In the fields of
physics,
classical
mechanics is one of the two major subfields 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 geometrically
distributed within a certain boundary under the action of a system
of forces. The other subfield is
quantum mechanics.
Classical mechanics 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,
engineering and
technology.
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;
general relativity is employed to handle
gravitation at a deeper level; and
quantum mechanics handles the
waveparticle duality of
atoms and
molecules.
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
natural philosophers,
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
socalled "
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. 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.
Description of the theory
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 realworld objects as
point particles, objects with
negligible size. The motion of a point particle
is characterized by a small number of
parameters: its position,
mass, and the
forces applied to
it. Each of these parameters is discussed in turn.
In reality, the kind of objects which classical mechanics can
describe always have a
nonzero size. (The
physics of
very small particles, such as the
electron, is more accurately described by
quantum mechanics). Objects with nonzero
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.
Position and its derivatives
The
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
preEinstein 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.
Velocity and speed
The
velocity, or the
rate of change of position with time, is defined as
the
derivative of the position with
respect to time or
 \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. 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} = u\vec{d} and the
velocity of the second object by the vector \vec{v} = v\vec{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:
 \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
 \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
observer in any state of
motion, 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 acceleration of the
observer and the requirement that all forces entering the
observer's physical laws originate in identifiable sources
(charges, gravitational bodies, and so forth). A noninertial
reference frame is one accelerating with respect to an inertial
one, and in such a noninertial frame a particle is subject to
acceleration by
fictitious forces
that enter the equations of motion solely as a result of its
accelerated motion, and do not originate in identifiable sources.
These fictitious forces are in addition to the real forces
recognized in an inertial frame. A key concept of inertial frames
is the method for identifying them. (See
inertial frame of reference for
a discussion.) 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 \scriptstyle{\vec{u}} to
S.
 \scriptstyle{\vec{v'} = \vec{v}  \vec{u}} (the velocity
\scriptstyle{\vec{v'}} of a particle from the perspective of
S' is slower by \scriptstyle{\vec{u}} than its velocity
\scriptstyle{\vec{v}} from the perspective of S)
 \scriptstyle{\vec{a'} = \vec{a}} (the acceleration of a
particle is the same in any inertial reference frame)
 \scriptstyle{\vec{F'} = \vec{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.
 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.
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":
 \vec{F} = {\mathrm{d}\vec{p} \over \mathrm{d}t} = {\mathrm{d}(m
\vec{v}) \over \mathrm{d}t}.
The quantity m\vec{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. Since the definition of acceleration is \vec{a} = \frac
{\mathrm{d} \vec{v}} {\mathrm{d}t}, the second law can be written
in the simplified and more familiar form
 \vec{F} = m \vec{a}.
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:
 \vec{F}_{\rm R} =  \lambda \vec{v}
with λ a positive constant. Then the equation of motion is
  \lambda \vec{v} = m \vec{a} = m {\mathrm{d}\vec{v} \over
\mathrm{d}t}.
This can be
integrated to
obtain
 \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. 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), slowing it down. This expression can be further
integrated to obtain the position \vec{r} of the particle as a
function of time.
Important forces include the
gravitational
force and the
Lorentz force for
electromagnetism. In addition,
Newton's third law can sometimes be used to deduce the forces
acting on a particle: if 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.
Energy
If a force \vec{F} is applied to a particle that achieves a
displacement \Delta\vec{r}, the
work done by the force is
defined as the scalar product of force and displacement vectors:
(noting that the displacement vector is the change in position
vector)
 W = \vec{F} \cdot \Delta \vec{r} .
If the mass of the particle is constant, and
W_{total} 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
E_{k} 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:
 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
E_{p}:
 \vec{F} =  \vec{\nabla} E_p.
If all the forces acting on a particle are conservative, and
E_{p} is the total
potential energy (which is defined as a
work of involved forces to rearrange mutual positions of bodies),
obtained by summing the potential energies corresponding to each
force

\vec{F} \cdot \Delta \vec{r} =  \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 nonpointlike 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 onedimensional
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 spacetime 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 spacetime 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 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.
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.
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.
An early experimental
scientific
method was introduced into
mechanics in the 11th century by
alBiruni, who along with
alKhazini 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. Concepts
related to
Newton's laws of
motion were also enunciated by several other
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 alHaytham
(Alhacen) and
Avicenna. The proportionality
between
force and
acceleration, an important principle in
classical mechanics, was first stated by
Hibat Allah Abu'lBarakat
alBaghdaadi,
(
cf.
