View source for Classical mechanics

{{Short description|Description of large objects' physics}}
{{otheruses|Classical Mechanics (disambiguation)}}
{{broader|Mechanics}}
{{pp-pc1}}
{{More citations needed|date=July 2022}}
[[File:Orbital motion.gif|thumb|right|261x261px|alt=animation of orbital velocity and centripetal acceleration|Diagram of orbital motion of a satellite around the Earth, showing perpendicular velocity and acceleration (force) vectors, represented through a classical interpretation]]
{{Classical mechanics}}

'''Classical mechanics''' is a [[Theoretical physics|physical theory]] describing the [[motion]] of objects such as [[projectile]]s, parts of [[Machine (mechanical)|machinery]], [[spacecraft]], [[planets]], [[star]]s, and [[galaxies]]. The development of classical mechanics involved [[Scientific Revolution|substantial change in the methods and philosophy]] of physics.<ref>{{citation |last=Ben-Chaim |first=Michael |year=2004 |publication-date=2004 |title = Experimental Philosophy and the Birth of Empirical Science: Boyle, Locke and Newton |location=Aldershot |publisher=Ashgate |isbn=0-7546-4091-4 |oclc=53887772 }}.</ref> The qualifier ''classical'' distinguishes this type of mechanics from physics developed after the [[History of physics#20th century: birth of modern physics|revolutions in physics of the early 20th century]], all of which revealed limitations in classical mechanics.<ref>{{citation |last=Agar |first=Jon |year=2012 |publication-date=2012 |title=Science in the Twentieth Century and Beyond |location=Cambridge |publisher=Polity Press |isbn=978-0-7456-3469-2 }}.</ref>

The earliest formulation of classical mechanics is often referred to as [[Newtonian mechanics]]. It consists of the physical concepts based on the 17th century foundational works of Sir [[Isaac Newton]], and the mathematical methods invented by Newton, [[Gottfried Wilhelm Leibniz]], [[Leonhard Euler]] and others to describe the motion of [[Physical body|bodies]] under the influence of [[force]]s. Later, methods based on [[energy]] were developed by Euler, [[Joseph-Louis Lagrange]], [[William Rowan Hamilton]] and others, leading to the development of [[analytical mechanics]] (which includes [[Lagrangian mechanics]] and [[Hamiltonian mechanics]]). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics.

If the present state of an object that obeys the laws of classical mechanics is known, it is possible to [[determinism|determine how it will move in the future]], and how it has moved in the past. [[Chaos theory]] shows that the long term predictions of classical mechanics are not reliable. Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching the [[speed of light]]. With objects  about the size of an atom's diameter, it becomes necessary to use [[quantum mechanics]]. To describe velocities approaching the speed of light, [[special relativity]] is needed. In cases where objects become extremely massive, [[general relativity]] becomes applicable. Some modern sources include [[relativistic mechanics]] in classical physics, as representing the field in its most developed and accurate form.

== Branches ==
=== Traditional division ===
Classical mechanics was traditionally divided into three main branches. 
''[[Statics]]'' is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in [[mechanical equilibrium|equilibrium]] with its environment.<ref name=Wright>{{cite book |title=Elements of Mechanics Including Kinematics, Kinetics and Statics: with applications |year=1896 |publisher=E. and F. N. Spon |author=Thomas Wallace Wright |page=85 |url=https://books.google.com/books?id=-LwLAAAAYAAJ&q=mechanics+kinetics}}</ref>  ''[[Kinematics]]'' describes the [[Motion (physics)|motion]] of points, [[Physical object|bodies]] (objects), and systems of bodies (groups of objects) without considering the [[force]]s that cause them to move.<ref name="Whittaker">{{cite book |title=[[A Treatise on the Analytical Dynamics of Particles and Rigid Bodies]]|author=Edmund Taylor Whittaker|author-link=E. T. Whittaker |at=Chapter 1 |year=1904 |publisher=Cambridge University Press |isbn=0-521-35883-3}}</ref><ref name=Beggs>{{cite book |title=Kinematics |author=Joseph Stiles Beggs |page=1 |url=https://books.google.com/books?id=y6iJ1NIYSmgC |isbn=0-89116-355-7 |year=1983 |publisher=Taylor & Francis}}</ref><ref name=Wright/> Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of [[mathematics]].<ref>{{cite book |title=Engineering Mechanics: Dynamics |author=Russell C. Hibbeler |chapter=Kinematics and kinetics of a particle |chapter-url=https://books.google.com/books?id=tOFRjXB-XvMC&pg=PA298 |page=298 |isbn=978-0-13-607791-6 |year=2009 |edition=12th |publisher=Prentice Hall}}</ref><ref>{{cite book |title=Dynamics of Multibody Systems |author=Ahmed A. Shabana |chapter=Reference kinematics |chapter-url=https://books.google.com/books?id=zxuG-l7J5rgC&pg=PA28 |edition=2nd |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-54411-5}}</ref><ref>{{cite book |title=Mechanical Systems, Classical Models: Particle Mechanics |chapter=Kinematics |page=287 |chapter-url=https://books.google.com/books?id=k4H2AjWh9qQC&pg=PA287 |author=P. P. Teodorescu |isbn=978-1-4020-5441-9 |year=2007 |publisher=Springer}}.</ref> [[Dynamics (mechanics)|''Dynamics'']] goes beyond merely describing objects' behavior and also considers the forces which explain it.
Some authors (for example, Taylor (2005)<ref name="Taylor">{{cite book| author=John Robert Taylor| title=Classical Mechanics|  publisher=University Science Books| url=https://books.google.com/books?id=P1kCtNr-pJsC&q=dynamics| isbn=978-1-891389-22-1| year=2005}}</ref> and Greenwood (1997)<ref name=Greenwood>
{{cite book |title=Classical Mechanics |author = Donald T Greenwood |page=1 |url=https://books.google.com/books?id=x7rj83I98yMC&q=classical+dynamics |isbn=0-486-69690-1 |publisher=Courier Dover Publications |year=1997 |edition=Reprint of 1977}}</ref>) include [[special relativity]] within classical dynamics.

