
Classical mechanics is a
physical theory describing the
motion of
macroscopic objects, from
projectiles to parts of
machinery, and
astronomical objects, such as
spacecraft,
planets,
stars, and
galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).
The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir
Isaac Newton, and the mathematical methods invented by
Gottfried Wilhelm Leibniz,
Joseph-Louis Lagrange,
Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of
bodies under the influence of a system of
forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as
Lagrangian mechanics and
Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of
analytical mechanics. They are, with some modification, also used in all areas of modern physics.
Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the
speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of
mechanics:
quantum mechanics. To describe velocities that are not small compared to the speed of light,
special relativity is needed. In cases where objects become extremely massive,
general relativity becomes applicable. However, a number of modern sources do include
relativistic mechanics in classical physics, which in their view represents classical mechanics in its most developed and accurate form.
Description of the theory
The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as
point particles (objects with negligible size). The motion of a point particle is characterized by a small number of
parameters: its position,
mass, and the
forces applied to it. Each of these parameters is discussed in turn.
In reality, the kind of objects that classical mechanics can describe always have a
non-zero size. (The physics of ''very'' small particles, such as the
electron, is more accurately described by
quantum mechanics.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional
degrees of freedom, e.g., 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 of a large number of collectively acting point particles. The
center of mass of a composite object behaves like a point particle.
Classical mechanics uses
common sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also
Action at a distance).
Position and its derivatives
The ''position'' of a
point particle is defined in relation to a
coordinate system centered on an arbitrary fixed reference point in
space called the origin ''O''. A simple coordinate system might describe the position of a
particle ''P'' with a
vector notated by an arrow labeled r that points from the origin ''O'' to point ''P''. In general, the point particle does not need to be stationary relative to ''O''. In cases where ''P'' is moving relative to ''O'', r is defined as a function of ''t'',
time. In pre-Einstein relativity (known as
Galilean relativity), time is considered an absolute, i.e., the
time interval that is observed to elapse 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:
:
.
In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at . However, from the perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as –10 km/h where the sign implies opposite direction. 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 and the velocity of the second object by the vector , where ''u'' is the speed of the first object, ''v'' is the speed of the second object, and d and e are
unit vectors in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is:
:
Similarly, the first object sees the velocity of the second object as:
:
When both objects are moving in the same direction, this equation can be simplified to:
:
Or, by ignoring direction, the difference can be given in terms of speed only:
:
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):
:
Acceleration represents the velocity's change over time. Velocity can change in either magnitude or direction, or both. Occasionally, a decrease in the magnitude of velocity "''v''" is referred to as ''deceleration'', but generally any change in the velocity over time, including deceleration, is simply referred to as acceleration.
Frames of reference
While the position, velocity and acceleration of a
particle can be described with respect to any
observer in any state of motion, classical mechanics assumes the existence of a special family of
reference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called
inertial frames. An inertial frame is an idealized frame of reference within which an object has no external force acting upon it. Because there is no external force acting upon it, the object has a constant velocity; that is, it is either at rest or moving uniformly in a straight line.
A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that do not accelerate with respect to
distant stars (an extremely distant point) are regarded as good approximations to inertial frames.
Non-inertial reference frames accelerate in relation to an existing inertial frame. They form the basis for Einstein's relativity. Due to the relative motion, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration. These forces are referred to as
fictitious forces, inertia forces, or pseudo-forces.
Consider two
reference frames ''S'' and
S'. For observers in each of the reference frames an event has space-time coordinates of (''x'',''y'',''z'',''t'') in frame ''S'' and (
x',
y',
z',
t') in frame
S'. Assuming time is measured the same in all reference frames, and if we require when , 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:
:
:
:
:
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 and
Coriolis force.
Forces and Newton's second law
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, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others.
Newton was the first to mathematically express the relationship between
force and
momentum. Some physicists interpret
Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":
:
The quantity ''m''v is called the (
canonical)
momentum. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is , the second law can be written in the simplified and more familiar form:
:
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:
:
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
:
This can be
integrated to obtain
:
where 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), and the particle is slowing down. This expression can be further integrated to obtain the position r of the particle as a function of time.
Important forces include the
gravitational force and the
Lorentz force for
electromagnetism. 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.
Work and energy
If a constant force F is applied to a particle that makes a displacement Δr,
[The displacement is the difference of the particle's initial and final positions: .] the ''work done'' by the force is defined as the
scalar product of the force and displacement vectors:
:
More generally, if the force varies as a function of position as the particle moves from r
1 to r
2 along a path ''C'', the work done on the particle is given by the
line integral
:
If the work done in moving the particle from r
1 to r
2 is the same no matter what path is taken, the force is said to be
conservative.
Gravity is a conservative force, as is the force due to an idealized
spring, as given by
Hooke's law. The force due to
friction is non-conservative.
The
kinetic energy ''E''
k of a particle of mass ''m'' travelling at speed ''v'' is given by
:
For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.
The
work–energy theorem states that for a particle of constant mass ''m'', the total work ''W'' done on the particle as it moves from position r
1 to r
2 is equal to the change in
kinetic energy ''E''
k of the particle:
:
Conservative forces can be expressed as the
gradient of a scalar function, known as the
potential energy and denoted ''E''
p:
:
If all the forces acting on a particle are conservative, and ''E''
p is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force
:
The decrease in the potential energy is equal to the increase in the kinetic energy
:
This result is known as ''conservation of energy'' and states that the total
energy,
:
is constant in time. It is often useful, because many commonly encountered forces are conservative.
Beyond Newton's laws
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. )
There are two important alternative formulations of classical mechanics:
Lagrangian mechanics and
Hamiltonian mechanics. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in
generalized coordinates. These are basically mathematical rewriting of Newton's laws, but complicated mechanical problems are much easier to solve in these forms. Also, analogy with quantum mechanics is more explicit in Hamiltonian formalism.
The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the
Poynting vector divided by ''c''
2, where ''c'' is the
speed of light in free space.
Limits of validity
Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being
general relativity and relativistic
statistical mechanics.
Geometric optics is an approximation to the
quantum theory of light, and does not have a superior "classical" form.
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.
The Newtonian approximation to special relativity
In special relativity, the momentum of a particle is given by
:
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''
2/''c''
2 is approximately zero, and so
:
Thus the Newtonian equation 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
:
where ''f''
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 non-relativistic particles, this wavelength is
:
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 V, had a wavelength 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.
History
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.
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.

