Classical mechanics
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{{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]], [[star]]s, and [[galaxies]]. It produces very accurate results within these domains, and is one of the oldest and largest subjects in [[science]] and [[technology]].
Besides this, many related specialties exist, dealing with [[gas]]es, [[liquid]]s, and [[solid]]s, 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.
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 law]]s governing and mathematically describing the motions of [[physical body|bodies]] and aggregates of bodies. The other sub-field is [[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 Galilei|Galileo]], but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "[[Theory of relativity|relativistic physics]]" from that category. However, a number of modern sources ''do'' include [[Theory of relativity|Einstein's mechanics]], which in their view represents ''classical mechanics'' in its most developed and most accurate form.
<!--The notion of “classical“ may be somewhat confusing, insofar as this term usually refers to the era of [[classical antiquity]] in [[European history]]. While many discoveries within the [[mathematics]] of that period remain in full force today, and of the greatest use, the same cannot be said about its "science". This in no way belittles the many important developments, especially within technology, which took place in antiquity and during the [[Middle Ages]] in Europe and elsewhere.
However, the emergence of classical mechanics was a decisive stage in the development of [[science]], in the modern sense of the term. What characterizes it, above all, is its insistence on [[mathematics]] (rather than [[speculation]]), and its reliance on [[experiment]] (rather than [[observation]]). With classical mechanics it was established how to formulate [[quantitative]] predictions in [[theory]], and how to test them by carefully designed [[measurement]]. The emerging globally cooperative endeavor increasingly provided for much closer scrutiny and testing, both of theory and experiment. This was, and remains, a key factor in establishing certain knowledge, and in bringing it to the service of society. History shows how closely the health and wealth of a society depends on nurturing this investigative and critical approach. -->
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 [[Isaac Newton|Newton]] himself, in parallel with [[Gottfried Wilhelm von Leibniz|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 [[Isaac Newton|Newton]].
==Description of the theory==
[[Image:Tir parabòlic.png|thumb|The analysis of projectile motion is a part of classical mechanics.]]
The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as [[point particle]]s, objects with [[negligible]] size. The motion of a point particle is characterized by a small number of [[parameter]]s: its [[position]], [[mass]], and the [[force]]s applied to it. Each of these parameters is discussed in turn.
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 (physics and chemistry)|degrees of freedom]]—for example, a [[baseball]] can [[rotation|spin]] while it is moving. However, the results for point particles can be used to study such objects by treating them as [[wiktionary:Composite|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.
===Displacement and its derivatives===
{|class="wikitable" align="right" border="1"
|-
|colspan="2"|The [[SI]] derived units with [[kilogram|kg]], [[meter|m]] and [[second|s]]
|-
|[[displacement]]||m
|-
|[[speed]]||m s<sup>−1</sup>
|-
|[[acceleration]]||m s<sup>−2</sup>
|-
|[[jerk]]||m s<sup>−3</sup>
|-
|[[specific energy]]||m² s<sup>−2</sup>
|-
|absorbed dose rate||m² s<sup>−3</sup>
|-
|[[moment of inertia]]||kg m²
|-
|[[momentum]]||kg m s<sup>−1</sup>
|-
|[[angular momentum]]||kg m² s<sup>−1</sup>
|-
|[[force]]||kg m s<sup>−2</sup>
|-
|[[torque]]||kg m² s<sup>−2</sup>
|-
|[[energy]]||kg m² s<sup>−2</sup>
|-
|[[Power (physics)|power]]||kg m² s<sup>−3</sup>
|-
|[[pressure]]||kg m<sup>−1</sup> s<sup>−2</sup>
|-
|[[surface tension]]||kg s<sup>−2</sup>
|-
|[[irradiance]]||kg s<sup>−3</sup>
|-
|[[kinematic viscosity]]||m² s<sup>−1</sup>
|-
|[[dynamic viscosity]]||kg m<sup>−1</sup> 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 (spatial)|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.<ref>[http://ocw.mit.edu/NR/rdonlyres/Physics/8-01Physics-IFall2003/B4144452-A6DE-464D-A0FA-D4D057AA9222/0/binder1.pdf MIT physics 8.01 lecture notes (page 12)] (PDF)</ref>
====Velocity and speed====
The ''[[velocity]]'', or the [[calculus|rate of change]] of position with time, is defined as the [[derivative]] of the position with respect to time or
: <math>\vec{v} = {\mathrm{d}\vec{r} \over \mathrm{d}t}\,\!</math>.
