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Ket

2014-01-11T18:47:40Z

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{{Infobox physical quantity
|bgcolour={default}
|name = Kinetic energy
|image=[[File:Wooden roller coaster txgi.jpg|220px]]
|caption=The cars of a [[roller coaster]] reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational [[potential energy]]. The sum of kinetic and potential energy in the system remains constant, ignoring losses to [[friction]].
|unit = [[joule]] (J)
|symbols = KE, ''E''k, or T
|derivations = ''E''k = ½''[[mass|m]][[velocity|v]]''2 
''E''k = ''E''t+''E''r
}}
{{Classical mechanics}}
In [[physics]], the '''kinetic energy''' of an object is the [[energy]] which it possesses due to its [[motion (physics)|motion]].<ref>{{cite book
|title=Textbook of Engineering Physics (Part I)
|first1=Mahesh C.
|last1=Jain
|publisher=PHI Learning Pvt. Ltd.
|year=2009
|isbn=81-203-3862-6
|page=9
|url=http://books.google.com/books?id=DqZlU3RJTywC}}, [http://books.google.com/books?id=DqZlU3RJTywC&pg=PA9 Chapter 1, p. 9]
</ref>
It is defined as the [[work (physics)|work]] needed to accelerate a body of a given mass from rest to its stated [[velocity]]. Having gained this energy during its [[acceleration]], the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest.

In [[classical mechanics]], the kinetic energy of a non-rotating object of [[mass]] ''m'' traveling at a [[speed]] ''v'' is ''½ mv²''. In [[Special relativity|relativistic mechanics]], this is only a good approximation when ''v'' is much less than the [[speed of light]].

==History and etymology==
The adjective ''kinetic'' has its roots in the [[Ancient Greek|Greek]] word ''κίνησις'' ([[-kinesis|kinesis]]) meaning ''motion''. The dichotomy between kinetic energy and [[potential energy]] can be traced back to [[Aristotle]]'s concepts of [[actuality and potentiality]].{{citation needed|date=July 2012}}

The principle in [[classical mechanics]] that ''E ∝ mv²'' was first developed by [[Gottfried Leibniz]] and [[Johann Bernoulli]], who described kinetic energy as the ''living force'', ''[[vis viva]]''. [[Willem 's Gravesande]] of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, [[Willem 's Gravesande]] determined that their penetration depth was proportional to the square of their impact speed. [[Émilie du Châtelet]] recognized the implications of the experiment and published an explanation.<ref>{{Cite book|author=Judith P. Zinsser |title=Emilie du Chatelet: Daring Genius of the Enlightenment|publisher=Penguin|year= 2007|isbn=0-14-311268-6}}</ref>

The terms ''kinetic energy'' and ''work'' in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to [[Gaspard-Gustave Coriolis]], who in 1829 published the paper titled ''Du Calcul de l'Effet des Machines'' outlining the mathematics of kinetic energy. [[William Thomson, 1st Baron Kelvin|William Thomson]], later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51.<ref>{{cite book| author=Crosbie Smith, M. Norton Wise|title=Energy and Empire: A Biographical Study of Lord Kelvin|publisher=Cambridge University Press|pages=866| isbn=0-521-26173-2}}</ref><ref>{{cite book|author=John Theodore Merz|title=A History of European Thought in the Nineteenth Century|publisher=Blackwood|year=1912|page= 139|isbn=0-8446-2579-5}}</ref>

==Introduction==
[[Energy]] occurs in many forms, including [[chemical energy]], [[thermal energy]], [[electromagnetic radiation]], [[gravitational energy]], [[electric energy]], [[elastic energy]], [[nuclear binding energy|nuclear energy]], and [[rest energy]]. These can be categorized in two main classes: [[potential energy]] and kinetic energy.

Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a [[cyclist]] uses [[food energy|chemical energy provided by food]] to accelerate a [[bicycle]] to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome [[drag (physics)|air resistance]] and [[friction]]. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist.

The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively the cyclist could connect a [[Bottle_dynamo|dynamo]] to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as [[heat]].

Like any physical quantity which is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's [[frame of reference]]. Thus, the kinetic energy of an object is not [[Galilean invariance|invariant]].

[[Spacecraft]] use chemical energy to launch and gain considerable kinetic energy to reach [[orbital speed|orbital velocity]]. In a perfectly circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is [[elliptic orbit|elliptical]] or [[hyperbolic trajectory|hyperbolic]], then throughout the orbit kinetic and [[potential energy]] are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Without loss or gain, however, the sum of the kinetic and potential energy remains constant.

Kinetic energy can be passed from one object to another. In the game of [[billiards]], the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically and the ball it collided with accelerates to a speed as the kinetic energy is passed on to it. [[Collisions]] in billiards are effectively [[elastic collision]]s, in which kinetic energy is preserved. In [[inelastic collision]]s, kinetic energy is dissipated in various forms of energy, such as heat, sound, binding energy (breaking bound structures).

