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Encyclopedia > Classical mechanics

Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science and technology. A projectile is any object sent through the air by the application of some force. ... A machine is any mechanical or electrical device that transmits or modifies energy to perform or assist in the performance of tasks. ... See lists of astronomical objects for a list of the various lists of astronomical objects in Wikipedia. ... The Space Shuttle Discovery as seen from the International Space Station. ... A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... STARS can mean: Shock Trauma Air Rescue Society Special Tactics And Rescue Service, a fictional task force that appears in Capcoms Resident Evil video game franchise. ... This article is about a celestial body. ... Part of a scientific laboratory at the University of Cologne. ... By the mid 20th century humans had achieved a mastery of technology sufficient to leave the surface of the Earth for the first time and explore space. ...


Besides this, many related specialties exist, dealing with gases, liquids, and solids, and so on. Classical mechanics is enhanced by special relativity for objects moving with high velocity, approaching the speed of light. Furthermore, general relativity is employed to handle gravitation at a deeper level. blah blah blah lalallalalalala In physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies. The other sub-field is quantum mechanics. For other uses, see Gas (disambiguation). ... For other uses, see Liquid (disambiguation). ... For other uses, see Solid (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... This article is about velocity in physics. ... The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... “Gravity” redirects here. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Mechanic (disambiguation). ... For a list of set rules, see Laws of science. ... A physical body is an object which can be described by the theories of classical mechanics, or quantum mechanics, and experimented upon by physical instruments. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...


The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century workers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "relativistic physics" from that category. However, a number of modern sources do include Einstein's mechanics, which in their view represents classical mechanics in its most developed and most accurate form. The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. This is further described in the following sections. More abstract, and general methods include Lagrangian mechanics and Hamiltonian mechanics. While the terms classical mechanics and Newtonian mechanics are usually considered equivalent (if relativity is excluded), much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... “Kepler” redirects here. ... Monument of Tycho Brahe and Johannes Kepler in Prague Tycho Brahe, born Tyge Ottesen Brahe (December 14, 1546 – October 24, 1601), was a Danish nobleman from the region of Scania (in modern-day Sweden), best known today as an early astronomer, though in his lifetime he was also well known... Galileo redirects here. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... It has been suggested that this article or section be merged with Classical mechanics. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ...

Contents

Description of the theory

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. A point particle is an idealized particle heavily used in physics. ... Although related to the more mathematical concepts of infinitesimal , the idea of something being negligible is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic engineering, computer programming and in everyday decision-making. ... The factual accuracy of this article is disputed. ... Look up position in Wiktionary, the free dictionary. ... This article or section is in need of attention from an expert on the subject. ... its made by jaypeeng magandang google wikepedia For other uses, see Force (disambiguation). ...


In reality, the kind of objects which 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—for example, a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle. For other uses, see Electron (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... This article is about the sport. ... A sphere rotating around its axis. ... Look up composite in Wiktionary, the free dictionary. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...


Displacement and its derivatives

The SI derived units with kg, m and s
displacement m
speed m s-1
acceleration m s-2
jerk m s-3
specific energy m2 s-2
absorbed dose rate m2 s-3
moment of inertia kg m2
momentum kg m s-1
angular momentum kg m2 s-1
force kg m s-2
torque kg m2 s-2
energy kg m2 s-2
power kg m2 s-3
pressure kg m-1 s-2
surface tension kg s-2
irradiance kg s-3
kinematic viscosity m2 s-1
dynamic viscosity kg m-1 s