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. 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 <math>\vec{u} = u\vec{d}</math> and the velocity of the second object by the vector <math>\vec{v} = v\vec{e}</math> where <math>u</math> is the speed of the first object, <math>v</math> is the speed of the second object, and <math>\vec{d}</math> and <math>\vec{e}</math> are [[unit vector]]s in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is:
:<math>\vec{u'} = \vec{u} - \vec{v}\,\!</math>
Similarly:
:<math>\vec{v'}= \vec{v} - \vec{u}\,\!</math>
When both objects are moving in the same direction, this equation can be simplified to:
:<math>\vec{u'} = ( u - v ) \vec{d}\,\!</math>
Or, by ignoring direction, the difference can be given in terms of speed only:
:<math> u' = u - v \,\!</math>
====Acceleration====
The ''[[acceleration]]'', or rate of change of velocity, is the [[derivative]] of the velocity with respect to time (the [[derivative|second derivative]] of the position with respect to time) or
: <math>\vec{a} = {\mathrm{d}\vec{v} \over \mathrm{d}t}</math>.
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, <math>v</math>, 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, <math>S</math> and <math>S'</math>, where <math>S'</math> is traveling at a relative velocity of <math>\vec{u}</math> to <math>S</math>.
* <math>\vec{v'} = \vec{v} - \vec{u}</math> (the velocity <math>\vec{v'}</math> of a particle from the perspective of ''S' is'' slowed by <math>\vec{u}</math> than its velocity <math>\vec{v}</math> from the perspective of ''S'')
* <math>\vec{a'}</math> = <math>\vec{a}</math> (the acceleration of a particle remains the same regardless of reference frame)
* <math>\vec{F'}</math> = <math>\vec{F}</math> (the force on a particle remains the same regardless of reference frame)
* '''F'''' = '''F''' (since '''F''' = ''m'''''a''', as long as the mass ''m'' stays constant) (the force on a particle remains the same regardless of reference frame; see [[Newton's laws of motion|Newton's law]])
* 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 [[Maxwells equations|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.
===Forces; Newton's Second Law===
[[Isaac Newton|Newton]] was the first to mathematically express the relationship between [[force]] and [[momentum]]. Some physicists interpret [[Newton's laws of motion|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":
: <math>\vec{F} = {\mathrm{d}\vec{p} \over \mathrm{d}t} = {\mathrm{d}(m \vec{v}) \over \mathrm{d}t}</math>.
The quantity <math>m\vec{v}</math> is called the ([[canonical momentum|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
: <math>\vec{F} = m \vec{a}</math>
where <math>\vec{a} = \frac {\mathrm{d} \vec{v}} {\mathrm{d}t}</math> 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.
Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for <math>\vec{F}</math>, 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:
: <math>\vec{F}_{\rm R} = - \lambda \vec{v}</math>
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 [[Mechanical work|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
: <math>- \lambda \vec{v} = m \vec{a} = m {\mathrm{d}\vec{v} \over \mathrm{d}t}</math>.
This can be [[antiderivative|integrated]] to obtain
: <math>\vec{v} = \vec{v}_0 e^{- \lambda t / m}</math>
where <math>\vec{v}_0</math> is the initial velocity. This means that the velocity of this particle [[exponential decay|decays exponentially]] to zero as time progresses. This expression can be further integrated to obtain the position <math>\vec{r}</math> of the particle as a function of time.