[[Flywheel]]s have been developed as a method of [[flywheel energy storage|energy storage]]. This illustrates that kinetic energy is also stored in rotational motion.

Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by [[Newtonian mechanics|Newtonian (classical) mechanics]] is suitable. However, if the speed of the object is comparable to the speed of light, [[special relativity|relativistic effects]] become significant and the relativistic formula is used. If the object is on the atomic or [[sub-atomic scale]], [[quantum mechanical]] effects are significant and a quantum mechanical model must be employed.

==Newtonian kinetic energy==

===Kinetic energy of rigid bodies===
In [[classical mechanics]], the kinetic energy of a ''point object'' (an object so small that its mass can be assumed to exist at one point), or a non-rotating [[rigid body]] depends on the [[mass]] of the body as well as its [[speed]]. The kinetic energy is equal to the mass multiplied by the square of the speed, multiplied by the constant 1/2. In formula form:

:<math>E_\text{k} =\tfrac{1}{2} mv^2 </math>

where <math>m</math> is the mass and <math>v</math> is the speed (or the velocity) of the body. In [[SI]] units (used for most modern scientific work), mass is measured in [[kilogram]]s, speed in [[metres per second]], and the resulting kinetic energy is in [[joule]]s.

For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as
:<math>E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12960 \,\text{J} = 12.96 \,\text{kJ}</math>

When you throw a ball, you do [[work (physics)|work]] on it to give it speed as it leaves your hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: '''net force × displacement = kinetic energy''', i.e.,

:<math>F s =\tfrac{1}{2} mv^2</math>

Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed.

The kinetic energy of an object is related to its [[momentum]] by the equation:
:<math>E_\text{k} = \frac{p^2}{2m}</math>

where:
:<math>p\;</math> is momentum
:<math>m\;</math> is mass of the body

For the ''translational kinetic energy,'' that is the kinetic energy associated with [[rectilinear motion]], of a [[rigid body]] with constant [[mass]] <math>m\;</math>, whose [[center of mass]] is moving in a straight line with speed <math>v\;</math>, as seen above is equal to

:<math> E_\text{t} =\tfrac{1}{2} mv^2 </math>

where:
:<math>m\;</math> is the mass of the body
:<math>v\;</math> is the speed of the [[center of mass]] of the body.

The kinetic energy of any entity depends on the reference frame in which it is measured. However the total energy of an isolated system, i.e. one which energy can neither enter nor leave, does not change in whatever reference frame it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the [[Oberth effect]]. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the [[center of momentum]] frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the [[invariant mass]] of the system as a whole.

====Derivation====
The work done accelerating a particle during the infinitesimal time interval ''dt'' is given by the dot product of ''force'' and ''displacement'':
:<math>\mathbf{F} \cdot d \mathbf{x} = \mathbf{F} \cdot \mathbf{v} d t = \frac{d \mathbf{p}}{d t} \cdot \mathbf{v} d t = \mathbf{v} \cdot d \mathbf{p} = \mathbf{v} \cdot d (m \mathbf{v})\,,</math>
where we have assumed the relationship '''p''' = ''m'' '''v'''. (However, also see the special relativistic derivation [[Kinetic energy#Relativistic kinetic energy of rigid bodies|below]].)

Applying the [[product rule]] we see that:
:<math> d(\mathbf{v} \cdot \mathbf{v}) = (d \mathbf{v}) \cdot \mathbf{v} + \mathbf{v} \cdot (d \mathbf{v}) = 2(\mathbf{v} \cdot d\mathbf{v}).</math>

Therefore (assuming constant mass so that ''dm''=0), the following can be seen:
:<math> \mathbf{v} \cdot d (m \mathbf{v}) = \frac{m}{2} d (\mathbf{v} \cdot \mathbf{v}) = \frac{m}{2} d v^2 = d \left(\frac{m v^2}{2}\right). </math>

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:
:<math> E_\text{k} = \int \mathbf{F} \cdot d \mathbf{x} = \int \mathbf{v} \cdot d (m \mathbf{v}) = \int d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}. </math>

This equation states that the kinetic energy (''E''k) is equal to the [[integral]] of the [[dot product]] of the [[velocity]] ('''v''') of a body and the [[infinitesimal]] change of the body's [[momentum]] ('''p'''). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

===Rotating bodies===
If a rigid body is rotating about any line through the center of mass then it has [[rotational energy|''rotational kinetic energy'']] (<math>E_\text{r}\,</math>) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

:<math> E_\text{r} = \int \frac{v^2 dm}{2} = \int \frac{(r \omega)^2 dm}{2} = \frac{\omega^2}{2} \int{r^2}dm = \frac{\omega^2}{2} I = \begin{matrix} \frac{1}{2} \end{matrix} I \omega^2 </math>

where:
*ω is the body's [[angular velocity]]
*''r'' is the distance of any mass ''dm'' from that line
*<math>I\,</math> is the body's [[moment of inertia]], equal to <math>\int{r^2}dm</math>.