The displacement, or position, of a point particle is defined with respect to an arbitrary fixed reference point, O, in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. It is defined as the vector r from O to the particle. In general, the point particle need not be stationary relative to O, 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, i.e., the time interval between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.[1] Look up si, Si, SI in Wiktionary, the free dictionary. ... KG, kg or Kg can refer to several things: Kilogram, the SI base unit of mass. ... For other uses, see M (disambiguation). ... Look up S, s in Wiktionary, the free dictionary. ... Look up displacement in Wiktionary, the free dictionary. ... This article does not cite any references or sources. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... Look up jerk, jolt, surge, lurch in Wiktionary, the free dictionary. ... The fuel value or relative energy density is the quantity of potential energy in fuel, food or other substance. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2), is the rotational analog of mass. ... This article is about momentum in physics. ... This gyroscope remains upright while spinning due to its angular momentum. ... its made by jaypeeng magandang google wikepedia For other uses, see Force (disambiguation). ... Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ... In physics, power (symbol: P) is the rate at which work is performed or energy is transferred. ... This article is about pressure in the physical sciences. ... Surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. ... Irradiance, radiant emittance, and radiant exitance are radiometry terms for the power of electromagnetic radiation at a surface, per unit area. ... Viscosity is a measure of the resistance of a fluid to deformation under shear stress. ... Viscosity is a measure of the resistance of a fluid to deformation under shear stress. ... Space has been an interest for philosophers and scientists for much of human history. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Look up time in Wiktionary, the free dictionary. ... In general, the principle of relativity is the requirement that the laws of physics be the same for all observers. ... The absolute time is a hypothetical time that either runs at the same rate for all the observers in the universe or the rate of time of each observer can be scaled to the absolute time by multiplying the rate by a constant. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ...


Velocity

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or This article is about velocity in physics. ... For other uses, see Calculus (disambiguation). ... For a non-technical overview of the subject, see Calculus. ...

vec{v} = {mathrm{d}vec{r} over mathrm{d}t},!.

In classical mechanics, velocities are directly additive and subtractive. For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, then from the perspective of the slower car, the faster car is traveling East at 60−50 = 10 km/h. Whereas, from the perspective of the faster car, the slower car is moving 10 km/h to the West. What if the car is traveling north? Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.


Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector vec{u} = uvec{d} and the velocity of the second object by the vector vec{v} = vvec{e} where u is the speed of the first object, v is the speed of the second object, and vec{d} and vec{e} are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is: In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

vec{u'} = vec{u} - vec{v},!

Similarly:

vec{v'}= vec{v} - vec{u},!

When both objects are moving in the same direction, this equation can be simplified to:

vec{u'} = ( u - v ) vec{d},!

Or, by ignoring direction, the difference can be given in terms of speed only:

 u' = u - v ,!

Acceleration

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) or Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... For a non-technical overview of the subject, see Calculus. ... For a non-technical overview of the subject, see Calculus. ...

vec{a} = {mathrm{d}vec{v} over mathrm{d}t}.

Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both. If only the magnitude, v, of the velocity decreases, this is sometimes referred to as deceleration, but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.


Frames of reference

While the position and velocity and acceleration of a particle can be referred to any arbitrary point of reference and accompanying coordinate system (reference frame), Classical Mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. They are characterized by the absence of accelerated motion between any two of them and the requirement of forces to produce accelerated motion of particles relative to any one of them. Any non-inertial reference frame would be accelerated with respect to an inertial one and relative to such a non-inertial frame a particle would, nevertheless, display accelerated motion. A weakness in the concept of inertial frames is the absence of any guaranteed method for identifying them. For practical purposes, reference frames that are unaccelerated with respect to the distant stars are regarded as good approximations to inertial frames.


The following consequences can be derived about the perspective of an event in two inertial reference frames, S and S', where S' is traveling at a relative velocity of vec{u} to S.

  • vec{v'} = vec{v} - vec{u} (the velocity vec{v'} of a particle from the perspective of S' is slower by vec{u} than its velocity vec{v} from the perspective of S)
  • vec{a'} = vec{a} (the acceleration of a particle remains the same regardless of reference frame)
  • vec{F'} = vec{F} (the force on a particle remains the same regardless of reference frame)
  • the speed of light is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics have a counterpart in classical mechanics.
  • the form of Maxwell's equations is not preserved across such inertial reference frames. However, in Einstein's theory of special relativity, the assumed constancy (invariance) of the vacuum speed of light alters the relationships between inertial reference frames so as to render Maxwell's equations invariant.