Important forces include the [[gravity|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 <math>\vec{F}</math> on another particle B, it follows that B must exert an equal and opposite ''reaction force'', -<math>\vec{F}</math>, on A. The strong form of Newton's third law requires that <math>\vec{F}</math> and -<math>\vec{F}</math> 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.
===Energy===
If a force <math>\vec{F}</math> is applied to a particle that achieves a displacement <math>\Delta\vec{s}</math>, the ''work done'' by the force is defined as the scalar product of force and displacement vectors:
: <math> W = \vec{F} \cdot \Delta \vec{s} </math>.
If the mass of the particle is constant, and ''W''<sub>total</sub> is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:
: <math> W_{\rm total} = \Delta E_k \,\!</math>,
where ''E<sub>k</sub>'' is called the [[kinetic energy]]. For a point particle, it is mathematically defined as the amount of [[mechanical work|work]] done to accelerate the particle from zero velocity to the given velocity v:
: <math> E_k = \begin{matrix} \frac{1}{2} \end{matrix} mv^2 </math>.
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 ''E<sub>p</sub>'':
: <math>\vec{F} = - \vec{\nabla} E_p</math>.
If all the forces acting on a particle are conservative, and ''E<sub>p</sub>'' 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
{|
|
| <math>\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 \,\!</math>.
|}
This result is known as ''conservation of energy'' and states that the total [[energy]],
: <math>\sum E = E_k + E_p \,\!</math>
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.
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, for describing mechanical systems.
===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:
:''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]].
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 (fictitious)|centrifugal force]] and [[Coriolis force]].
==History==
{{Main|History of classical mechanics}}
{{See also|Timeline of classical mechanics}}
Some [[Greek philosophy|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.
An early experimental [[scientific method]] was introduced into [[Islamic science#Mechanics|mechanics]] in the 11th century by [[al-Biruni]], who along with [[al-Khazini]] in the 12th century, unified [[statics]] and [[dynamics]] into the [[science]] of mechanics, and combined the fields of [[hydrostatics]] with dynamics to create the field of [[hydrodynamics]].<ref>Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 614-642 [642], [[Routledge]], London and New York</ref> Concepts related to [[Newton's laws of motion]] were also enunciated by several other [[Islamic physics|Muslim physicists]] during the [[Middle Ages]]. Early versions of the law of [[inertia]], known as Newton's first law of motion, and the concept relating to [[momentum]], part of Newton's second law of motion, were described by [[Ibn al-Haytham]] (Alhacen)<ref>[[Abdus Salam]] (1984), "Islam and Science". In C. H. Lai (1987), ''Ideals and Realities: Selected Essays of Abdus Salam'', 2nd ed., World Scientific, Singapore, p. 179-213.</ref><ref>Seyyed [[Hossein Nasr]], "The achievements of Ibn Sina in the field of science and his contributions to its philosophy", ''Islam & Science'', December 2003.</ref> and [[Avicenna]].<ref name=Espinoza>Fernando Espinoza (2005). "An analysis of the historical development of ideas about motion and its implications for teaching", ''Physics Education'' '''40''' (2), p. 141.</ref><ref>Seyyed [[Hossein Nasr]], "Islamic Conception Of Intellectual Life", in Philip P. Wiener (ed.), ''Dictionary of the History of Ideas'', Vol. 2, p. 65, Charles Scribner's Sons, New York, 1973-1974.</ref> The proportionality between [[force]] and [[acceleration]], an important principle in classical mechanics, was first stated by [[Hibat Allah Abu'l-Barakat al-Baghdaadi]],<ref>{{cite encyclopedia
| last = [[Shlomo Pines]]
| title = Abu'l-Barakāt al-Baghdādī, Hibat Allah
| encyclopedia = [[Dictionary of Scientific Biography]]
| volume = 1
| pages = 26-28
| publisher = Charles Scribner's Sons
| location = New York
| date = 1970
| isbn = 0684101149
}}