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
ellipses. This break with
ancient thought was happening around the same time that
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
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.
Newton founded his principles of natural philosophy on three proposed
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 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 had 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 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 so-called
Newton's rings (a
wave interference phenomenon) he maintained his own
corpuscular theory of light.

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.

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.
Black-body 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
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. Also, it has been extended into the
complex domain where complex classical mechanics exhibits behaviors very similar to quantum mechanics.
Complex Elliptic Pendulum
Carl M. Bender, Daniel W. Hook, Karta Kooner i
Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. I
/ref>
Branches
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:
* Newtonian mechanics
* Lagrangian mechanics
* Hamiltonian mechanics
Alternatively, a division can be made by region of application:
* Celestial mechanics, relating to stars, planets and other celestial bodies
* Continuum mechanics, for materials modelled as a continuum, e.g., solids and fluids (i.e., liquids and gases).
* Relativistic mechanics (i.e. including the special and 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 thermodynamic properties of materials.
See also
* Dynamical systems
* History of classical mechanics
* List of equations in classical mechanics
* List of publications in classical mechanics
* List of textbooks on classical and quantum mechanics
* Molecular dynamics
* Newton's laws of motion
* Special theory of relativity
* Quantum Mechanics
* Quantum Field Theory
Notes
References
Further reading
*
*
*
*
*
*
*
* *
*
*
External links
* Crowell, Benjamin
Light and Matter
(an introductory text, uses algebra with optional sections involving calculus)
* Fitzpatrick, Richard
(uses calculus)
* Hoiland, Paul (2004)
Preferred Frames of Reference & Relativity
* Horbatsch, Marko, "
'".
* Rosu, Haret C., "
Classical Mechanics
'". Physics Education. 1999. rxiv.org : physics/9909035* Shapiro, Joel A. (2003)
Classical Mechanics
* Sussman, Gerald Jay & Wisdom, Jack & Mayer, Meinhard E. (2001)
Structure and Interpretation of Classical Mechanics
* Tong, David
(Cambridge lecture notes on Lagrangian and Hamiltonian formalism)
Kinematic Models for Design Digital Library (KMODDL)
br /> Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes a
e-book library
of classic texts on mechanical design and engineering.
MIT OpenCourseWare 8.01: Classical Mechanics
Free videos of actual course lectures with links to lecture notes, assignments and exams.
* Alejandro A. Torassa
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