Механикаи классикӣ

Аз Википедиа

Механикаи классикӣ, ки ҳамчун Механикаи Ньютонӣ низ ном бурда мешавад як ҷабҳаи физика буда, ҳаракати ҷирмҳои макроскопиро меомӯзад. Ба ҷирмҳои макроскопӣ аз тири гулӯла сар карда то қисмҳои машинасозӣ ва ҷирмҳои астрономие монанди киштии кайҳонӣ, сайёраҳо, ситораҳо ва галактикаҳо медароянд. Ин ҷабҳаи қадимтарини илму текнолоҷӣ буда, дар маҳдудияти ҳодисаҳои макроскопӣ натиҷаҳои хеле муайяну саҳеҳро медиҳад.

Ба ғайр аз ин, ихтисосҳои бисьёри дигар ҳастанд, ки бевосита ба механикаи классикӣ иртибот доранд. Барои ҷирмҳое ки бо суръати баланд (наздик ба суръати рӯшноӣ) ҳаракат мекунанд механикаи классикӣ бо нисбияти махсус ҳамҷоя омӯхта мешавад. Боз ҳам барои чуқуртар омӯхтани қувваи ҷозиба дар механикаи классикӣ аз нисбияти умумӣ низ истифода бурда мешавад.

Дар физика, механикаи классикӣ яке аз ду ҷабҳаест, ки илми механикиро меомӯзанд. Илми механикӣ гуфта, маҷмӯаи қонунҳои физикиро гӯянд, ки ҳаракати ҷисмҳои физикро бо роҳи математикӣ тавсиф мекунанд. Ҷабҳаи дуввумин механикаи квантист.

Мундариҷа

1 Тавсифи назарӣ
2 History
3 Limits of validity
- 3.1 The Newtonian approximation to special relativity
- 3.2 The classical approximation to quantum mechanics
4 Notes
5 References
6 Ин ҷоро низ нигаред
- 6.1 Шохаҳо
7 External links

[вироиш] Тавсифи назарӣ

Таҳлили тири гулӯла яке аз масъалаҳои механикаи классикист.

Мафҳумҳои асосии механикаи классикӣ чунинанд. Одатан, ҷирмҳои ҷаҳони реал дар механикаи классикӣ бо зарраҳои нуқтавӣ иваз карда мешаванд. Заррааҳои нуқтавӣ, ё ин ки нуқтаҳои материалӣ гуфта он ҷирмҳоеро менонамнд, ки ҳангоми ҳаракат ченакашон ба назар гирифта намешавад. Ҳаракати нуқтаи материалӣ бо чанде аз параметрҳо тавсиф сохта мешавад: мавқеъи он, масса ва қуввае ки ба он таъсир мекунад.

Дар ҳақиқат, ҳамаи ҷирмҳое ки механикаи классикӣ тавсиф мекунад ҳамеша ченаки носифрии муайянеро дороанд. Физикаи ҷирмҳои ченакашон хурд бошад бо механикаи квантӣ саҳееҳтар омӯхта мешавад. Ҷирмҳои ченакашон носифрӣ табиати мукаммалтар доранд, вале ба ҳар ҳол натиҷаи донише ки механикаи классикӣ барои нуқтаҳои материалӣ медиҳад дар омӯхтани он ҷирмҳои пур аз маҷмӯъи нуқтаҳои материалӣ ба кор меояд. Маркази массаи ҷирмҳои мукаммал мисли нуқаи материалӣ ҳаракат мекунад.

[вироиш] Кӯчиш ва ҳосилаҳои он

Кӯчиш, ё мавқеъи нуқтаи материалӣ аз рӯи нуқтаи муқаррари ихтиёрии О дар фазо, ки одатан нуқтаи ибтидои системаи сарҳисоб ба шумор меравад, муайян намуда мешавад. Он ҳамчу вектори r аз O то ба нуқта муайян карда мешавад. Умуман, худи нуқтаи материалӣ нисбат ба O метавонад стационар (муқаррар) набошад, аз ин рӯ r функцияи вақти гузашта нисбат ба вақти ихтиёрии ибтидоӣ, t, аст. Дар назарияи нисбияти пеш аз Эйнштейн (ҳамчу нисбияти Галилео маъруф аст), вақт чун бузургии мутлақ ба ҳисоб мерафт, яъне фосилаи вақти байни ду ҷуфти ҳодиса байни ҳамаи мушоҳидон баробар буд. Илова бар мафҳуми вақти мутлақ, механикаи классикӣ фарзияи ҳандасаи Уқлидусро барои сохти фазо истифода мебарад.