=== Forces vs. energy ===
Another division is based on the choice of mathematical formalism. Classical mechanics can be mathematically presented in multiple different ways. The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to a particular formalism based on [[Newton's laws of motion]]. Newtonian mechanics in this sense emphasizes force as a [[vector (physics)|vector]] quantity.<ref name=Lanczos>{{cite book |title=The variational principles of mechanics |last=Lanczos |first=Cornelius |page=Introduction, pp. xxi–xxix |edition=4th |publisher=Dover Publications Inc. |location= New York |isbn=0-486-65067-7 |year=1970 |url=https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4 |no-pp=true}}</ref>

In contrast, [[analytical mechanics]] uses ''[[Scalar (physics)|scalar]]'' properties of motion representing the system as a whole—usually its [[kinetic energy]] and [[potential energy]]. The [[equations of motion]] are derived from the scalar quantity by some underlying principle about the scalar's [[calculus of variations|variation]]. Two dominant branches of analytical mechanics are [[Lagrangian mechanics]], which uses generalized coordinates and corresponding generalized velocities in [[Configuration space (physics)|configuration space]], and [[Hamiltonian mechanics]], which uses coordinates and corresponding momenta in [[phase space]]. Both formulations are equivalent by a [[Legendre transformation#Hamilton–Lagrange mechanics|Legendre transformation]] on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as [[Hamilton–Jacobi theory]], [[Routhian mechanics]], and [[Appell's equation of motion]]. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the [[principle of least action]]. One result is [[Noether's theorem]], a statement which connects [[conservation law]]s to their associated [[Symmetry (physics)|symmetries]].

=== By region of application ===
Alternatively, a division can be made by region of application:
* [[Celestial mechanics]], relating to [[star]]s, [[planet]]s and other celestial bodies
* [[Continuum mechanics]], for materials modelled as a continuum, e.g., [[solid]]s and [[fluid]]s (i.e., [[liquid]]s and [[gas]]es).
* [[Relativistic mechanics]] (i.e. including the [[Special relativity|special]] and [[general relativity|general]] theories of relativity), for bodies whose speed is close to the speed of light.
* [[Statistical mechanics]], which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk [[thermodynamics|thermodynamic]] properties of materials.

== Description of objects and their motion ==
[[File:Tir parabòlic.svg|thumb|alt=diagram of parabolic projectile motion|The analysis of [[projectile motion]] is a part of classical mechanics.]]
For simplicity, classical mechanics often models real-world objects as [[point particle]]s, that is, objects with negligible size. The motion of a point particle is determined by a small number of [[parameter]]s: its position, [[mass]], and the [[force]]s applied to it. 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". (These generalizations/extensions are derived from Newton's laws, say, by decomposing a solid body into a collection of points.)

In reality, the kind of objects that classical mechanics can describe always have a [[0 (number)|non-zero]] size. (The behavior 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]], e.g., a [[Baseball (ball)|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 [[wikt:composite|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 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]]).
{{clearright}}