<br>([[cf.]] Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", ''Journal of the History of Ideas'' '''64''' (4), p. 521-546 [528]</ref> and theories on gravity were developed by [[Ja'far Muhammad ibn Mūsā ibn Shākir]],<ref>[[Robert Briffault]] (1938). ''The Making of Humanity'', p. 191.</ref> [[Ibn al-Haytham]],<ref>Nader El-Bizri (2006), "Ibn al-Haytham or Alhazen", in Josef W. Meri (2006), ''Medieval Islamic Civilization: An Encyclopaedia'', Vol. II, p. 343-345, [[Routledge]], New York, London.</ref> and [[al-Khazini]].<ref>Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., ''Encyclopaedia of the History of Arabic Science'', Vol. 2, p. 622. London and New York: Routledge.</ref> It is known that [[Galileo Galilei]]'s mathematical treatment of [[acceleration]] and his concept of [[Inertia#Early understanding of motion|impetus]]<ref>Galileo Galilei, ''Two New Sciences'', trans. Stillman Drake, (Madison: Univ. of Wisconsin Pr., 1974), pp 217, 225, 296-7.</ref> grew out of earlier medieval analyses of [[Motion (physics)|motion]], especially those of [[Avicenna]],<ref name=Espinoza/> [[Ibn Bajjah]],<ref>Ernest A. Moody (1951). "Galileo and Avempace: The Dynamics of the Leaning Tower Experiment (I)", ''Journal of the History of Ideas'' '''12''' (2), p. 163-193.</ref> and [[Jean Buridan]].
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 [[Ancient philosophy|ancient thought]] was happening around the same time that [[Galileo Galilei|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 [[Leaning Tower of Pisa|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.
As foundation for his principles of natural philosophy, Newton proposed three [[Newton's laws of motion|laws of motion]], the [[law of inertia]], his second law of acceleration, mentioned above, and the law of [[action and reaction]], and hence laying the foundations for classical mechanics. Both Newtons second and third laws were given proper scientific and mathematical treatment in Newton's [[Philosophiæ Naturalis Principia Mathematica]], which distinguishes them from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. [[Newton]] also enunciated the principles of [[conservation of momentum]] and [[angular momentum]]. In Mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of [[gravity]] in [[Newton's law of universal gravitation]]. The combination of Newton's laws of motion and gravitation provide the fullest and most accurate description of [[classical mechanics]]. 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.
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.
Newton, and most of his contemporaries, with the notable exception of [[Christiaan Huygens|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]].
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]].
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 [[theory of relativity|relativistic]] [[quantum mechanics]] or [[quantum field theory]].<ref>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.</ref> It is the non-relativistic, non-quantum mechanical limit for massive particles.
==Limits of validity==
[[image:physicsdomains.jpg|380px|thumb|Domain of validity for Classical Mechanics]]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 optics|quantum theory of light]], and does not have a superior "classical" form.
===The Newtonian approximation to special relativity===
Newtonian, or non-relativistic classical momentum
:<math>p = m_0 v</math>
is the result of the [[Orders of approximation|first order]] [[Taylor series|Taylor approximation]] of the relativistic expression:
:<math>p = \frac{m_0 v}{ \sqrt{1-v^2/c^2}} = m_0 v \left(1+\frac{1}{2}\frac{v^2}{c^2} + ... \right)</math>
when expanded about
:<math>\frac{v}{c}=0</math>
so it is only valid when the velocity is much less than the speed of light. Quantitatively speaking, the approximation is good so long as
:<math>\left(\frac{v}{c}\right)^2 << 1 </math>
For example, the relativistic cyclotron frequency of a [[cyclotron]], [[gyrotron]], or high voltage [[magnetron]] is given by <math>f=f_c\frac{m_0}{m_0+T/c^2}</math>, where
<math>f_c</math> is the classical frequency of an electron (or other charged particle) with kinetic energy <math>T</math> and (rest) mass <math>m_0</math> 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.
