'Classical mechanics' (also called 'Newtonian 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 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. Furthermore,
general relativity is employed to handle
gravitation at a deeper level.
In
physics, 'classical mechanics' is one of the two major sub-fields of study in the science of
mechanics, which is concerned with the set of
physical laws governing and mathematically describing the motions of
bodies and aggregates of bodies. The other sub-field is
quantum mechanics.
The term 'classical mechanics' was coined in the early 20th century to describe the system of mathematical physics begun by
Isaac Newton and many contemporary 17th century workers, building upon the earlier astronomical theories of
Johannes Kepler, which in turn were based on the precise observations of
Tycho Brahe and the studies of terrestrial projectile motion of
Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "
relativistic physics" from that category. However, a number of modern sources ''do'' include
Einstein's mechanics, which in their view represents ''classical mechanics'' in its most developed and most accurate form.
The initial stage in the development of classical mechanics is often referred to as
Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by
Newton himself, in parallel with
Leibniz, and others. This is further described in the following sections. More abstract, and general methods include
Lagrangian mechanics and
Hamiltonian mechanics. While the terms 'classical mechanics' and 'Newtonian mechanics' are usually considered equivalent (if relativity is excluded), much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of
Newton.
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 which classical mechanics can describe always have a non-zero size. (The physics of ''very'' small particles, such as the
electron, is more accurately described by
quantum mechanics). Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional
degrees of freedom—for example, a
baseball can
spin while it is moving. However, the results for point particles can be used to study such objects by treating them as
composite objects, made up of a large number of interacting point particles. The
center of mass of a composite object behaves like a point particle.
Displacement and its derivatives
The ''displacement'', or ''position'', of a point particle is defined with respect to an arbitrary fixed reference point, 'O', in
space, usually accompanied by a coordinate system, with the reference point located at the ''origin'' of the coordinate system. It is defined as the
vector 'r' from 'O' to the particle. In general, the point particle need not be stationary relative to 'O', so 'r' is a function of ''t'', the
time elapsed since an arbitrary initial time. In pre-Einstein relativity (known as
Galilean relativity), time is considered an absolute, i.e., the time interval between any given pair of events is the same for all observers. In addition to relying on
absolute time, classical mechanics assumes
Euclidean geometry for the structure of space.
[1]
Velocity
The ''
velocity'', or the
rate of change of position with time, is defined as the
derivative of the position with respect to time or
:
.
In classical mechanics, velocities are directly additive and subtractive. For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, then from the perspective of the slower car, the faster car is traveling East at 60−50 = 10 km/h. Whereas, from the perspective of the faster car, the slower car is moving 10 km/h to the West. What if the car is traveling north? Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.
Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector
and the velocity of the second object by the vector
where
is the speed of the first object,
is the speed of the second object, and
and
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:
:
Similarly:
:
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) or
:
.
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,
, of the velocity decreases, this is sometimes referred to as ''deceleration'', but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.
Frames of reference
While the position and velocity and acceleration of a particle can be referred to any arbitrary point of reference and accompanying coordinate system (reference frame), Classical Mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. They are characterized by the absence of accelerated motion between any two of them and the requirement of forces to produce accelerated motion of particles relative to any one of them. Any non-inertial reference frame would be accelerated with respect to an inertial one and relative to such a non-inertial frame a particle would, nevertheless, display accelerated motion. A weakness in the concept of inertial frames is the absence of any guaranteed method for identifying them. For practical purposes, reference frames that are unaccelerated with respect to the distant stars are regarded as good approximations to inertial frames.
The following consequences can be derived about the perspective of an event in two inertial reference frames,
and
, where
is traveling at a relative velocity of
to
.
★
(the velocity
of a particle from the perspective of ''S' is'' slower by
than its velocity
from the perspective of ''S'')
★
=
(the acceleration of a particle remains the same regardless of reference frame)
★
=
(the force on a particle remains the same regardless of reference frame)
★ the
speed of light is not a constant 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":
:
.
The quantity
is called the (
canonical)
momentum. The net force on a particle is, thus, equal to rate change of
momentum of the particle with time. Typically, the mass ''m'' is constant in time, and Newton's law can be written in the simplified form
:
where
is the acceleration. It is not always the case that ''m'' is independent of ''t''. For example, the mass of a
rocket decreases as its propellant is ejected. Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used.
Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for
, obtained by considering the particular physical entities with which the particle is interacting. For example, a typical
resistive force may be modelled as a function of the velocity of the particle, for example:
:
with λ a positive constant (although this relation is known to be incorrect for drag in dense air, for example, it is accurate enough for elementary
work). Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an
ordinary differential equation, which is called the ''equation of motion''. Continuing the example, assume that friction is the only force acting on the particle. Then the equation of motion is
:
.
This can be
integrated to obtain
:
where
is the initial velocity. This means that the velocity of this particle
decays exponentially to zero as time progresses. This expression can be further integrated to obtain the position
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
on another particle B, it follows that B must exert an equal and opposite ''reaction force'', -
, on A. The strong form of Newton's third law requires that
and -
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
is applied to a particle that achieves a displacement
, the ''work done'' by the force is defined as the scalar product of force and displacement vectors:
:
.
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:
:
,
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:
:
.
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'':
:
.
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
{|
|
|
.
|}
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 includes descriptions of the complex motions of extended non-pointlike objects. The concepts of
angular momentum rely on the same
calculus used to describe one-dimensional motion.
There are two important alternative formulations of classical mechanics:
Lagrangian mechanics and
Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are often more useful for solving problems. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems.