=== Kinematics ===
{{Main|Kinematics}}
{| class="wikitable" style="float:right; margin:0 0 1em 1em;"
|+ The [[SI]] derived "mechanical"<br/>(that is, not [[Electromagnetism|electromagnetic]] or [[Thermal physics|thermal]])<br/>units with kg, m and [[second|s]]
|-
|position||m
|-
|angular position/[[angle]]||unitless (radian)
|-
|[[velocity]]||m·s<sup>−1</sup>
|-
|[[angular velocity]]||s<sup>−1</sup>
|-
|[[acceleration]]||m·s<sup>−2</sup>
|-
|[[angular acceleration]]||s<sup>−2</sup>
|-
|[[Jerk (physics)|jerk]]||m·s<sup>−3</sup>
|-
|"angular jerk"||s<sup>−3</sup>
|-
|[[specific energy]]||m<sup>2</sup>·s<sup>−2</sup>
|-
|absorbed dose rate||m<sup>2</sup>·s<sup>−3</sup>
|-
|[[moment of inertia]]||kg·m<sup>2</sup>
|-
|[[momentum]]||kg·m·s<sup>−1</sup>
|-
|[[angular momentum]]||kg·m<sup>2</sup>·s<sup>−1</sup>
|-
|[[force]]||kg·m·s<sup>−2</sup>
|-
|[[torque]]||kg·m<sup>2</sup>·s<sup>−2</sup>
|-
|[[energy]]||kg·m<sup>2</sup>·s<sup>−2</sup>
|-
|[[Power (physics)|power]]||kg·m<sup>2</sup>·s<sup>−3</sup>
|-
|[[pressure]] and [[energy density]]||kg·m<sup>−1</sup>·s<sup>−2</sup>
|-
|[[surface tension]]||kg·s<sup>−2</sup>
|-
|[[spring constant]]||kg·s<sup>−2</sup>
|-
|[[irradiance]] and [[energy flux]]||kg·s<sup>−3</sup>
|-
|[[kinematic viscosity]]||m<sup>2</sup>·s<sup>−1</sup>
|-
|[[dynamic viscosity]]||kg·m<sup>−1</sup>·s<sup>−1</sup>
|-
|[[density]] (mass density)||kg·m<sup>−3</sup>
|-
|[[specific weight]] (weight density)||kg·m<sup>−2</sup>·s<sup>−2</sup>
|-
|[[number density]]||m<sup>−3</sup>
|-
|[[Action (physics)|action]]||kg·m<sup>2</sup>·s<sup>−1</sup>
|}
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 (geometric)|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.e., the [[time interval]] that is observed to elapse between any given pair of events is the same for all observers.<ref>{{cite book |title=Elements of Newtonian Mechanics |edition=illustrated |first1=Jens M. |last1=Knudsen |first2=Poul |last2=Hjorth |publisher=Springer Science & Business Media |year=2012 |isbn=978-3-642-97599-8 |page=30 |url=https://books.google.com/books?id=rkP1CAAAQBAJ}} [https://books.google.com/books?id=rkP1CAAAQBAJ&pg=PA30 Extract of page 30]</ref> In addition to relying on [[absolute time]], classical mechanics assumes [[Euclidean geometry]] for the structure of space.<ref>[http://ocw.mit.edu/courses/physics/8-01-physics-i-fall-2003/lecture-notes/binder1.pdf MIT physics 8.01 lecture notes (page 12)]. {{webarchive|url=http://webarchive.loc.gov/all/20130709154423/http%3A//ocw.mit.edu/courses/physics/8%2D01%2Dphysics%2Di%2Dfall%2D2003/lecture%2Dnotes/binder1.pdf |date=2013-07-09 }} (PDF)</ref>

==== Velocity and speed ====
{{Main|Velocity|speed}}
The ''[[velocity]]'', or the [[calculus|rate of change]] of displacement with time, is defined as the [[derivative]] of the position with respect to time:
: <math>\mathbf{v} = {\mathrm{d}\mathbf{r} \over \mathrm{d}t}\,\!</math>.

In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60&nbsp;km/h and passes another car traveling in the same direction at 50&nbsp;km/h, the slower car perceives the faster car as traveling east at {{nowrap|60 − 50 {{=}} 10 km/h}}. However, from the perspective of the faster car, the slower car is moving 10&nbsp;km/h to the west, often denoted as −10&nbsp;km/h where the sign implies opposite direction. Velocities are directly additive as [[b:Physics with Calculus/Mechanics/Scalar and Vector Quantities|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 {{nowrap|'''u''' {{=}} ''u'''''d'''}} and the velocity of the second object by the vector {{nowrap|'''v''' {{=}} ''v'''''e'''}}, where ''u'' is the speed of the first object, ''v'' is the speed of the second object, and '''d''' and '''e''' are [[unit vector]]s in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is:
: <math>\mathbf{u}' = \mathbf{u} - \mathbf{v} \, .</math>

Similarly, the first object sees the velocity of the second object as:
: <math>\mathbf{v'}= \mathbf{v} - \mathbf{u} \, .</math>

When both objects are moving in the same direction, this equation can be simplified to:
: <math>\mathbf{u}' = ( u - v ) \mathbf{d} \, .</math>

Or, by ignoring direction, the difference can be given in terms of speed only:
: <math>u' = u - v \, .</math>

==== Acceleration ====
{{Main|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):
: <math>\mathbf{a} = {\mathrm{d}\mathbf{v} \over \mathrm{d}t} = {\mathrm{d^2}\mathbf{r} \over \mathrm{d}t^2}.</math>

Acceleration represents the velocity's change over time.  Velocity can change in magnitude, 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 referred to as acceleration.