(In this equation the moment of [[inertia]] must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).

===Kinetic energy of systems===
A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the [[Solar System]] the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains.

A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's [[center of momentum]]) may have various kinds of [[internal energy]] at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However all internal energies of all types contribute to body's mass, inertia, and total energy.

===Frame of reference===

The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable [[inertial frame of reference]]. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary from the point of view of an observer moving with the same velocity as the bullet, and so has zero kinetic energy.<ref>{{cite book
|title=Introduction to the theory of relativity
|first1=Francis Weston
|last1=Sears
|first2=Robert W.
|last2=Brehme
|publisher=Addison-Wesley
|year=1968
|page=127
}}, [http://books.google.com/books?ei=uLlaTKiSF5DuOaqf3JYP&ct=result&id=cpzvAAAAMAAJ&dq=%22in+its+own+rest+frame%22+%22kinetic+energy%22&q=%22in+its+own+rest+frame%22 Snippet view of page 127]
</ref> By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's [[invariant mass]], which is independent of the reference frame.

The total kinetic energy of a system depends on the [[inertial frame of reference]]: it is the sum of the total kinetic energy in a [[center of momentum frame]] and the kinetic energy the total mass would have if it were concentrated in the [[center of mass]].

This may be simply shown: let <math>\textstyle\mathbf{V}</math> be the relative velocity of the center of mass frame ''i'' in the frame ''k''.
Since <math>\textstyle v^2 = (v_i + V)^2 = (\mathbf{v}_i + \mathbf{V}) \cdot (\mathbf{v}_i + \mathbf{V}) = \mathbf{v}_i \cdot \mathbf{v}_i + 2 \mathbf{v}_i \cdot \mathbf{V} + \mathbf{V} \cdot \mathbf{V} = v_i^2 + 2 \mathbf{v}_i \cdot \mathbf{V} + V^2</math>,

:<math>E_\text{k} = \int \frac{v^2}{2} dm = \int \frac{v_i^2}{2} dm + \mathbf{V} \cdot \int \mathbf{v}_i dm + \frac{V^2}{2} \int dm. </math>

However, let <math> \int \frac{v_i^2}{2} dm = E_i </math> the kinetic energy in the center of mass frame, <math> \int \mathbf{v}_i dm </math> would be simply the total momentum which is by definition zero in the center of mass frame, and let the total mass: <math> \int dm = M </math>. Substituting, we get:<ref>[http://www.phy.duke.edu/~rgb/Class/intro_physics_1/intro_physics_1/node64.html Physics notes - Kinetic energy in the CM frame]. [[Duke University|Duke]].edu. Accessed 2007-11-24.</ref>

:<math> E_\text{k} = E_i + \frac{M V^2}{2}. </math>

Thus the kinetic energy of a system is lowest with respect to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the [[center of mass frame]] or any other [[center of momentum frame]]). In any other frame of reference there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the [[center of momentum frame]] is a quantity which is both invariant (all observers see it to be the same) and is conserved (in an isolated system, it cannot change value, no matter what happens inside the system).

===Rotation in systems===
It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass ([[rotational energy]]):

:<math> E_\text{k} = E_t + E_\text{r} \, </math>

where:
:''E''k is the total kinetic energy
:''E''t is the translational kinetic energy
:''E''r is the ''rotational energy'' or ''angular kinetic energy'' in the rest frame

Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.

==Relativistic kinetic energy of rigid bodies==
{{See also|Mass in special relativity|Tests of relativistic energy and momentum}}

In [[special relativity]], we must change the expression for linear momentum.

Using ''m'' for [[rest mass]], '''v''' and ''v'' for the object's velocity and speed respectively, and ''c'' for the speed of light in vacuum, we assume for linear momentum that <math>\mathbf{p}=m\gamma \mathbf{v}</math>, where <math>\gamma = 1/\sqrt{1-v^2/c^2}</math>.