Дар ҳосилаи воҳидҳои СИ бо кг, м ва с
кӯчиш	м
суръат	м с^-1
шитоб	м с^-2
шитобнокӣ	м с^-3
моменти инерция	кг м²
импульс	кг м с^-1
моменти импульс	кг м² с^-1
қувва	кг м с^-2
моменти қувва	кг м² с^-2
энергия	кг м² с^-2
тавоноӣ - иқтидор	кг м² с^-3
фишор	кг м^-1 с^-2
шиддати сатҳӣ	кг с^-2

[вироиш] Суръат ва зудӣ

Суръат, ёхуд зудии тағйири мавқеъ бо вақт, ҳамчу ҳосилаи мавқеь нисбат ба вақт ё

$\vec{v} = {\mathrm{d}\vec{r} \over \mathrm{d}t}\,\!$

муайян карда мешавад.

Дар механикаи классикӣ, суръат хосияти ҷамъшавандагӣ ёки тарҳшавандагиро дорад. Масалан, агар машинае ки бо суръати 60 км/соат ба самти Шарқ ҳаракат дорад, аз назди машинаи дигари бо суръати 50км/соат ба самти Шарқ ҳаракат истода гузарад, аз нуқтаи назари машинаи оҳиста, машинаи тез бо суръати 60 − 50 = 10 км/соат ба самти шарқ ҳаракат дорад. Валекин аммо, аз нуқтаи назари машинаи тез ҳаракаткардаистода, машинаи оҳиста бо суръати 10 км/соат ба самти Ғарб ҳаракат дорад. Суръатҳо ҳамчу бузургиҳои векторӣ ҷамъ карда мешаванд, яьне ҳангоми бо он сарукор доштан аз таҳлили векторҳо бояд истифода бурд.

Аз ҷиҳати математикӣ гӯем, агар суръати ҷирми аввалиеро ки дар боло мубоҳиса кардем бо вектори $\vec{u} = u\vec{d}$ ва суръати ҷирми дуввумро бо вектори $\vec{v} = v\vec{e}$ ишорат намоем, пас суръати ҷирми аввал аз нуқтаи назари ҷирми дуввум ба

$\vec{u'} = \vec{u} - \vec{v}\,\!$

ва худи ҳамин хел:

$\vec{v'}= \vec{v} - \vec{u}\,\!$

баробар аст. Ин ҷо $u$ зудии ҷирми аввал, $v$ зудии ҷирми сонӣ, $\vec{d}$ ва $\vec{e}$ векторҳои воҳидиенад, ки мутаносибан дар самти ҳаракати ҳар ду ҷирм ҳастанд.

Агар ҳар ду ҷирм ба як сӯ ҳаракат кунанд, он гоҳ ин муодиларо метавон мухтасар навишт:

$\vec{u'} = ( u - v ) \vec{d}\,\!$

Ё ин ки агар самтро ба назар нагирем, фарқи онҳоро метавон танҳо дар асоси зудиҳо навишт:

$u' = u - v \,\!$

[вироиш] Шитоб

Шитоб, ёхуд зудии тағйири суръат, ин ҳосилаи суръат аз вақт (ҳосилаи тартиби дуввуми мавқеъ аз вақт) аст ё

$\vec{a} = {\mathrm{d}\vec{v} \over \mathrm{d}t}$ .

Шитоб метавонад аз тағйири бузургии суръат бо гузашти вақт, ё аз тағйири самти суръат бо гузашти вақт ва ё аз ҳисоби ҳарду пайдо шавад. Агар танҳо бузургии суръат, $v$ , кам шавад, инро баъзан даранг меноманд, аммо умуман ҳар гуна тағйири суръат бо гузашти вақт, ҳамчунин даранг, ба таври оддӣ шитоб номида мешавад.

