In physics , CLASSICAL MECHANICS (also known as NEWTONIAN MECHANICS)
is one of two major sub-fields of mechanics . The other sub-field is
quantum mechanics .
The term classical mechanics was coined in the early 20th century. It
describes the system of physics started by
The earliest development of classical mechanics is often referred to
as Newtonian mechanics. It consists of the physical concepts employed
by and the mathematical methods invented by Newton, Leibniz and
others. Later, more abstract and general methods were developed,
leading to the reformulations of classical mechanics known as
CONTENTS * 1 Description of the theory * 1.1 Position and its derivatives * 1.1.1
* 1.2 Forces; Newton\'s second law * 1.3 Work and energy * 1.4 Beyond Newton\'s laws * 2 Limits of validity * 2.1 The Newtonian approximation to special relativity * 2.2 The classical approximation to quantum mechanics * 3 History * 4 Branches * 5 See also * 6 Notes * 7 References * 8 Further reading * 9 External links 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 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.
POSITION AND ITS DERIVATIVES Main article:
The SI derived "mechanical" (that is, not electromagnetic or thermal ) units with kg, m and s position m angular position/angle unitless (radian) velocity m·s−1 angular velocity s−1 acceleration m·s−2 angular acceleration s−2 jerk m·s−3 "angular jerk" s−3 specific energy m2·s−2 absorbed dose rate m2·s−3 moment of inertia kg·m2 momentum kg·m·s−1 angular momentum kg·m2·s−1 force kg·m·s−2 torque kg·m2·s−2 energy kg·m2·s−2 power kg·m2·s−3 pressure and energy density kg·m−1·s−2 surface tension kg·s−2 spring constant kg·s−2 irradiance and energy flux kg·s−3 kinematic viscosity m2·s−1 dynamic viscosity kg·m−1·s−1 density (mass density) kg·m−3 density (weight density) kg·m−2·s−2 number density m−3 action kg·m2·s−1 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
Main articles:
The velocity , or the rate of change of position with time, is defined as the derivative of the position with respect to time: v = d r d t {displaystyle mathbf {v} ={mathrm {d} mathbf {r} over mathrm {d} t},!} . 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 60 − 50 = 10 km/h. 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 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 object respectively, then the velocity of the first object as seen by the second object is u = u v . {displaystyle mathbf {u} '=mathbf {u} -mathbf {v} ,.} Similarly, the first object sees the velocity of the second object as v = v u . {displaystyle mathbf {v'} =mathbf {v} -mathbf {u} ,.} When both objects are moving in the same direction, this equation can be simplified to u = ( u v ) d . {displaystyle mathbf {u} '=(u-v)mathbf {d} ,.} Or, by ignoring direction, the difference can be given in terms of speed only: u = u v . {displaystyle u'=u-v,.} Acceleration Main article:
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): a = d v d t = d 2 r d t 2 . {displaystyle mathbf {a} ={mathrm {d} mathbf {v} over mathrm {d} t}={mathrm {d^{2}} mathbf {r} over mathrm {d} t^{2}}.}
Frames Of Reference Main articles:
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 a frame of reference within which an object interacting with no forces (an idealized situation) appears either at rest or moving uniformly in a straight line. This is the fundamental definition of an inertial frame. These are characterized by the requirement that all forces entering the observer's physical laws originate from identifiable sources caused by fields , such as electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), gravitational field (caused by mass), and so forth. 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 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 u t {displaystyle x'=x-ut,} y = y {displaystyle y'=y,} z = z {displaystyle z'=z,} t = t . {displaystyle t'=t,.} This set of formulas defines a group transformation known as the
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
FORCES; NEWTON\'S SECOND LAW Main articles:
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
The quantity mV 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 A = dV/dt, the second law can be written in the simplified and more familiar form: F = m a . {displaystyle mathbf {F} =mmathbf {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: F R = v , {displaystyle mathbf {F} _{rm {R}}=-lambda mathbf {v} ,,} 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 v = m a = m d v d t . {displaystyle -lambda mathbf {v} =mmathbf {a} =m{mathrm {d} mathbf {v} over mathrm {d} t},.} This can be integrated to obtain v = v 0 e t m {displaystyle mathbf {v} =mathbf {v} _{0}e^{frac {-lambda t}{m}}} where V0 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
WORK AND ENERGY Main articles:
If a constant force F is applied to a particle that makes a displacement ΔR, the work done by the force is defined as the scalar product of the force and displacement vectors: W = F r . {displaystyle W=mathbf {F} cdot Delta mathbf {r} ,.} More generally, if the force varies as a function of position as the particle moves from R1 to R2 along a path C, the work done on the particle is given by the line integral W = C F ( r ) d r . {displaystyle W=int _{C}mathbf {F} (mathbf {r} )cdot mathrm {d} mathbf {r} ,.} If the work done in moving the particle from R1 to R2 is the same no
matter what path is taken, the force is said to be conservative .