===The classical approximation to quantum mechanics===
The ray approximation of classical mechanics breaks down when the [[De Broglie hypothesis|de Broglie wavelength]] is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is
:<math>\lambda=\frac{h}{p}</math>
where ''h'' is [[Planck's constant]] and ''p'' is the momentum.
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.
More practical examples of the failure of classical mechanics on an engineering scale are conduction by [[quantum tunneling]] in [[tunnel diode]]s and very narrow [[transistor]] [[gate (transistor)|gates]] in [[integrated circuit]]s.
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.
==Notes==
{{reflist}}
==References==
* {{cite book | author=Feynman, Richard | title=Six Easy Pieces | publisher=Perseus Publishing | year=1996 | id=ISBN 0-201-40825-2}}
* {{cite book | author=Feynman, Richard; Phillips, Richard | title=Six Easy Pieces | publisher=Perseus Publishing | year=1998 | id=ISBN 0-201-32841-0}}
* {{cite book | author=Feynman, Richard | title=Lectures on Physics | publisher=Perseus Publishing | year=1999 | id=ISBN 0-7382-0092-1}}
* {{cite book | author=Landau, L. D.; Lifshitz, E. M. | title=Mechanics Course of Theoretical Physics , Vol. 1 | publisher=Franklin Book Company, Inc. | year=1972 | id=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
==See also==
{{portal|Physics}}
* [[History of classical mechanics]]
* [[Dynamical systems]]
* [[List of equations in classical mechanics]]
* [[List of publications in physics#Classical mechanics|List of publications in classical mechanics]]
* [[Molecular dynamics]]
*[[Newton's laws of motion]]
*[[Special theory of relativity]]
===Branches===
<div style="-moz-column-count:2; column-count:2;">
* [[Celestial Mechanics]]
* [[Continuum mechanics|Continuum Mechanics]]
* [[General Relativity]]
* [[Geometric Optics]]
* [[Hamiltonian Mechanics]]
* [[Lagrangian Mechanics]]
* [[Newtonian Mechanics]]
* [[Special Relativity]]
* [[Statistical Mechanics]]
* [[Thermodynamics]]
</div>
==External links==
{{commonscat|Classical mechanics}}
* Binney, James. [http://www-thphys.physics.ox.ac.uk/users/JamesBinney/ Classical Mechanics] (Lagrangian and Hamiltonian formalisms)
* Crowell, Benjamin. [http://www.lightandmatter.com/area1book1.html Newtonian Physics] (an introductory text, uses algebra with optional sections involving calculus)
* Fitzpatrick, Richard. [http://farside.ph.utexas.edu/teaching/301/301.html Classical Mechanics] (uses calculus)
* Hoiland, Paul (2004). [http://doc.cern.ch//archive/electronic/other/ext/ext-2004-126.pdf Preferred Frames of Reference & Relativity]
* Horbatsch, Marko, "''[http://www.yorku.ca/marko/PHYS2010/index.htm Classical Mechanics Course Notes]''".
* Rosu, Haret C., "''[http://arxiv.org/abs/physics/9909035 Classical Mechanics]''". Physics Education. 1999. [arxiv.org : physics/9909035]
* Schiller, Christoph. [http://www.motionmountain.net Motion Mountain] (an introductory text, uses some calculus; see also [[Motion_Mountain]])
* Sussman, Gerald Jay & Wisdom, Jack & Mayer,Meinhard E. (2001). [http://mitpress.mit.edu/SICM/ Structure and Interpretation of Classical Mechanics]
* Tong, David. [http://www.damtp.cam.ac.uk/user/tong/dynamics.html Classical Dynamics] (Cambridge lecture notes on Lagrangian and Hamiltonian formalism)
{{Physics-footer}}
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[[Category:Fundamental physics concepts]]
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