[[Integration by parts|Integrating by parts]] gives
:<math>E_\text{k} = \int \mathbf{v} \cdot d \mathbf{p}= \int \mathbf{v} \cdot d (m \gamma \mathbf{v}) = m \gamma \mathbf{v} \cdot \mathbf{v} - \int m \gamma \mathbf{v} \cdot d \mathbf{v} = m \gamma v^2 - \frac{m}{2} \int \gamma d (v^2)</math>
Remembering that <math>\gamma = (1 - v^2/c^2)^{-1/2}\!</math>, we get:
:<math>\begin{align}
E_\text{k} &= m \gamma v^2 - \frac{- m c^2}{2} \int \gamma d (1 - v^2/c^2) \\
&= m \gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} - E_0
\end{align}</math>
where ''E''0 serves as an integration constant.
Thus:
:<math>\begin{align}
E_\text{k} &= m \gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\
&= m \gamma (v^2 + c^2 - v^2) - E_0 \\
&= m \gamma c^2 - E_0
\end{align}</math>
The constant of integration ''E''0 is found by observing that, when <math>\mathbf{v }= 0 , \ \gamma = 1\!</math> and <math> E_\text{k} = 0 \!</math>, giving
:<math>E_0 = m c^2 \,</math>
and giving the usual formula:
:<math>E_\text{k} = m \gamma c^2 - m c^2 = \frac{m c^2}{\sqrt{1 - v^2/c^2}} - m c^2</math>

If a body's speed is a significant fraction of the [[speed of light]], it is necessary to use relativistic mechanics (the [[Relativity theory|theory of relativity]] as developed by [[Albert Einstein]]) to calculate its kinetic energy.

For a relativistic object the momentum p is equal to:

:<math> p = \frac{m v}{\sqrt{1 - (v/c)^2}} </math>.

Thus the work expended accelerating an object from rest to a relativistic speed is:

:<math>E_\text{k} = \frac{m c^2}{\sqrt{1 - (v/c)^2}} - m c^2 </math>.

The equation shows that the energy of an object approaches infinity as the velocity ''v'' approaches the speed of light ''c'', thus it is impossible to accelerate an object across this boundary.

The mathematical by-product of this calculation is the [[mass-energy equivalence]] formula—the body at rest must have energy content equal to:

:<math>E_\text{rest} = E_0 = m c^2 \!</math>

At a low speed (v<<c), the relativistic kinetic energy may be approximated well by the classical kinetic energy. This is done by [[binomial approximation]]. Indeed, taking [[Taylor expansion]] for the reciprocal square root and keeping first two terms we get:

:<math>E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2\right) - m c^2 = \frac{1}{2} m v^2 </math>,

So, the total energy E can be partitioned into the energy of the rest mass plus the traditional Newtonian kinetic energy at low speeds.

When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the approximation is small for low speeds, and can be found by extending the expansion into a Taylor series by one more term:

:<math> E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2 + \frac{3}{8} v^4/c^4\right) - m c^2 = \frac{1}{2} m v^2 + \frac{3}{8} m v^4/c^2 </math>.

For example, for a speed of {{convert|10|km/s|mph|abbr=on}} the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg), etc.

For higher speeds, the formula for the relativistic kinetic energy<ref>In Einstein's original [http://www.uni-kiel.de/ub/digiport/ab1800/G4378.html Über die spezielle und die allgemeine Relativitätstheorie] (Zu Seite 41) and in most translations (e.g. [http://bartleby.com/173/15.html Relativity - The Special and General Theory]) kinetic energy is defined as <math>m c^2 / \sqrt{1 - v^2/c^2}</math>.</ref> is derived by simply subtracting the rest mass energy from the total energy:

:<math> E_\text{k} = m \gamma c^2 - m c^2 = m c^2\left(\frac{1}{\sqrt{1 - (v/c)^2}} - 1\right) </math>.

The relation between kinetic energy and [[momentum]] is more complicated in this case, and is given by the equation:

:<math>E_\text{k} = \sqrt{p^2 c^2 + m^2 c^4} - m c^2</math>.

This can also be expanded as a [[Taylor series]], the first term of which is the simple expression from Newtonian mechanics.

What this suggests is that the formulas for energy and momentum are not special and axiomatic, but rather concepts which emerge from the equation of mass with energy and the principles of relativity.

===General relativity===
{{see also|Schwarzschild geodesics}}
Using the convention that
:<math>g_{\alpha \beta} \, u^{\alpha} \, u^{\beta} \, = \, - c^2 </math>

where the [[four-velocity]] of a particle is
:<math>u^{\alpha} \, = \, \frac{d x^{\alpha}}{d \tau} </math>

and <math>\tau \,</math> is the [[proper time]] of the particle, there is also an expression for the kinetic energy of the particle in [[general relativity]].