The kinetic energy Ek of a particle of mass m travelling at speed v is given by E k = 1 2 m v 2 . {displaystyle E_{mathrm {k} }={tfrac {1}{2}}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. 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 R1 to R2 is equal to the change in kinetic energy Ek of the particle: W = E k = E k , 2 E k , 1 = 1 2 m ( v 2 2 v 1 2 ) . {displaystyle W=Delta E_{mathrm {k} }=E_{mathrm {k,2} }-E_{mathrm {k,1} }={tfrac {1}{2}}mleft(v_{2}^{,2}-v_{1}^{,2}right),.} Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted Ep: F = E p . {displaystyle mathbf {F} =-mathbf {nabla } E_{mathrm {p} },.} If all the forces acting on a particle are conservative, and Ep 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 F r = E p r = E p . {displaystyle mathbf {F} cdot Delta mathbf {r} =-mathbf {nabla } E_{mathrm {p} }cdot Delta mathbf {r} =-Delta E_{mathrm {p} },.} The decrease in the potential energy is equal to the increase in the kinetic energy E p = E k ( E k + E p ) = 0 . {displaystyle -Delta E_{mathrm {p} }=Delta E_{mathrm {k} }Rightarrow Delta (E_{mathrm {k} }+E_{mathrm {p} })=0,.} This result is known as conservation of energy and states that the total energy , E = E k + E p , {displaystyle sum E=E_{mathrm {k} }+E_{mathrm {p} },,} is constant in time. It is often useful, because many commonly encountered forces are conservative. BEYOND NEWTON\'S LAWS
There are two important alternative formulations of classical
mechanics:
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 c2, where c is the speed of light in free space. LIMITS OF VALIDITY Domain of validity for Classical
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.
THE NEWTONIAN APPROXIMATION TO SPECIAL RELATIVITY In special relativity, the momentum of a particle is given by p = m v 1 v 2 / c 2 , {displaystyle mathbf {p} ={frac {mmathbf {v} }{sqrt {1-v^{2}/c^{2}}}},,} where m is the particle's rest mass, V its velocity, and c is the speed of light. If v is very small compared to c, v2/c2 is approximately zero, and so p m v . {displaystyle mathbf {p} approx mmathbf {v} ,.} Thus the Newtonian equation P = mV 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 f = f c m 0 m 0 + T / c 2 , {displaystyle f=f_{mathrm {c} }{frac {m_{0}}{m_{0}+T/c^{2}}},,} where fc is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest ) mass m0 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 = h p {displaystyle 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
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 .
HISTORY Main article:
Some
In his Elementa super demonstrationem ponderum, medieval
mathematician
The first published causal explanation of the motions of planets was
Johannes Kepler's
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
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. Hamilton 's greatest
contribution is perhaps the reformulation of
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. Several re-formulations
progressively allowed finding solutions to a far greater number of
problems. The first notable re-formulation was in 1788 by Joseph Louis
Lagrange .
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
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, the place of classical mechanics
in physics has been no longer that of 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
BRANCHES
*
Another division is based on the choice of mathematical formalism: *
Alternatively, a division can be made by region of application: *
SEE ALSO *
*
NOTES * ^ The notion of "classical" may be somewhat confusing, since this
term usually refers to the era of classical antiquity in European
history . While many discoveries within the mathematics of that period
are applicable today and of great use, much of the science that
emerged from that time has since been superseded by more accurate
models. This in no way detracts from the science of that time as most
of modern physics is built directly upon those developments. The
emergence of classical mechanics was a decisive stage in the
development of science , in the modern sense of the term. Above all,
it is characterized by an insistence that more rigor be used to
describe the behavior of bodies. Such an exacting foundation is only
available through mathematical treatment and reliance on experiment ,
rather than speculation .
REFERENCES * ^ Bettini, Alessandro (2016). A Course in Classical Physics
1—Mechanics. Springer. p. vii. ISBN 978-3-319-29256-4 .
* ^ French, A.P. (1971). Newtonian Mechanics. New York: W. W.
Norton & Company. p. 3. ISBN 0393099709 .
* ^ Kleppner, Daniel; Kolenkow, Robert (2014). An Introduction to
FURTHER READING * Alonso, M.; Finn, J. (1992). Fundamental University Physics.
Addison-Wesley.
* Feynman, Richard (1999). The Feynman Lectures on
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