The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

Forces; Newton's Second Law

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 to be a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law": Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... its made by jaypeeng magandang google wikepedia For other uses, see Force (disambiguation). ... This article is about momentum in physics. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

vec{F} = {mathrm{d}(m vec{v}) over mathrm{d}t}= {mathrm{d}vec{p} over mathrm{d}t}.

The quantity mvec{v} is called the (canonical) momentum. The net force on a particle is, thus, equal to rate change of momentum of the particle with time. Typically, the mass m is constant in time, and Newton's law can be written in the simplified form In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... This article is about momentum in physics. ... This article is about momentum in physics. ...

vec{F} = m vec{a}

where vec{a} = frac {mathrm{d} vec{v}} {mathrm{d}t} is the acceleration. 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. This article is about vehicles powered by rocket engines. ...


Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for vec{F}, obtained by considering the particular physical entities with which the particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, for example: In physics, a resistive force is a force that acts on a body due to its motion relative to other bodies with which it is in contact, whose direction is opposite to the velocity of the body (or in static friction, opposite to the sum of the other forces). ...

vec{F}_{rm R} = - lambda vec{v}

with λ a positive constant (although this relation is known to be incorrect for drag in dense air, for example, it is accurate enough for elementary work). 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. Continuing the example, assume that friction is the only force acting on the particle. Then the equation of motion is In physics, mechanical work is the amount of energy transferred by a force. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

- lambda vec{v} = m vec{a} = m {mathrm{d}vec{v} over mathrm{d}t}.

This can be integrated to obtain In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

vec{v} = vec{v}_0 e^{- lambda t / m}

where vec{v}_0 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 vec{r} of the particle as a function of time. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...


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 it is known that particle A exerts a force vec{F} on another particle B, it follows that B must exert an equal and opposite reaction force, -vec{F}, on A. The strong form of Newton's third law requires that vec{F} and -vec{F} act along the line connecting A and B, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces. Gravity is a force of attraction that acts between bodies that have mass. ... Lorentz force. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...


Energy

If a force vec{F} is applied to a particle that achieves a displacement Deltavec{s}, the work done by the force is defined as the scalar product of force and displacement vectors:

 W = vec{F} cdot Delta vec{s} .

If the mass of the particle is constant, and Wtotal is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:

 W_{rm total} = Delta E_k ,!,

where Ek is called the kinetic energy. For a point particle, it is mathematically defined as the amount of work done to accelerate the particle from zero velocity to the given velocity v: The kinetic energy of an object is the extra energy which it possesses due to its motion. ... In physics, mechanical work is the amount of energy transferred by a force. ...

 E_k = begin{matrix} frac{1}{2} end{matrix} mv^2 .

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.


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 Ep: For other uses, see Gradient (disambiguation). ... Potential energy can be thought of as energy stored within a physical system. ...

vec{F} = - vec{nabla} E_p.

If all the forces acting on a particle are conservative, and Ep 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 Potential energy can be thought of as energy stored within a physical system. ...

vec{F} cdot Delta vec{s} = - vec{nabla} E_p cdot Delta vec{s} = - Delta E_p Rightarrow - Delta E_p = Delta E_k Rightarrow Delta (E_k + E_p) = 0 ,!.

This result is known as conservation of energy and states that the total energy,

sum E = E_k + E_p ,!

is constant in time. It is often useful, because many commonly encountered forces are conservative.


Beyond Newton's Laws

Classical mechanics also includes descriptions of the complex motions of extended non-pointlike objects. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. This gyroscope remains upright while spinning due to its angular momentum. ... For other uses, see Calculus (disambiguation). ...


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. Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...