If the particle has momentum
:<math>p_{\beta} \, = \, m \, g_{\beta \alpha} \, u^{\alpha} </math>

as it passes by an observer with four-velocity ''u''obs, then the expression for total energy of the particle as observed (measured in a local inertial frame) is
:<math>E \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} </math>

and the kinetic energy can be expressed as the total energy minus the rest energy:
:<math>E_{k} \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} \, - \, m \, c^2 \, .</math>

Consider the case of a metric which is diagonal and spatially isotropic (''g''tt,''g''ss,''g''ss,''g''ss). Since
:<math>u^{\alpha} = \frac{d x^{\alpha}}{d t} \frac{d t}{d \tau} = v^{\alpha} u^{t} \,</math>

where ''v''α is the ordinary velocity measured w.r.t. the coordinate system, we get
:<math>-c^2 = g_{\alpha \beta} u^{\alpha} u^{\beta} = g_{t t} (u^{t})^2 + g_{s s} v^2 (u^{t})^2 \,.</math>

Solving for ''u''t gives
:<math>u^{t} = c \sqrt{\frac{-1}{g_{t t} + g_{s s} v^2}} \,.</math>

Thus for a stationary observer (''v''= 0)
:<math>u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{t t}}} \,</math>

and thus the kinetic energy takes the form
:<math>E_\text{k} = - m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,.</math>

Factoring out the rest energy gives:
:<math>E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,.</math>

This expression reduces to the special relativistic case for the flat-space metric where
:<math>g_{t t} = -c^2 \,</math>
:<math>g_{s s} = 1 \,.</math>

In the Newtonian approximation to general relativity
:<math>g_{t t} = - \left( c^2 + 2 \Phi \right) \,</math>
:<math>g_{s s} = 1 - \frac{2 \Phi}{c^2} \,</math>

where Φ is the Newtonian [[gravitational potential]]. This means clocks run slower and measuring rods are shorter near massive bodies.

==Kinetic energy in quantum mechanics==

{{further2|[[Hamiltonian (quantum mechanics)]]}}

In [[quantum mechanics]], observables like kinetic energy are represented as operators. For one particle of mass ''m'', the kinetic energy operator appears as a term in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and is defined in terms of the more fundamental momentum operator <math>\hat p</math> as

:<math>\hat T = \frac{\hat p^2}{2m}.</math>

Notice that this can be obtained by replacing <math>p</math> by <math>\hat p</math> in the classical expression for kinetic energy in terms of [[momentum]],
:<math>E_\text{k} = \frac{p^2}{2m}.</math>

In the [[Schrödinger picture]], <math>\hat p</math> takes the form <math>-i\hbar\nabla </math> where the derivative is taken with respect to position coordinates and hence

:<math>\hat T = -\frac{\hbar^2}{2m}\nabla^2.</math>

The expectation value of the electron kinetic energy, <math>\langle\hat{T}\rangle</math>, for a system of ''N'' electrons described by the [[Wave function|wavefunction]] <math>\vert\psi\rangle</math> is a sum of 1-electron operator expectation values:
:<math>\langle\hat{T}\rangle = \bigg\langle\psi \bigg\vert \sum_{i=1}^N \frac{-\hbar^2}{2 m_\text{e}} \nabla^2_i \bigg\vert \psi \bigg\rangle = -\frac{\hbar^2}{2 m_\text{e}} \sum_{i=1}^N \bigg\langle\psi \bigg\vert \nabla^2_i \bigg\vert \psi \bigg\rangle</math>
where <math>m_\text{e}</math> is the mass of the electron and <math>\nabla^2_i</math> is the [[Laplacian]] operator acting upon the coordinates of the ''i''th electron and the summation runs over all electrons.

The [[Density functional theory|density functional]] formalism of quantum mechanics requires knowledge of the electron density ''only'', i.e., it formally does not require knowledge of the wavefunction. Given an electron density <math>\rho(\mathbf{r})</math>, the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as
:<math> T[\rho] = \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r </math>
where <math>T[\rho]</math> is known as the [[Carl Friedrich von Weizsäcker|von Weizsäcker]] kinetic energy functional.

==See also==
{{Portal|Energy}}
* [[Escape velocity]]
* [[Joule]]
* [[KE-Munitions]]
* [[Projectile#Typical_projectile_speeds|Kinetic energy per unit mass of projectiles]]
* [[Projectile#Kinetic projectiles|Kinetic projectile]]
* [[Parallel axis theorem]]
* [[Potential energy]]
* [[Recoil]]

==Notes==
{{reflist}}

==References==
* [http://www.kineticenergys.com kinetic energy]—What it is and how it works.
* [[Oxford Dictionary]] 1998
* {{cite web | url = http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Coriolis.html | title = Biography of Gaspard-Gustave de Coriolis (1792-1843) | accessdate = 2006-03-03 | author = School of Mathematics and Statistics, University of St Andrews | year = 2000 }}
* {{cite book | last = Serway | first = Raymond A. | coauthors = Jewett, John W. | title = Physics for Scientists and Engineers | edition = 6th | publisher = Brooks/Cole | year = 2004 | isbn = 0-534-40842-7 }}
* {{cite book | last = Tipler | first = Paul | title = Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics | edition = 5th | publisher = W. H. Freeman | year = 2004 | isbn = 0-7167-0809-4 }}
* {{cite book | last = Tipler | first = Paul | coauthors = Llewellyn, Ralph | title = Modern Physics | edition = 4th | publisher = W. H. Freeman | year = 2002 | isbn = 0-7167-4345-0 }}