[вироиш] Системаҳои сарҳисоб

Вақте ки гап дар бораи мавқеъ, суръат ва шитоби ҷисм дар нуқтаи ихтиёрии сарҳисоб ва дар системаи сарҳисоби ҳамроҳ меравад, механикаи классикӣ мавҷудияти оилаи махсуси системаи сарҳисобро фарз мекунад, ки дар онҳо қонунҳои механикӣ шакли қиёсан оддитарро мегиранд. Ин гуна системаҳои сарҳисоби махсусро системаҳои сарҳисоби инерциалӣ меноманд. Системаҳои сарҳисоби инерсиалӣ гуфта ситемаҳоеро меноманд, ки байни ҳар дутои онҳо ҳаракати ростхатта вуҷуд дорад ва дар натиҷаи қувва гузоштан ба ин ё он ҷисм дар системаи сарҳисоби инерсиалӣ он ҳаракати собитшитоб мекунад. Ҳар гуна системаҳои сарҳисоби ноинерсиалӣ нисбат ба системаҳои сарҳисоби инерсиалӣ бошитоб ҳаракат хоҳад кард ва ҷисми дар ин система ҷойгиршуда новобаста аз таъсири қувваи ба он додашуда ҳаракати бошитоб хоҳад кард. Заифии мафҳуми системаҳои сарҳисоби инерсиалӣ дар набудани ягон услуби кафолатдиҳандаи дарёфтани системаҳои инерсиалӣ аст. Барои мақсадҳои амалӣ, системаҳои сарҳисобе ки нисбат ба ситораҳои дар масофаи дур ҷойгиршуда бешитоб ҳаракат доранд, шабеҳии хуб ба системаҳои инерсиалӣ ҳастанд.

Дар бораи нисбияти ҳодисае дар ду системаи инерсиалӣ, $S$ ва $S'$ , ки яке аз дигаре бо суръати нисбии $\vec{u}$ ҳаракат дорад, метавон чунин натиҷаҳоро ҷамъбаст кард.

$\vec{v'} = \vec{v} - \vec{u}$ (суръати зарра $\vec{v'}$ аз назари S' нисбат ба суръати он $\vec{v}$ аз назари S ба миқдори $\vec{u}$ паст мешавад)
$\vec{a'}$ = $\vec{a}$ (новобаста аз системаи сарҳисоб шитоби ҷисм доимӣ боқӣ мемонад)
$\vec{F'}$ = $\vec{F}$ (новобаста аз системаи сарҳисоб қувваи ба ҷисм таъсиркунанда доимӣ боқӣ мемонад)
F' = F (чунки F = ma, ва агар m доимӣ боқӣ монад)
дар механикаи классикӣ суръати рӯшноӣ доимӣ нест, ва ҳамчунон дар механикаи нисбият низ ба суръати рӯшноӣ мавқеъи хоссае дода нашудааст
шакли муодилаҳои Максвелл аз як системаи сарҳисоби инерсиалӣ ба дигараш бетағйир боқӣ намемонад. Ба ҳар ҳол, дар назарияи нисбияти Эйнштейн фарзияи доимӣ (инвариант) будани суръати рӯшноӣ муносибати байни системаҳои инерсиалиро чунон тағйир медиҳад, ки дар натииҷа муодилаҳои Махвелл инвариант боқӣ мемонанд.

[вироиш] Қувваҳо; Қонуни дуюми Нютон

Нютон аввалин нафарест, муносибати байни қувва ва импулсро дар шакли математикӣ ифода намудааст. Аз рӯи таъбири баъзе физикҳо қонуни дуввуми Нютон ин таърифи қувва ва масса аст ва бархи дигар дар ақидаи онанд ки ин постулати фундаменталӣ буда, қонуни табиат аст. Ҳарду таъбир оқибати ягонаи математикӣ доранд, ки таърихан ҳамчу "Қонуни дуввуми Нютон" ном бурда мешавад:

$\vec{F} = {\mathrm{d}\vec{p} \over \mathrm{d}t} = {\mathrm{d}(m \vec{v}) \over \mathrm{d}t}$ .

Бузургии $m\vec{v}$ импульси каноникӣ ном бурда мешавад. Баробартаъсиркунандаи қувваҳо ба ҷисм он гаҳ ба тағйири импульси ҷисм бо вақт баробар аст. Одатан, массаи ҷисм m бо гузашти вақт доимӣ буда, қонуни дуввуми Нютон дар шакли соддатар чунин навишта мешавад

$\vec{F} = m \vec{a}$

дар ин ҷо $\vec{a} = \frac {\mathrm{d} \vec{v}} {\mathrm{d}t}$ бузургии шитоб аст. Вале ҳама вақт ҳам m аз t новобаста нест. Масалан, массаи ракета бо хориҷшавии сӯзишвориаш кам шудан мегирад. Дар ин гуна ҳолатҳо муодилаи болоӣ хато буда шакли пурраи қонуни дуввуми Нютон бояд истифода бурда шавад.