Classical transformations

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' . Assuming time is measured the same in all reference frames, and if we require x = x' when t = 0, then the relation between the space-time coordinates of the same event observed from the reference frames S' and S, which are moving at a relative velocity of u in the x direction is: A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

x' = x - ut
y' = y
z' = z
t' = t

This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). This type of transformation is a limiting case of special relativity when the velocity u is very small compared to c, the speed of light. In mathematics, groups are often used to describe symmetries of objects. ... The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...


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. Centrifugal force (from Latin centrum centre and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ...


History

Some Greek philosophers of antiquity, among them Aristotle, may have been the first to maintain the idea that "everything happens for a reason" and that theoretical principles can assist in the understanding of nature. While, to a modern reader, many of these preserved ideas come forth as eminently reasonable, there is a conspicuous lack of both mathematical theory and controlled experiment, as we know it. These both turned out to be decisive factors in forming modern science, and they started out with classical mechanics. The Greeks, and Aristotle in particular, were the first to propose that there are abstract principles governing nature. ... Wikipedia does not yet have an article with this exact name. ... Aristotle (Greek: Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ...


The first published causal explanation of the motions of planets was Johannes Kepler's Astronomia nova published in 1609. He concluded, based on Tycho Brahe's observations of the orbit of Mars, that the orbits were ellipses. This break with medieval thought was happening around the same time that Galilei was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannon balls of different masses from the tower of Pisa, showing that they both hit the ground at the same time. The reality of this experiment is disputed, but, more importantly, he did carry out quantitative experiments by rolling balls on an inclined plane. His theory of accelerated motion derived from the results of such experiments, and forms a cornerstone of classical mechanics. A causal system is a system that depends only on the current and previous inputs. ... A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... Astronomia nova (A new astronomy), written by Johannes Kepler and published in 1609, set out the evidence for what came to be known as Keplers laws of planetary motion. ... Monument of Tycho Brahe and Johannes Kepler in Prague Tycho Brahe, born Tyge Ottesen Brahe (December 14, 1546 – October 24, 1601), was a Danish nobleman from the region of Scania (in modern-day Sweden), best known today as an early astronomer, though in his lifetime he was also well known... Adjectives: Martian Atmosphere Surface pressure: 0. ... The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times. ... Galileo redirects here. ... The Leaning Tower of Pisa (Italian: ) or simply The Tower of Pisa (La Torre di Pisa) is the campanile, or freestanding bell tower, of the cathedral of the Italian city of Pisa. ... The inclined plane is one of the classical simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. ...


As foundation for his principles of natural philosophy, Newton proposed three laws of motion, the law of inertia, his second law, mentioned above, and the law of action and reaction. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of Kepler's laws of motion of the planets. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... In classical mechanics, Newtons third law states that forces occur in pairs, one called the action and the other the reaction (actio et reactio in Latin). ... Johannes Keplers primary contributions to astronomy/ astrophysics were the three laws of planetary motion. ...


Newton previously invented the calculus, of mathematics, and used it to perform the mathematical calculations. For acceptability, his book, the Principia, was formulated entirely in terms of the long established geometric methods, which were soon to be eclipsed by his calculus. However it was Leibniz who developed the notation of the derivative and integral preferred today. For other uses, see Calculus (disambiguation). ... Newtons own copy of his Principia, with hand written corrections for the second edition. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... For a non-technical overview of the subject, see Calculus. ... This article is about the concept of integrals in calculus. ...


Newton, and most of his contemporaries, with the notable exception of Huygens, worked on the assumption that classical mechanics would be able to explain all phenomena, including light, in the form of geometric optics. Even when discovering the so-called Newton's rings (a wave interference phenomenon) his explanation remained with his own corpuscular theory of light. Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 – July 8, 1698), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... This article does not cite any references or sources. ... See also list of optical topics. ... Newtons rings (created by green monochromatic light) The phenomenon of Newtons rings, named after Isaac Newton, is an interference pattern caused by the reflection of light between two surfaces - a spherical surface and an adjacent flat surface. ... In communications, interference is anything which alters, modifies, or disrupts a message; as it travels along a channel, between a source and a receiver. ... According to the Corpuscular theory of light, set forward by Sir Isaac Newton, light is made up of small discrete particles called corpuscles (little particles). ...