{{Footer energy}}

[[Category:Forms of energy]]
[[Category:Kinetic energy]]
[[Category:Dynamics]]

[[ml:ഗതികോര്‍ജ്ജം]]

Weierstrass transform

2014-01-11T17:53:51Z

109.152.240.81: /* The inverse */

In cryptography, the '''dining cryptographers problem''' studies how to perform a [[secure multi-party computation]] of the boolean-OR function. [[David Chaum]] first proposed this problem in 1988, and used it as an illustrative example to show it was possible to send anonymous messages with unconditional sender and recipient untraceability.<ref>{{cite journal | author=David Chaum | title=The Dining Cryptographers Problem: Unconditional Sender and Recipient Untraceability | journal=Journal of Cryptology | volume=1 | issue=1 | year=1988 | pages=65–75 | doi=10.1007/BF00206326 | url=http://www.cs.cornell.edu/People/egs/herbivore/dcnets.html}}</ref> Anonymous communication networks based on this problem are often referred to as '''DC-nets'''.

Despite the word ''dining'', the dining cryptographers problem is unrelated to the [[dining philosophers problem]].

== Description ==
[[File:Dining Cryptographers.svg|thumb|Dining cryptographers problem illustration|600px|right]]Three cryptographers gather around a table for dinner. The waiter informs them that the meal has been paid by someone, who could be one of the cryptographers or the [[National Security Agency]] (NSA). The cryptographers respect each other's right to make an anonymous payment, but want to find out whether the NSA paid. So they decide to execute a two-stage protocol.

In the first stage, every two cryptographers establish a shared one-bit secret, say by tossing a coin behind a menu so that only two cryptographers see the outcome in turn for each two cryptographers. Suppose, after the coin tossing, cryptographer A and B share a secret bit <math>\scriptstyle 1</math>, A and C share <math>\scriptstyle 0</math>, and B and C share <math>\scriptstyle 1</math>.

In the second stage, each cryptographer publicly announces a bit, which is
* if they didn't pay the meal, the [[Exclusive OR]] (XOR) of the two shared bits they hold with their two neighbours
* if they did pay the meal, the opposite of that XOR.

Suppose none of the cryptographers paid, then A would announce <math>\scriptstyle 1 \,\oplus\, 0 \;=\; 1</math>, B would announce <math>\scriptstyle 1 \,\oplus\, 1 \;=\; 0</math>, and C would announce <math>\scriptstyle 0 \,\oplus\, 1 \;=\; 1</math>. On the other hand, if A paid, he would announce <math>\scriptstyle \lnot{(1 \,\oplus\, 0)} \;=\; 0</math>.

After the second stage is the truth revealing. One simply performs XOR of all the announced bits. If the result is 0, then it implies that none of the cryptographers paid (so NSA must have paid). Otherwise, it would imply one of the cryptographers paid, but their identity remains unknown to the other cryptographers.

The above protocol was named by David Chaum as the Dining Cryptographers network, or DC-net.

==Limitations==
The DC-net protocol is simple and elegant. It has several limitations, however, some solutions to which have been explored in follow-up research (see the References section below).

1. '''Collision''' - If two cryptographers paid for the dinner, their messages will cancel each other out, and the final XOR result will be <math>\scriptstyle 0</math>. This is called a collision, and allows only one participant to transmit at a time using this protocol. In a more general case, a collision happens as long as any even number of participants send messages.

2. '''Disruption''' - Any malicious cryptographer who does not want the group to communicate successfully can jam the protocol so that the final XOR result is useless, simply by sending random bits instead of the correct result of the XOR. This problem occurs because the original protocol was designed without using any [[public key]] technology, and lacks reliable mechanisms to check whether participants honestly follow the protocol.

3. '''Complexity''' - The protocol requires pair-wise shared secret keys between the participants, which may be problematic if there are many participants. Also, though the DC-net protocol is "unconditionally secure", it actually depends on the assumption that "unconditionally secure" channels already exist between pairs of the participants, which is not easy to achieve in practice.

A related [[anonymous veto network]] algorithm computes the logical OR of several users' inputs, rather than a logical XOR as in DC-nets, which may be useful in applications to which a logical OR combining operation is naturally suited.

==Generalizations==

DC-nets generalizes readily to allow transmissions of more than one information bit per round, to groups larger than three participants, and to arbitrary "alphabets" other than the binary digits 0 and 1, as described below.