Abel B.
Franco (October 2003).
"Avempace, Projectile Motion, and Impetus Theory",
Journal of
the History of Ideas 64 (4), p.
521546 [528] and theories on gravity were developed by
Ja'far
Muhammad ibn Mūsā ibn Shākir,
Ibn
alHaytham, and
alKhazini. It is
known that
Galileo Galilei's
mathematical treatment of
acceleration
and his concept of
impetus grew out of
earlier medieval analyses of
motion, especially those of
Avicenna,
Ibn Bajjah, 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 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.
As foundation for his principles of natural philosophy, Newton
proposed three
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 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
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 socalled
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. Some of these
difficulties related to compatibility with
electromagnetic theory, and the
famous
MichelsonMorley
experiment. The resolution of these problems led to the
special theory of
relativity, often included in the term 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 welldefined quantity.
Blackbody radiation was not explained
without the introduction of
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
photoelectric effect. The effort at
resolving these problems led to the development of
quantum mechanics.
Since the end of the 20th century, the place of classical mechanics
in
physics has been no longer that of an
independent theory. 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 for the study of
the motion of nonquantum mechanical, lowenergy particles in weak
gravitational fields.
Limits of validity
Domain of validity for Classical
Mechanics
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.
The Newtonian approximation to special relativity
Newtonian, or nonrelativistic classical momentum
 \vec{p} = m_0 \vec{v}
is the result of the
first
order Taylor approximation of the
relativistic expression:
 \vec{p} = \frac{m_0 \vec{v}}{ \sqrt{1v^2/c^2}} = m_0 \vec{v}
\left(1+\frac{1}{2}\frac{v^2}{c^2} + ... \right), where
v=\vec{v}
when expanded about
 \frac{v}{c}=0
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
 \left(\frac{v}{c}\right)^2 << 1=""
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For example, the relativistic cyclotron frequency of a
cyclotron,
gyrotron, or
high voltage
magnetron is given by
f=f_c\frac{m_0}{m_0+T/c^2}, wheref_c is the classical frequency of
an electron (or other charged particle) with kinetic energy T and
(rest) mass m_0 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 wavelength is not
much smaller than other dimensions of the system. For
nonrelativistic particles, this wavelength is
 \lambda=\frac{h}{p}
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 diodes and very narrow
transistor gates
in
integrated circuits.
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.
Branches
Branches of mechanics
Classical mechanics was traditionally divided into three main
branches:
 Statics, the study of equilibrium and its relation to
forces
 Dynamics, the study of
motion and its relation to forces
 Kinematics, dealing with the
implications of observed motions without regard for circumstances
causing them
Another division is based on the choice of mathematical formalism:
Alternatively, a division can be made by region of application:
See also
Notes
 MIT physics 8.01 lecture notes (page 12)
(PDF)
 Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in
Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 2, p. 614642 [642],
Routledge, London
and New York
 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. 179213.
 Seyyed Hossein
Nasr, "The achievements of Ibn Sina in the field of science and
his contributions to its philosophy", Islam & Science,
December 2003.
 Fernando Espinoza (2005). "An analysis of the historical
development of ideas about motion and its implications for
teaching", Physics Education 40 (2), p.
141.
 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, 19731974.
 Robert
Briffault (1938). The Making of Humanity, p. 191.
 Nader ElBizri (2006), "Ibn alHaytham or Alhazen", in Josef W.
Meri (2006), Medieval Islamic Civilization: An
Encyclopaedia, Vol. II, p. 343345, Routledge, New York, London.
 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.
 Galileo Galilei, Two New Sciences, trans.
Stillman
Drake, (Madison: Univ. of Wisconsin Pr., 1974), pp 217, 225,
2967.
 Ernest A. Moody (1951). "Galileo and Avempace: The Dynamics of
the Leaning Tower Experiment (I)", Journal of the History of
Ideas 12 (2), p. 163193.
 Page 210 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.
References
Further reading
External links
 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)
 Kinematic Models for Design Digital Library
(KMODDL)
Movies and photos of hundreds of working mechanicalsystems models
at Cornell University. Also includes an ebook library of classic texts on mechanical design
and engineering.
 MIT Open Course Ware: 8.01 Physics I: Classical
Mechanics 1999
nb:Klassisk mekanikk