==== Frames of reference ====
{{Main|Inertial frame of reference|Galilean transformation}}
While the position, velocity and acceleration of a [[particle]] can be described with respect to any [[observer (special relativity)|observer]] in any state of motion, classical mechanics assumes the existence of a special family of [[frame of reference|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 with zero net force acting upon it moves with a constant velocity; that is, it is either at rest or moving uniformly in a straight line. In an inertial frame Newton's law of motion, <math>F=ma</math>, is valid.<ref name=Goldstein>Goldstein, Herbert (1950). Classical Mechanics (1st ed.). Addison-Wesley.</ref>{{rp|185}}

[[Non-inertial reference frame]]s accelerate in relation to another inertial frame. A body rotating with respect to an inertial frame is not an inertial frame.<ref name=Goldstein></ref> When viewed from an inertial frame, 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 force]]s, inertia forces, or pseudo-forces.

Consider two [[reference frames]] ''S'' and <var>S'</var>. For observers in each of the reference frames an event has space-time coordinates of (''x'',''y'',''z'',''t'') in frame ''S'' and (<var>x'</var>,<var>y'</var>,<var>z'</var>,<var>t'</var>) in frame <var>S'</var>. Assuming time is measured the same in all reference frames, if we require {{nowrap|''x'' {{=}} <var>x'</var>}} when {{nowrap|''t'' {{=}} 0}}, then the relation between the space-time coordinates of the same event observed from the reference frames <var>S'</var> and ''S'', which are moving at a relative velocity ''u'' in the ''x'' direction, is:
: <math> \begin{align}
x' &= x - tu,\\
y' &= y, \\
z' &= z, \\
t' &= t. 
\end{align}</math>

This set of formulas defines a [[group transformation]] known as the [[Galilean transformation]] (informally, the ''Galilean transform''). This group is a limiting case of the [[Poincaré group]] used in [[special relativity]]. The limiting case applies when the velocity ''u'' is very small compared to ''c'', the [[speed of light]].

The transformations have the following consequences:
* '''v'''′ = '''v''' − '''u''' (the velocity '''v'''′ of a particle from the perspective of ''S''′ is slower by '''u''' than its velocity '''v''' from the perspective of ''S'')
* '''a'''′ = '''a''' (the acceleration of a particle is the same in any inertial reference frame)
* '''F'''′ = '''F''' (the force on a particle is the same in any inertial 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.

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]].

== Newtonian mechanics ==
{{Main|Force|Newton's laws of motion}}

A force in physics is any action that causes an object's velocity to change; that is, to accelerate.  A force originates from within a [[field (physics)|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.

[[Isaac Newton|Newton]] was the first to mathematically express the relationship between [[force]] and [[momentum]]. Some physicists interpret [[Newton's second law|Newton's second law of motion]] as a definition of force and mass, while others consider it a fundamental postulate, a law of nature.<ref>{{cite book|last1=Thornton|first1=Stephen T.|last2=Marion|first2=Jerry B.|title=Classical dynamics of particles and systems|url=https://archive.org/details/classicaldynamic00thor|url-access=limited|date=2004|publisher=Brooks/Cole|location=Belmont, CA|isbn=978-0-534-40896-1|pages=[https://archive.org/details/classicaldynamic00thor/page/n67 50]|edition=5.}}</ref> Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":
: <math>\mathbf{F} = {\mathrm{d}\mathbf{p} \over \mathrm{d}t} = {\mathrm{d}(m \mathbf{v}) \over \mathrm{d}t}.</math>

The quantity ''m'''''v''' is called the ([[canonical momentum|canonical]]) [[momentum]]. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is {{nowrap|'''a''' {{=}} d'''v'''/d''t''}}, the second law can be written in the simplified and more familiar form:
: <math>\mathbf{F} = m \mathbf{a} \, .</math>

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:
: <math>\mathbf{F}_{\rm R} = - \lambda \mathbf{v} \, ,</math>

where ''λ'' is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is
: <math>- \lambda \mathbf{v} = m \mathbf{a} = m {\mathrm{d}\mathbf{v} \over \mathrm{d}t} \, .</math>

This can be [[antiderivative|integrated]] to obtain
: <math>\mathbf{v} = \mathbf{v}_0 e^{{-\lambda t}/{m}}</math>

where '''v'''<sub>0</sub> is the initial velocity. This means that the velocity of this particle [[exponential decay|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 [[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 '''F''' on another particle ''B'', it follows that ''B'' must exert an equal and opposite ''reaction force'', −'''F''', on ''A''. The strong form of Newton's third law requires that '''F''' and −'''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.{{clarify|date=January 2016}}

=== Work and energy ===
{{Main|Work (physics)|kinetic energy|potential energy}}

If a constant force '''F''' is applied to a particle that makes a displacement Δ'''r''',{{NoteTag|The displacement {{math|Δ'''r'''}} is the difference of the particle's initial and final positions: {{math|1=Δ'''r''' = '''r'''<sub>final</sub> − '''r'''<sub>initial</sub>}}.}} the ''work done'' by the force is defined as the [[scalar product]] of the force and displacement vectors:
: <math>W = \mathbf{F} \cdot \Delta \mathbf{r} \, .</math>

