characterized by a small number of parameters : its position, mass , and the forces applied to it.

Each of these parameters is discussed in turn. In reality, the kind of objects that classical mechanics can describe always have a non-zero size. The physics of very small particles, such as the electron , is more accurately described by quantum mechanics. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom , e.

However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles. The center of mass of a composite object behaves like a point particle. Classical mechanics uses common sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as location in space and speed.

Non-relativistic mechanics also assumes that forces act instantaneously see also Action at a distance. The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O.

In cases where P is moving relative to O , r is defined as a function of t , time. In pre-Einstein relativity known as Galilean relativity , time is considered an absolute, i. The velocity , or the rate of change of position with time, is defined as the derivative of the position with respect to time:.

In classical mechanics, velocities are directly additive and subtractive. Velocities are directly additive as vector quantities ; they must be dealt with using vector analysis. The acceleration , or rate of change of velocity, is the derivative of the velocity with respect to time the second derivative of the position with respect to time :.

Acceleration represents the velocity's change over time. Velocity can change in either magnitude or direction, or both. Occasionally, a decrease in the magnitude of velocity " v " is referred to as deceleration , but generally any change in the velocity over time, including deceleration, is simply referred to as acceleration.

While the position, velocity and acceleration of a particle can be described with respect to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. An inertial frame is an idealized frame of reference within which an object has no external force acting upon it.

Because there is no external force acting upon it, the object has a constant velocity; that is, it is either at rest or moving uniformly in a straight line. A key concept of inertial frames is the method for identifying them.

For practical purposes, reference frames that do not accelerate with respect to distant stars an extremely distant point are regarded as good approximations to inertial frames. Non-inertial reference frames accelerate in relation to an existing inertial frame.

They form the basis for Einstein's relativity. Due to the relative motion, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration.

These forces are referred to as fictitious forces , inertia forces, or pseudo-forces. Consider two reference frames S and S'. For observers in each of the reference frames an event has space-time coordinates of x , y , z , t in frame S and x' , y' , z' , t' in frame S'. This set of formulas defines a group transformation known as the Galilean transformation informally, the Galilean transform. The limiting case applies when the velocity u is very small compared to c , the speed of light.

For some problems, it is convenient to use rotating coordinates reference frames. Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious centrifugal force and Coriolis force. A force in physics is any action which causes an object's velocity to change; that is, to accelerate. A force originates from within a field , such as an electro-static field caused by static electrical charges , electro-magnetic field caused by moving charges , or gravitational field caused by mass , among others.

Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. The quantity m v is called the canonical momentum. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle.

Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation , which is called the equation of motion. As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example:. Then the equation of motion is. This means that the velocity of this particle decays exponentially to zero as time progresses.

In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction which converts it to heat energy in accordance with the conservation of energy , and the particle is slowing down. 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. Illustrations of the weak form of Newton's third law are often found for magnetic forces. More generally, if the force varies as a function of position as the particle moves from r 1 to r 2 along a path C , the work done on the particle is given by the line integral.

If the work done in moving the particle from r 1 to r 2 is the same no matter what path is taken, the force is said to be conservative. Gravity is a conservative force, as is the force due to an idealized spring , as given by Hooke's law.

The force due to friction is non-conservative. The kinetic energy E k of a particle of mass m travelling at speed v is given by. For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles. The work—energy theorem states that for a particle of constant mass m , the total work W done on the particle as it moves from position r 1 to r 2 is equal to the change in kinetic energy E k of the particle:.

Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted E p :. If all the forces acting on a particle are conservative, and E p is the total potential energy which is defined as a work of involved forces to rearrange mutual positions of bodies , obtained by summing the potential energies corresponding to each force.

This result is known as conservation of energy and states that the total energy ,. It is often useful, because many commonly encountered forces are conservative.

Classical mechanics also describes the more complex motions of extended non-pointlike objects. Euler's laws provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. The rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass". There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics.

These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates.

These are basically mathematical rewriting of Newton's laws, but complicated mechanical problems are much easier to solve in these forms. Also, analogy with quantum mechanics is more explicit in Hamiltonian formalism. The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector divided by c 2 , where c is the speed of light in free space.

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. Geometric optics is an approximation to the quantum theory of light , and does not have a superior "classical" form.

When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory QFT is of use. QFT deals with small distances, and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level, statistical mechanics becomes useful.

Statistical mechanics describes the behavior of large but countable numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in thermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics.

In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. In case that objects become extremely heavy i. In that case, General relativity GR becomes applicable. Open navigation menu. Close suggestions Search Search. User Settings. Skip carousel. Carousel Previous.

Haynes ManualsThe Haynes Intended for advanced undergraduates and beginning graduate students, the volumes in the series provide not only a complete survey of classical theoretical physics but also an enormous number of worked examples and problems to show students clearly how to apply the abstract principles to realistic problems. Categories: Physics Theory of Relativity and Gravitation. Publisher : Springer New York. File Info : pdf 4 Mb. No Comments Jan 26,

## Classical Mechanics

He usually stayed about half an hour; when he had no auditors he commonly returned in a quarter of that time. This is a second course in classical mechanics, given to final year undergraduates. They were last updated in January Individual chapters and problem sheets are available below.

Haynes ManualsThe Haynes Intended for advanced undergraduates and beginning graduate students, the volumes in the series provide not only a complete survey of classical theoretical physics but also an enormous number of worked examples and problems to show students clearly how to apply the abstract principles to realistic problems. Categories: Physics Theory of Relativity and Gravitation. Publisher : Springer New York.

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