=== Transmissions of longer messages ===

To enable an anonymous sender to transmit more than one bit of information per DC-nets round, the group of cryptographers can simply repeat the protocol as many times as desired to create a desired number of bits worth of transmission bandwidth. These repetitions need not be performed serially. In practical DC-net systems, it is typical for pairs of participants to agree up-front on a single shared "master" secret, using [[Diffie–Hellman key exchange]] for example. Each participant then locally feeds this shared master secret into a [[pseudorandom number generator]], in order to produce as many shared "coin flips" as desired to allow an anonymous sender to transmit multiple bits of information.

=== Larger group sizes ===

The protocol can be generalized to a group of <math>\scriptstyle n</math> participants, each with a shared secret key in common with each other participant. In each round of the protocol, if a participant wants to transmit an untraceable message to the group, they invert their publicly announced bit. The participants can be visualized as a [[Complete graph|fully connected graph]] with the vertices representing the participants and the edges representing their shared secret keys.

=== Sparse secret sharing graphs ===

The protocol may be run with less than fully connected secret sharing graphs, which can improve the performance and scalability of practical DC-net implementations, at the potential risk of reducing anonymity if colluding participants can split the secret sharing graph into separate connected components. For example, an intuitively appealing but less secure generalization to <math>\scriptstyle n > 3</math> participants using a [[ring topology]], where each cryptographer sitting around a table shares a secret ''only'' with the cryptographer to their immediate left and right, and ''not'' with every other cryptographer. Such a topology is appealing because each cryptographer needs to coordinate two coin flips per round, rather than <math>\scriptstyle n</math>. However, if Adam and Charlie are actually NSA agents sitting immediately to the left and right of Bob, an innocent victim, and if Adam and Charlie secretly collude to reveal their secrets to each other, then they can determine with certainty whether or not Bob was the sender of a 1 bit in a DC-net run, regardless of how many participants there are in total. This is because the colluding participants Adam and Charlie effectively "split" the secret sharing graph into two separate disconnected components, one containing only Bob, the other containing all other honest participants.

Another compromise secret sharing DC-net topology, employed in the [http://dedis.cs.yale.edu/2010/anon/ Dissent] system for scalability,<ref name="WCFJ12">{{cite conference | author= David Isaac Wolinsky, Henry Corrigan-Gibbs, Bryan Ford, and Aaron Johnson | title= Dissent in Numbers: Making Strong Anonymity Scale | conference=10th USENIX Symposium on Operating Systems Design and Implementation (OSDI) | location=Hollywood, CA, USA | date=October 8–10, 2012 | url=http://dedis.cs.yale.edu/2010/anon/papers/osdi12-abs}}</ref> may be described as a ''client/server'' or ''user/trustee'' topology. In this variant, we assume there are two types of participants playing different roles: a potentially large number ''n ''of users who desire anonymity, and a much smaller number <math>\scriptstyle m</math> of ''trustees'' whose role is to help the users obtain that anonymity. In this topology, each of the <math>\scriptstyle n</math> users shares a secret with each of the <math>\scriptstyle m</math> trustees - but users share no secrets directly with other users, and trustees share no secrets directly with other trustees - resulting in an <math>\scriptstyle n \times m</math> secret sharing matrix. If the number of trustees <math>\scriptstyle m</math> is small, then each user needs to manage only a few shared secrets, improving efficiency for users in the same way the ring topology does. However, as long as ''at least one trustee'' behaves honestly and does not leak his or her secrets or collude with other participants, then that honest trustee forms a "hub" connecting all honest users into a single fully connected component, regardless of which or how many other users and/or trustees might be dishonestly colluding. Users need not know or guess which trustee is honest; their security depends only on the ''existence'' of at least one honest, non-colluding trustee.

=== Alternate alphabets and combining operators ===

Though the simple DC-nets protocol uses [[Bit|binary digits]] as its transmission alphabet, and uses the XOR operator to combine cipher texts, the basic protocol generalizes to any alphabet and combining operator suitable for [[one-time pad]] encryption. This flexibility arises naturally from the fact that the secrets shared between the many pairs of participants are, in effect, merely one-time pads combined together symmetrically within a single DC-net round.