More generally, if the force varies as a function of position as the particle moves from '''r'''<sub>1</sub> to '''r'''<sub>2</sub> along a path ''C'', the work done on the particle is given by the [[line integral]]
: <math>W = \int_C \mathbf{F}(\mathbf{r}) \cdot \mathrm{d}\mathbf{r} \, .</math>

If the work done in moving the particle from '''r'''<sub>1</sub> to '''r'''<sub>2</sub> is the same no matter what path is taken, the force is said to be [[Conservative force|conservative]]. [[Gravity]] is a conservative force, as is the force due to an idealized [[Spring (device)|spring]], as given by [[Hooke's law]]. The force due to [[friction]] is non-conservative.

The [[kinetic energy]] ''E''<sub>k</sub> of a particle of mass ''m'' travelling at speed ''v'' is given by
: <math>E_\mathrm{k} = \tfrac{1}{2}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.

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'''<sub>1</sub> to '''r'''<sub>2</sub> is equal to the change in [[kinetic energy]] ''E''<sub>k</sub> of the particle:
: <math>W = \Delta E_\mathrm{k} = E_\mathrm{k_2} - E_\mathrm{k_1} = \tfrac{1}{2} m \left(v_2^{\, 2} - v_1^{\, 2}\right) .</math>

Conservative forces can be expressed as the [[gradient]] of a scalar function, known as the [[potential energy]] and denoted ''E''<sub>p</sub>:
: <math>\mathbf{F} = - \mathbf{\nabla} E_\mathrm{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>\mathbf{F} \cdot \Delta \mathbf{r} = - \mathbf{\nabla} E_\mathrm{p} \cdot \Delta \mathbf{r} = - \Delta E_\mathrm{p} \, .</math>

The decrease in the potential energy is equal to the increase in the kinetic energy
: <math>-\Delta E_\mathrm{p} = \Delta E_\mathrm{k} \Rightarrow \Delta (E_\mathrm{k} + E_\mathrm{p}) = 0 \, .</math>

This result is known as ''conservation of energy'' and states that the total [[energy]],
: <math>\sum E = E_\mathrm{k} + E_\mathrm{p} \, ,</math>

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

== Lagrangian mechanics ==
{{main|Lagrangian mechanics}}
[[Lagrangian mechanics]] is a formulation of classical mechanics founded on the [[stationary-action principle]] (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer [[Joseph-Louis Lagrange]] in his presentation to the Turin Academy of Science in 1760<ref>{{cite journal|last1=Fraser |first1=Craig |title=J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics |journal=Archive for History of Exact Sciences |volume=28 |number=3 |year=1983 |pages=197–241 |doi=10.1007/BF00328268 |jstor=41133689}}</ref> culminating in his 1788 grand opus, ''[[Mécanique analytique]]''. Lagrangian mechanics describes a mechanical system as a pair <math display="inline">(M,L)</math> consisting of a [[configuration space (physics)|configuration space]] <math display="inline">M</math> and a smooth function <math display="inline">L</math> within that space called a Lagrangian. For many systems, <math display="inline">L = T - V,</math> where <math display="inline">T</math> and <math>V</math> are the [[Kinetic energy|kinetic]] and [[Potential energy|potential]] energy of the system, respectively. The stationary action principle requires that the [[Action_(physics)#Action_(functional)|action functional]] of the system derived from <math display="inline">L</math> must remain at a stationary point (a [[Maximum and minimum|maximum]], [[Maximum and minimum|minimum]], or [[Saddle point|saddle]]) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.<ref>{{cite book|title=Analytical Mechanics|edition=2nd|last1=Hand|first1=L. N.|last2=Finch|first2=J. D.|date=1998|url=https://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA141|publisher=Cambridge University Press|isbn=9780521575720 |pages=18–20, 23, 46, 51}}</ref>

== Hamiltonian mechanics ==
{{main|Hamiltonian mechanics}}
[[Hamiltonian mechanics]] emerged in 1833 as a reformulation of [[Lagrangian mechanics]]. Introduced by [[Sir William Rowan Hamilton]],<ref>{{Cite book |author=Hamilton, William Rowan |url=http://worldcat.org/oclc/68159539 |title=On a general method of expressing the paths of light, & of the planets, by the coefficients of a characteristic function. |date=1833 |publisher=Printed by P.D. Hardy |oclc=68159539}}</ref> Hamiltonian mechanics replaces (generalized) velocities <math>\dot q^i</math> used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, [[symplectic geometry]] and [[Poisson structure]]s) and serves as a link between classical and [[quantum mechanics]].

In this formalism, the dynamics of a system are governed by Hamilton's equations, which express the time derivatives of position and momentum variables in terms of [[partial derivative]]s of a function called the Hamiltonian:
<math display="block">
\frac{\mathrm{d}\boldsymbol{q}}{\mathrm{d}t} = \frac{\partial \mathcal H}{\partial \boldsymbol{p}},\quad
\frac{\mathrm{d}\boldsymbol{p}}{\mathrm{d}t} = -\frac{\partial \mathcal H}{\partial \boldsymbol{q}}.
</math>
The Hamiltonian is the [[Legendre transform]] of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system.