After Newton, classical mechanics became a principal field of study in mathematics as well as physics.


Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. When combined with thermodynamics, classical mechanics leads to the Gibbs paradox of classical statistical mechanics, in which entropy is not a well-defined quantity. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms. The effort at resolving these problems led to the development of quantum mechanics. Similarly, the different behaviour of classical electromagnetism and classical mechanics under coordinate transformations (between differently moving frames of reference), eventually led to the theory of relativity. Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... For a less technical and generally accessible introduction to the topic, see Introduction to entropy. ... An energy level is a quantified stable energy, which a physical system can have; the term is most commonly used in reference to the electron configuration of electrons, in atoms or molecules. ... Properties For alternative meanings see atom (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... Two-dimensional analogy of space-time curvature described in General Relativity. ...


Since the end of the 20th century, the place of classical mechanics in physics has been no longer that of an independent theory. Along with classical electromagnetism, it has become embedded in relativistic quantum mechanics or quantum field theory.[2] It is the non-relativistic, non-quantum mechanical limit for massive particles. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Quantum field theory (QFT) is the quantum theory of fields. ...


Limits of validity

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. For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... See also list of optical topics. ... Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ...


The Newtonian approximation to special relativity

Newtonian, or non-relativistic classical mechanics approximates the relativistic momentum frac{m_0 v}{ sqrt{1-v^2/c^2}} with m0v, so it is only valid when the velocity is much less than the speed of light.


For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by f=f_cfrac{m_0}{m_0+T/c^2}, where fc is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m0 circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV. direct current accelerating voltage. A pair of Dee electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science. ... Gyrotrons are high powered electron tubes which emit a millimeter wave beam by bunching electrons with cyclotron motion in a strong magnetic field. ... A cavity magnetron is a high-powered vacuum tube that generates coherent microwaves. ...


The classical approximation to quantum mechanics

The ray approximation of classical mechanics breaks down when the de Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). ...

lambda=frac{h}{p}

where h is Planck's constant and p is the momentum. A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...


Again, this happens with electrons before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 volts, had a wave length of 0.167 nm, which was long enough to exhibit a single diffraction side lobe when reflecting from the face of a nickel crystal with atomic spacing of 0.215 nm. With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution from around a radian to a milliradian and see quantum diffraction from the periodic patterns of integrated circuit computer memory. Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... Clinton Joseph Davisson (22 October 1881–1 February 1958), was an American physicist. ... Lester Germer (full name Lester Halbert Germer; 1896–1971), American physicist. ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... In antenna engineering, the parts of the radiation pattern that are not the main lobe. ... For other uses, see Crystal (disambiguation). ... A large vacuum chamber. ... Angular resolution describes the resolving power of any optical device such as a telescope, a microscope, a camera, or an eye. ... Integrated circuit of Atmel Diopsis 740 System on Chip showing memory blocks, logic and input/output pads around the periphery Microchips with a transparent window, showing the integrated circuit inside. ...


More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated circuits. Quantum tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. ... Tunnel diode schematic symbol A tunnel diode or Esaki diode is a type of semiconductor diode which is capable of very fast operation, well into the microwave region GHz, by utilizing quantum mechanical effects. ... Assorted discrete transistors A transistor is a semiconductor device, commonly used as an amplifier or an electrically controlled switch. ... Large power N-channel field effect transistor The field-effect transistor (FET) is a type of transistor that relies on an electric field to control the shape and hence the conductivity of a channel in a semiconductor material. ... Integrated circuit of Atmel Diopsis 740 System on Chip showing memory blocks, logic and input/output pads around the periphery Microchips with a transparent window, showing the integrated circuit inside. ...


Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies. A high frequency approximation (or high energy approximation) for scattering or other wave propagation problems, in physics or engineering, is an approximation whose accuracy increases with the size of features on the scatterer or medium relative to the wavelength of the scattered particles. ... See also list of optical topics. ... The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ...


Notes

  1. ^ MIT physics 8.01 lecture notes (page 12) (PDF)
  2. ^ Page 2-10 of the Feynman Lectures on Physics says "For already in classical mechanics there was indeterminability from a practical point of view." The past tense here implies that classical physics is no longer fundamental.

The Feynman Lectures on Physics, by Richard Feynman, is perhaps his most accessible technical work for anyone with an interest in physics and today is considered to be the classic introduction to modern physics, including lectures on mathematics, electromagnetism, Newtonian physics, quantum physics, and even the relation of physics to...

References

  • Feynman, Richard (1996). Six Easy Pieces. Perseus Publishing. ISBN 0-201-40825-2. 
  • Feynman, Richard; Phillips, Richard (1998). Six Easy Pieces. Perseus Publishing. ISBN 0-201-32841-0. 
  • Feynman, Richard (1999). Lectures on Physics. Perseus Publishing. ISBN 0-7382-0092-1. 
  • Landau, L. D.; Lifshitz, E. M. (1972). Mechanics Course of Theoretical Physics , Vol. 1. Franklin Book Company, Inc.. ISBN 0-08-016739-X. 
  • Kleppner, D. and Kolenkow, R. J., An Introduction to Mechanics, McGraw-Hill (1973). ISBN 0-07-035048-5
  • Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics, MIT Press (2001). ISBN 0-262-19455-4}
  • Herbert Goldstein, Charles P. Poole, John L. Safko, Classical Mechanics (3rd Edition), Addison Wesley; ISBN 0-201-65702-3
  • Robert Martin Eisberg, Fundamentals of Modern Physics, John Wiley and Sons, 1961
  • M. Alonso, J. Finn, "Fundamental university physics", Addison-Wesley

// Gerald Jay Sussman is the Panasonic Professor of Electrical Engineering at the Massachusetts Institute of Technology (MIT). ... Jack Wisdom is a Professor of Planetary Sciences at the Massachusetts Institute of Technology. ... Structure and Interpretation of Classical Mechanics (SICM) is a textbook about classical mechanics, written by Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer. ... Herbert Goldstein (June 26, 1922 – January 12, 2005) was an American physicist and the author of the standard graduate textbook Classical Mechanics, widely considered to be one of the best books on the subject. ...

See also

Physics Portal

Image File history File links Portal. ... The Greeks, and Aristotle in particular, were the first to propose that there are abstract principles governing nature. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... This page gives a summary of important equations in classical mechanics. ... This is a list of important publications in physics, organized by field. ...

Branches

Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... See also list of optical topics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... It has been suggested that this article or section be merged with Classical mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

External links

  • Binney, James. Classical Mechanics (Lagrangian and Hamiltonian formalisms)
  • Crowell, Benjamin. Newtonian Physics (an introductory text, uses algebra with optional sections involving calculus)
  • Fitzpatrick, Richard. Classical Mechanics (uses calculus)
  • Hoiland, Paul (2004). Preferred Frames of Reference & Relativity
  • Horbatsch, Marko, "Classical Mechanics Course Notes".
  • Rosu, Haret C., "Classical Mechanics". Physics Education. 1999. [arxiv.org : physics/9909035]
  • Schiller, Christoph. Motion Mountain (an introductory text, uses some calculus)
  • Sussman, Gerald Jay & Wisdom, Jack & Mayer,Meinhard E. (2001). Structure and Interpretation of Classical Mechanics
  • Tong, David. Classical Dynamics (Cambridge lecture notes on Lagrangian and Hamiltonian formalism)

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