Classical transformations
Consider two
reference frames ''S'' and ''S' ''. For observers in each of the reference frames an event has space-time coordinates of (''x'',''y'',''z'',''t'') in frame ''S'' and (''x' '',''y' '',''z' '',''t' '') in frame ''S' ''. Assuming time is measured the same in all reference frames, and if we require ''x'' = ''x' when ''t'' = 0, then the relation between the space-time coordinates of the same event observed from the reference frames ''S' '' and ''S'', which are moving at a relative velocity of ''u'' in the ''x'' direction is:
:''x' = ''x'' - ''ut''
:''y' = ''y''
:''z' = ''z''
:''t' = ''t''
This set of formulas defines a
group transformation known as the
Galilean transformation (informally, the ''Galilean transform''). This type of transformation is a limiting case of
special relativity when the velocity u is very small compared to c, the
speed of light.
For some problems, it is convenient to use rotating coordinates (reference frames). Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious
centrifugal force 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.
The first published
causal explanation of the motions of
planets was Johannes Kepler's
Astronomia nova published in 1609. He concluded, based on
Tycho Brahe's observations of the orbit of
Mars, that the orbits were ellipses. This break with
medieval thought was happening around the same time that
Galilei was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannon balls of different masses from the
tower of Pisa, showing that they both hit the ground at the same time. The reality of this experiment is disputed, but, more importantly, he did carry out quantitative experiments by rolling balls on an
inclined plane. His theory of accelerated motion derived from the results of such experiments, and forms a cornerstone of classical mechanics.
As foundation for his principles of natural philosophy, Newton proposed three
laws of motion, the
law of inertia, his second law, mentioned above, and the law of
action and reaction. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of
Kepler's laws of motion of the planets.
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 so-called
Newton's rings (a
wave interference phenomenon) his explanation remained with his own
corpuscular theory of light.
After Newton, classical mechanics became a principal field of study in mathematics as well as physics.
Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. When combined with
thermodynamics, classical mechanics leads to the
Gibbs paradox of classical
statistical mechanics, in which
entropy is not a well-defined quantity. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the
energy levels and sizes of
atoms. The effort at resolving these problems led to the development of
quantum mechanics. Similarly, the different behaviour of classical
electromagnetism and classical mechanics under coordinate transformations (between differently moving
frames of reference), eventually led to the
theory of relativity.
Since the end of the 20th century, the place of classical mechanics in
physics has been no longer that of an independent theory. Along with classical
electromagnetism, it has become embedded in
relativistic quantum mechanics or
quantum field theory.
[2] It is the non-relativistic, non-quantum mechanical limit for massive particles.
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.
The Newtonian approximation to special relativity
Newtonian, or non-relativistic classical mechanics approximates the relativistic momentum
with
, so it is only valid when the velocity is much less than the speed of light.
For example, the relativistic cyclotron frequency of a
cyclotron,
gyrotron, or high voltage
magnetron is given by
, where
is the classical frequency of an electron (or other charged particle) with kinetic energy
and (rest) mass
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 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.
Notes
1. MIT physics 8.01 lecture notes (page 12) (PDF)
2. Page 2-10 of the ''Feynman Lectures on Physics'' says "For already in classical mechanics there was indeterminability from a practical point of view." The past tense here implies that classical physics is no longer fundamental.
References
★
Six Easy Pieces, Feynman, Richard, , , Perseus Publishing, 1996, ISBN 0-201-40825-2
★
Six Easy Pieces, Feynman, Richard; Phillips, Richard, , , Perseus Publishing, 1998, ISBN 0-201-32841-0
★
Lectures on Physics, Feynman, Richard, , , Perseus Publishing, 1999, ISBN 0-7382-0092-1
★
Mechanics Course of Theoretical Physics , Vol. 1, Landau, L. D.; Lifshitz, E. M., , , Franklin Book Company, Inc., 1972, ISBN 0-08-016739-X
★ Kleppner, D. and Kolenkow, R. J., ''An Introduction to Mechanics'', McGraw-Hill (1973). ISBN 0-07-035048-5
★
Gerald Jay Sussman and
Jack Wisdom, ''
Structure and Interpretation of Classical Mechanics'', MIT Press (2001). ISBN 0-262-19455-4}
★
Herbert Goldstein, Charles P. Poole, John L. Safko, ''Classical Mechanics (3rd Edition)'', Addison Wesley; ISBN 0-201-65702-3
★ Robert Martin Eisberg, ''Fundamentals of Modern Physics'', John Wiley and Sons, 1961
★ M. Alonso, J. Finn, "Fundamental university physics", Addison-Wesley
See also
★
History of classical mechanics
★
Dynamical systems
★
List of equations in classical mechanics
★
List of publications in classical mechanics
Branches
★
Celestial Mechanics
★
Continuum Mechanics
★
General Relativity
★
Geometric Optics
★
Hamiltonian Mechanics
★
Lagrangian Mechanics
★
Newtonian Mechanics
★
Special Relativity
★
Statistical Mechanics
★
Thermodynamics
External links
★ Binney, James.
Classical Mechanics (Lagrangian and Hamiltonian formalisms)
★ Crowell, Benjamin.
Newtonian Physics (an introductory text, uses algebra with optional sections involving calculus)
★ Fitzpatrick, Richard.
Classical Mechanics (uses calculus)
★ Hoiland, Paul (2004).
Preferred Frames of Reference & Relativity
★ Horbatsch, Marko, "''
Classical Mechanics Course Notes''".
★ Rosu, Haret C., "''
Classical Mechanics''". Physics Education. 1999. [arxiv.org : physics/9909035]
★ Schiller, Christoph.
Motion Mountain (an introductory text, uses some calculus)
★ Sussman, Gerald Jay & Wisdom, Jack & Mayer,Meinhard E. (2001).
Structure and Interpretation of Classical Mechanics
★ Tong, David.
Classical Dynamics (Cambridge lecture notes on Lagrangian and Hamiltonian formalism)
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