== Limits of validity ==
[[File:Modernphysicsfields 2.svg|upright=1.75|thumb|alt=two by two chart of mechanics for size by speed|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.

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.e., their [[Schwarzschild radius]] is not negligibly small for a given application), deviations from [[Newtonian mechanics]] become apparent and can be quantified by using the [[parameterized post-Newtonian formalism]]. In that case, [[general relativity]] (GR) becomes applicable. However, until now there is no theory of [[quantum gravity]] unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.<sup>[[Classical mechanics#cite note-Hamber 2009-4|[4]]][[Classical mechanics#cite note-5|[5]]]</sup>

=== Newtonian approximation to special relativity ===
In special relativity, the momentum of a particle is given by
: <math>\mathbf{p} = \frac{m \mathbf{v}}{ \sqrt{1 - \frac{v^2}{c^2}}} \, ,</math>
where ''m'' is the particle's rest mass, '''v''' its velocity, ''v'' is the modulus of '''v''', and ''c'' is the speed of light.

If ''v'' is very small compared to ''c'', ''v''<sup>2</sup>/''c''<sup>2</sup> is approximately zero, and so
: <math>\mathbf{p} \approx m\mathbf{v} \, .</math>
Thus the Newtonian equation {{nowrap|'''p''' {{=}} ''m'''''v'''}} is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light.

For example, the relativistic cyclotron frequency of a [[cyclotron]], [[gyrotron]], or high voltage [[magnetron]] is given by
: <math>f = f_\mathrm{c}\frac{m_0}{m_0 + \frac{T}{c^2}} \, ,</math>
where ''f''<sub>c</sub> is the classical frequency of an electron (or other charged particle) with kinetic energy ''T'' and ([[Invariant mass|rest]]) mass ''m''<sub>0</sub> circling in a magnetic field. The (rest) mass of an electron is 511&nbsp;keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11&nbsp;kV direct current accelerating voltage.

=== 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
: <math>\lambda=\frac{h}{p}</math>
where ''h'' is the [[Planck 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&nbsp;V, had a wavelength of 0.167&nbsp;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&nbsp;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.

== History ==
{{Main|History of classical mechanics}}
{{For timeline}}
The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in [[science]], [[engineering]], and [[technology]]. The development of classical mechanics lead to the development of many areas of mathematics.<ref>{{Cite book |last1=Doran |first1=Chris |title=Geometric algebra for physicists |last2=Lasenby |first2=Anthony N. |date=2003 |publisher=Cambridge university press |isbn=978-0-521-48022-2 |location=Cambridge New York}}</ref>{{rp|54}} 

Some [[Greek philosophers]] of antiquity, among them [[Aristotle]], founder of [[Aristotelian physics]], 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 later became decisive factors in forming modern science, and their early application came to be known as classical mechanics. In his ''Elementa super demonstrationem ponderum'', medieval mathematician [[Jordanus de Nemore]] introduced the concept of "positional [[gravity]]" and the use of component [[forces]].<!--looks like Jagged misuse of sources: An early mathematical and 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 [[Analytical dynamics|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, pp.&nbsp;614–642 [642], [[Routledge]], London and New York</ref>{{Verify source|date=September 2010}} 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 (Alhazen)<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, pp.&nbsp;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.&nbsp;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.&nbsp;65, Charles Scribner's Sons, New York, 1973–1974.</ref>{{Verify source|date=September 2010}} The proportionality between [[force]] and [[acceleration]], an important principle in classical mechanics, was first stated by [[Hibat Allah Abu'l-Barakat al-Baghdaadi|Abu'l-Barakat]],<ref>
{{cite encyclopedia
 |last=Shlomo Pines
 |author-link=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
 |year=1970
 |isbn=0-684-10114-9
}}
<br/>([[cf.]] Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", ''Journal of the History of Ideas'' '''64''' (4), pp.&nbsp;521–546 [528]</ref> and [[Ibn Bajjah]] also developed the concept of a [[Reaction (physics)|reaction]] force.<ref>[[Shlomo Pines]] (1964), "La dynamique d’Ibn Bajja", in ''Mélanges Alexandre Koyré'', I, 442–468 [462, 468], Paris.
<br>([[cf.]] Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", ''Journal of the History of Ideas'' '''64''' (4), pp.&nbsp;521–546 [543].)</ref>{{Verify source|date=September 2010}} Theories on gravity were developed by [[Banū Mūsā]],<ref>[[Robert Briffault]] (1938). ''The Making of Humanity'', p.&nbsp;191.</ref> Alhazen,<ref>Nader El-Bizri (2006), "Ibn al-Haytham or Alhazen", in Josef W. Meri (2006), ''Medieval Islamic Civilization: An Encyclopaedia'', Vol. II, pp.&nbsp;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.&nbsp;622. London and New York: Routledge.</ref> It is known{{Whom?|date=September 2010}} 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, 296–297.</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), pp.&nbsp;163–193.</ref> and [[Jean Buridan]].<ref>''[https://books.google.com/books?id=vPT-JubW-7QC&pg=PA87&dq&hl=en#v=onepage&q=&f=false A history of mechanics]''. René Dugas (1988). p. 87. {{ISBN|0-486-65632-2}}</ref>-->