One useful alternate choice of DC-nets alphabet and combining operator is to use a [[finite group]] suitable for public-key cryptography as the alphabet - such as a [[Schnorr group]] or [[elliptic curve]] - and to use the associated group operator as the DC-net combining operator. Such a choice of alphabet and operator makes it possible for clients to use [[zero-knowledge proof]] techniques to prove correctness properties about the DC-net ciphertexts that they produce - such as that the participant is not "jamming" the transmission channel - without compromising the anonymity offered by the DC-net. This technique was first suggested by Golle and Juels,<ref name="GJ04">{{cite conference | author= Philippe Golle and Ari Juels | title=Dining Cryptographers Revisited | conference=Eurocrypt 2004 | date=May 2–6, 2004 | location=Interlaken, Switzerland | url=http://china.rsa.com/rsalabs/staff/bios/ajuels/publications/pdfs/dc-revisited.pdf}}</ref> further developed by Franck,<ref>{{cite thesis |degree=M.Sc. |first=Christian |last=Franck |title=New Directions for Dining Cryptographers |year=2008 | url=http://secan-lab.uni.lu/images/stories/christian_franck/FRANCK_Christian_Master_Thesis.pdf}}</ref> and later implemented in [http://dedis.cs.yale.edu/2010/anon/papers/verdict-abs Verdict], a cryptographically verifiable implementation of the [http://dedis.cs.yale.edu/2010/anon/ Dissent] system.<ref name="CWF13">{{cite conference | author= Henry Corrigan-Gibbs, David Isaac Wolinsky, and Bryan Ford | title= Proactively Accountable Anonymous Messaging in Verdict | conference=22nd USENIX Security Symposium | date=August 14–16, 2013 | location=Washington, DC, USA | url=http://dedis.cs.yale.edu/2010/anon/papers/verdict-abs }}</ref>

== Handling or Avoiding Collisions ==

The measure originally suggested by David Chaum to avoid collisions is to re-transmit the message once a collision is detected, but the paper does not explain exactly how to arrange the re-transmission.

[http://dedis.cs.yale.edu/2010/anon/ Dissent] avoids the possibility of unintentional collisions by using a verifiable shuffle to establish a DC-nets transmission schedule, such that each participant knows exactly which bits in the schedule correspond to his own transmission slot, but does not know who owns other transmission slots.<ref name="CF10">{{cite conference | author=Henry Corrigan-Gibbs and Bryan Ford | title=Dissent: Accountable Group Anonymity | conference=17th ACM Conference on Computer and Communications Security (CCS) | date=October 2010 | location=Chicago, IL, USA | url=http://dedis.cs.yale.edu/2010/anon/papers/ccs10/}}</ref>

== Countering disruption attacks ==

[http://www.cs.cornell.edu/people/egs/herbivore/documentation.html Herbivore] divides a large anonymity network into smaller DC-net groups, enabling participants to evade disruption attempts by leaving a disrupted group and joining another group, until the participant finds a group free of disruptors.<ref name="SGRE04">{{cite conference | author=Emin Gün Sirer, Sharad Goel, Mark Robson, and Doğan Engin | title=Eluding Carnivores: File Sharing with Strong Anonymity | conference=ACM SIGOPS European workshop | date= September 19–22, 2004 | location= Leuven, Belgium | url=http://www.cs.cornell.edu/People/egs/714-spring05/herbivore-esigops.pdf}}</ref> This evasion approach introduces the risk that an adversary who owns many nodes could ''selectively'' disrupt only groups the adversary has not ''completely'' compromised, thereby "herding" participants toward groups that may be functional precisely because they are completely compromised.<ref name="BDMT07">{{cite conference | author=Nikita Borisov and George Danezis and Prateek Mittal and Parisa Tabriz | title=Denial of Service or Denial of Security? How Attacks on Reliability can Compromise Anonymity | conference=ACM Conference on Computer and Communications Security (CCS) | date=October 2007 | location=Alexandria, VA, USA | url=http://hostmaster.freehaven.net/anonbib/cache/ccs07-doa.pdf}}</ref>

[http://dedis.cs.yale.edu/2010/anon/ Dissent] implements several schemes to counter disruption. The original protocol<ref name="CF10" /> used a verifiable [[shuffling|cryptographic shuffle]] to form a DC-net transmission schedule and distribute "transmission assignments", allowing the correctness of subsequent DC-nets ciphertexts to be verified with a simple [[cryptographic hash function|cryptographic hash]] check. This technique required a fresh verifiable before every DC-nets round, however, leading to high latencies. A later, more efficient scheme allows a series of DC-net rounds to proceed without intervening shuffles in the absence of disruption, but in response to a disruption event uses a shuffle to distribute anonymous ''accusations'' enabling a disruption victim to expose and prove the identity of the perpetrator.<ref name="WCFJ12" /> Finally, more recent versions support fully verifiable DC-nets - at substantial cost in computation efficiency due to the use of [[public-key cryptography]] in the DC-net - as well as a ''hybrid'' mode that uses efficient XOR-based DC-nets in the normal case and verifiable DC-nets only upon disruption, to distribute accusations more quickly than is feasible using verifiable shuffles.<ref name="CWF13" />

== References ==
{{reflist}}

[[Category:Cryptography]]
[[Category:Mathematical problems]]
[[Category:Zero-knowledge protocols]]