Encyclopedia
Classical mechanics is used to describe 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.
Place in physics
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 developed in the 400 years since the groundbreaking works of
Brahe,
Kepler, and
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 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, the conventional content of classical mechanics was created in the 19th century and differs considerably from the work of
Newton.
Description of the theory
The following introduces the basic concepts of classical mechanics. For simplicity, it uses 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. True point particles, such as the
electron, are normally better 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.
Position and its derivatives
The
position of a point particle is defined with respect to an arbitrary fixed point in space, which is sometimes called the
origin,
O. It is defined as the vector
r from
O to the particle. In general, the point particle need not be stationary, so
r is a function of
t, the
time elapsed since an arbitrary initial time. In pre-Einstein relativity , time is considered an absolute in all
reference frames. In addition to relying on absolute time, classical mechanics uses
Euclidean geometry .
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
u =
ud and the velocity of the second object by the vector
v =
ve 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 particle respectively, then the velocity of the first object as seen by the second object is:
- u' = u - v
Similarly:
- v' = v - u
When both objects are moving in the same direction, this equation can be simplified to:
- u' = 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 or
- .
The acceleration vector can be changed by changing its magnitude, changing its direction, or both. If the magnitude of v decreases, this is sometimes referred to as deceleration
or retardation
, but generally any change in the velocity, including deceleration, is simply referred to as acceleration.
Frames of reference
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 u to S.
- v' = v - u
- a' = a
- F' = F
- the speed of light is not a constant in classical mechanics
- the form of Maxwell's equations is not preserved across reference frames
Forces; Newton's second law
Newton was first to mathematically define force as the rate of change of momentum: F=dp/dt. Despite that this is simply an accurate definition of force , it is historically regarded as "Newton's second law":
- .
The quantity
mv is called the 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
F, 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 ? a positive constant . 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
v0 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
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.
Energy
If a force
F is applied to a particle that achieves a displacement ?
s, 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 ?
Wtotal is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:
- ,
where
Ek 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
Ep:
- .
If all the forces acting on a particle are conservative, and
Ep is the total potential energy , 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 in frame
S and 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 -
uty = yz = zt = t
This set of formulas defines a group transformation known as the Galilean transformation . 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 . Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictituous
Centrifugal force and
Coriolis force.
History
Main article: History of classical mechanics
The
Greeks, and
Aristotle in particular, were the first to propose that there are abstract principles governing nature.
One of the first scientists who suggested abstract laws was
Galileo Galilei who may have performed the famous experiment of dropping two cannon balls of different masses from the
tower of Pisa. Though the reality of this experiment is disputed, he did carry out quantitative experiments by rolling balls on an
inclined plane; his correct theory of accelerated motion was apparently derived from the results of the experiments.
Sir Isaac Newton was the first to propose the three
laws of motion , and to prove that these laws govern both everyday objects and celestial objects.
Newton and most of his contemporaries, with the notable exception of
Christiaan Huygens hoped that classical mechanics would be able to explain all entities, including light. When he discovered
Newton's rings, Newton's own explanation avoided wave principles and resembled more the explanation for the decay of the neutral
Kaons, K
0 and K
0 bar. That is, he supposed that the light particles were altered or excited by the glass and resonated.
Newton also developed the
calculus which is necessary to perform the mathematical calculations involved in classical mechanics. However it was
Gottfried Leibniz who developed the notation of the
derivative and
integral which are used to this day.
After Newton the field became more mathematical and more abstract.
Although classical mechanics is largely compatible with other "classical physics" theories such as classical electrodynamics and
thermodynamics, some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. When combined with classical thermodynamics, classical mechanics leads to the Gibbs paradox 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 velocity transformations led to the
theory of relativity.
By the end of the 20th century, the place of classical mechanics in
physics is no longer that of an independent theory. Along with classical electromagnetism, it has become imbedded in relativistic
quantum mechanics or
quantum field theory. 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. The two 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.
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 with kinetic energy and mass circling in a magnetic field.
The 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 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 wave length is
where is Plank's Constant and 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 wave lengths than massless particles, such as light, with the same kinetic energies.
See also
- History of classical mechanics
- Dynamical systems
- List of equations in classical mechanics
- List of publications in classical mechanics
Branches
Notes
References
- Kleppner, D. and Kolenkow, R. J., An Introduction to Mechanics, McGraw-Hill . ISBN 0-07-035048-5
- Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics, MIT Press . ISBN 0-262-019455-4
- Herbert Goldstein, Charles P. Poole, John L. Safko, Classical Mechanics , 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
External links
- Binney, James.
- Crowell, Benjamin.
- Fitzpatrick, Richard.
- Hoiland, Paul .
- Horbatsch, Marko, "".
- Rosu, Haret C., "". Physics Education. 1999. [arxiv.org : physics/9909035]
- Schiller, Christoph.
- Sussman, Gerald Jay & Wisdom, Jack .
- Tong, David.