[[File:Theory of impetus.svg|left|thumb|180px|alt=a diagram of Theory of impetus of Albert of Saxony with a b c d |Three stage [[Theory of impetus]] according to [[Albert of Saxony (philosopher)|Albert of Saxony]]]]
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 on the orbit of [[Mars]], that the planet's orbits were [[ellipse]]s. This break with [[Ancient philosophy|ancient thought]] was happening around the same time that [[Galileo Galilei|Galileo]] was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannonballs of different weights from the [[Leaning Tower of Pisa|tower of Pisa]], showing that they both hit the ground at the same time. The reality of that particular experiment is disputed, but he did carry out quantitative experiments by rolling balls on an [[inclined plane]]. His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. In 1673 [[Christiaan Huygens]] described in his ''[[Horologium Oscillatorium]]'' the first two [[Newton's laws of motion|laws of motion]].<ref>{{cite book|author=Rob Iliffe & George E. Smith |title= The Cambridge Companion to Newton|date=2016 |publisher=Cambridge University Press |isbn= 9781107015463 |page=75}}</ref> The work is also the first modern treatise in which a physical problem (the [[Acceleration|accelerated motion]] of a falling body) is [[Mathematical model|idealized by a set of parameters]] then analyzed mathematically and constitutes one of the seminal works of [[applied mathematics]].<ref name=":0">{{Cite book|last=Yoder|first=Joella G.|author-link=Joella Yoder |url=https://www.cambridge.org/core/books/unrolling-time/1427509C7A14C464B08209322E42ABB6|title=Unrolling Time: Christiaan Huygens and the Mathematization of Nature|date=1988|publisher=Cambridge University Press|isbn=978-0-521-34140-0|location=Cambridge}}</ref> 
[[File:Portrait of Sir Isaac Newton, 1689.jpg|thumb|240px|alt=portrait of Isaac Newton with long hair looking left|Sir [[Isaac Newton]] (1643–1727), an influential figure in the history of physics and whose [[Newton's laws of motion|three laws of motion]] form the basis of classical mechanics]]
Newton founded his principles of natural philosophy on three proposed [[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 laid the foundations for classical mechanics. Both Newton's second and third laws were given the proper scientific and mathematical treatment in Newton's ''[[Philosophiæ Naturalis Principia Mathematica]]''. Here they are distinguished 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<!--is this redundant?--> scientific and mathematical formulation of [[gravity]] in [[Newton's law of universal gravitation]]. The combination of Newton's laws of motion and gravitation provides 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 had previously invented the [[calculus]]; however, the ''Principia'' was formulated entirely in terms of long-established geometric methods in emulation of [[Euclid]]. 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) he maintained his own [[corpuscular theory of light]].

[[File:Joseph Louis Lagrange2.jpg|left|thumb|180px|alt=Painting of Joseph-Louis Lagrange|[[Joseph-Louis Lagrange|Lagrange]]'s  contribution was realising Newton's ideas in the language of modern mathematics, now called [[Lagrangian mechanics]].]]
After Newton, classical mechanics became a principal field of study in mathematics as well as physics. Mathematical formulations progressively allowed finding solutions to a far greater number of problems. The first notable mathematical treatment was in 1788 by [[Joseph Louis Lagrange]]. Lagrangian mechanics was in turn re-formulated in 1833 by [[William Rowan Hamilton]].
[[File:WilliamRowanHamilton.jpeg|right|thumb|180px|alt=photograph of William Rowan Hamilton in looking left|[[William Rowan Hamilton|Hamilton]] developed an alternative to [[Lagrangian mechanics]] now called [[Hamiltonian mechanics]].]]

Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. Some of these difficulties related to compatibility with [[electromagnetic theory]], and the famous [[Michelson–Morley experiment]]. The resolution of these problems led to the [[special theory of relativity]], often still considered a part of classical mechanics.

A second set of difficulties were related to thermodynamics. When combined with [[thermodynamics]], classical mechanics leads to the [[Gibbs paradox]] of classical [[statistical mechanics]], in which [[entropy]] is not a well-defined quantity. [[Planck's law|Black-body radiation]] was not explained without the introduction of [[quantum|quanta]]. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the [[energy levels]] and sizes of [[atoms]] and the [[photo-electric effect]]. The effort at resolving these problems led to the development of [[quantum mechanics]].

Since the end of the 20th century, classical mechanics in [[physics]] has no longer been an independent theory. Instead, classical mechanics is now considered an approximate theory to the more general quantum mechanics. Emphasis has shifted to understanding the fundamental forces of nature as in the [[Standard Model]] and its more modern extensions into a unified [[theory of everything]]. Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields.

== See also ==
{{Portal|Physics}}
{{div col|colwidth=30em}}
* [[Dynamical system]]
* [[List of equations in classical mechanics]]
* [[List of textbooks on classical mechanics and quantum mechanics#Classical mechanics|List of publications in classical mechanics]]
* [[List of textbooks on classical mechanics and quantum mechanics]]
* [[Molecular dynamics]]
* [[Newton's laws of motion]]
* [[Special relativity]]
* [[Quantum mechanics]]
* [[Quantum field theory]]
{{div col end}}

== Notes ==
{{reflist|group=note|2}}

== References ==
{{reflist}}

== Further reading ==
* {{cite book |author1= Alonso, M. |author2= Finn, J. |title=Fundamental University Physics | year= 1992 |publisher=Addison-Wesley}}
* {{cite book |author=Feynman, Richard |title=The Feynman Lectures on Physics |publisher=Perseus Publishing |year=1999 | author-link= Richard Feynman |isbn=978-0-7382-0092-7|title-link=The Feynman Lectures on Physics }}
* {{cite book |author1=Feynman, Richard |author2=Phillips, Richard |title=Six Easy Pieces |publisher=Perseus Publishing |year=1998 |isbn=978-0-201-32841-7}}
* {{cite book |author1=Goldstein, Herbert |author2=Charles P. Poole |author3=John L. Safko |title=Classical Mechanics |publisher=Addison Wesley |year=2002 |edition=3rd |isbn=978-0-201-65702-9|author-link1= Herbert Goldstein |title-link=Classical Mechanics (Goldstein book) }}
* {{cite book |title=Classical Mechanics (5th ed.) |publisher=[[Imperial College Press]] |year=2004 |isbn=978-1-86094-424-6 |last1 = Kibble|first1 = Tom W.B.|author1-link=Tom Kibble|last2 = Berkshire|first2=Frank H.|author2-link=Frank H. Berkshire|title-link=Classical Mechanics (Kibble and Berkshire) }}
* {{cite book |author1=Kleppner, D. |author2=Kolenkow, R.J. |title=An Introduction to Mechanics |publisher=McGraw-Hill |year=1973 |isbn=978-0-07-035048-9 |url=https://archive.org/details/introductiontome00dani }}
* {{cite book |author1=Landau, L.D. |author2=Lifshitz, E.M. |title= Course of Theoretical Physics, Vol. 1 – Mechanics |publisher=Franklin Book Company |year=1972 |isbn=978-0-08-016739-8|title-link=Course of Theoretical Physics }}
* {{cite book|last=Morin|first=David|title=Introduction to Classical Mechanics: With Problems and Solutions|year=2008|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-87622-3|url=https://archive.org/details/introductiontocl00mori|edition=1st}}
* {{cite book |author1=Gerald Jay Sussman |author2=Jack Wisdom |title=Structure and Interpretation of Classical Mechanics |publisher=MIT Press |year=2001 |isbn=978-0-262-19455-6|author-link1=Gerald Jay Sussman |author-link2=Jack Wisdom |title-link=Structure and Interpretation of Classical Mechanics }}
* {{cite book|title=Essential Dynamics and Relativity|author=O'Donnell, Peter J. |publisher=CRC Press|year=2015|isbn=978-1-4665-8839-4}}
* {{cite book |author1=Thornton, Stephen T. |author2=Marion, Jerry B. |title=Classical Dynamics of Particles and Systems (5th ed.) |publisher=Brooks Cole |year=2003 |isbn=978-0-534-40896-1}}

== External links ==
{{Commons category}}
{{Wikiquote}}
* Crowell, Benjamin. [http://www.lightandmatter.com/lm Light and Matter] (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., "''[https://arxiv.org/abs/physics/9909035 Classical Mechanics]''". Physics Education. 1999. [arxiv.org : physics/9909035]
* Shapiro, Joel A. (2003). [http://www.physics.rutgers.edu/ugrad/494/bookr03D.pdf Classical Mechanics]
* Sussman, Gerald Jay & Wisdom, Jack &  Mayer, Meinhard E. (2001). [https://web.archive.org/web/20120920024409/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)
* [http://kmoddl.library.cornell.edu/index.php Kinematic Models for Design Digital Library (KMODDL)]<br /> Movies and photos of hundreds of working mechanical-systems models at [[Cornell University]]. Also includes an [http://kmoddl.library.cornell.edu/e-books.php e-book library] of classic texts on mechanical design and engineering.
* [https://web.archive.org/web/20190327102351/https://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/ MIT OpenCourseWare 8.01: Classical Mechanics] Free videos of actual course lectures with links to lecture notes, assignments and exams.
* Alejandro A. Torassa, [http://torassa.tripod.com/paper.htm On Classical Mechanics]

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