Қонуни дуввуми Нютон барои тавсифи ҳаракати зарра қаноатбахш нест. Вай бар замми ин, миқдори $\vec{F}$ ро ки аз таъсири ҷисм бармеояд, талаб мекунад. For example, a typical resistive force may be modelled as a function of the velocity of the particle, for example:

$\vec{F}_{\rm R} = - \lambda \vec{v}$

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

$- \lambda \vec{v} = m \vec{a} = m {\mathrm{d}\vec{v} \over \mathrm{d}t}$ .

This can be integrated to obtain

$\vec{v} = \vec{v}_0 e^{- \lambda t / m}$

where $\vec{v}_0$ 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 $\vec{r}$ 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 $\vec{F}$ on another particle B, it follows that B must exert an equal and opposite reaction force, - $\vec{F}$ , on A. The strong form of Newton's third law requires that $\vec{F}$ and - $\vec{F}$ 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.

[вироиш] Energy

If a force $\vec{F}$ is applied to a particle that achieves a displacement $\Delta\vec{s}$ , the work done by the force is defined as the scalar product of force and displacement vectors:

$W = \vec{F} \cdot \Delta \vec{s}$ .

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:

$W_{\rm total} = \Delta E_k \,\!$ ,

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:

$E_k = \begin{matrix} \frac{1}{2} \end{matrix} mv^2$ .

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.

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:

$\vec{F} = - \vec{\nabla} E_p$ .

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

$\vec{F} \cdot \Delta \vec{s} = - \vec{\nabla} E_p \cdot \Delta \vec{s} = - \Delta E_p \Rightarrow - \Delta E_p = \Delta E_k \Rightarrow \Delta (E_k + E_p) = 0 \,\!$ .

This result is known as conservation of energy and states that the total energy,

$\sum E = E_k + E_p \,\!$

is constant in time. It is often useful, because many commonly encountered forces are conservative.

[вироиш] 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.

[вироиш] 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.

[вироиш] History

Мақолаи асосӣ: History of classical mechanics

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.

An early experimental scientific method was introduced into mechanics by al-Biruni in the 11th century,^[1] and concepts relating to Newton's laws of motion were enunciated by several Muslim physicists during the Middle Ages. Early versions of the law of inertia, known as Newton's first law of motion, and the concept relating to momentum, part of Newton's second law of motion, were described by Ibn al-Haytham (Alhacen)^[2]^[3] and Avicenna.^[4]^[5] The proportionality between force and acceleration, a "fundamental law of classical mechanics" foreshadowing Newton's second law of motion, was discovered by Hibat Allah Abu'l-Barakat al-Baghdaadi,^[6] while the concept of reaction, foreshadowing Newton's third law of motion, was given an early scientific treatment by Ibn Bajjah (Avempace).^[7] Theories foreshadowing Newton's law of universal gravitation were developed by Ja'far Muhammad ibn Mūsā ibn Shākir,^[8] Ibn al-Haytham,^[9] and al-Khazini.^[10] It is known that Galileo Galilei's mathematical treatment of acceleration and his concept of impetus^[11] grew out of earlier medieval analyses of motion, especially those of Avicenna,^[4] Ibn Bajjah,^[12] and Jean Buridan.

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 ancient 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 of acceleration, mentioned above, and the law of action and reaction, and hence laying the foundations for classical mechanics. Both Newtons second and third laws were given proper scientific and mathematical treatment in Newton's Philosophiæ Naturalis Principia Mathematica, which distinguishes them from earlier attempts at explaining similar phenomenon, which were either incomplete, incorrect, or given little accurate mathematical expression. Newton also enunciated the principles of conservation of momentum and angular momentum. In Mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of gravity in Newton's law of universal gravitation. The combination of Newton's laws of motion and gravitation provide the fullest and most accurate description of classical mechanics. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of Kepler's laws of motion of the planets.

Newton 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.^[13] It is the non-relativistic, non-quantum mechanical limit for massive particles.

[вироиш] 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.

[вироиш] The Newtonian approximation to special relativity

Newtonian, or non-relativistic classical mechanics approximates the relativistic momentum $\frac{m_0 v}{ \sqrt{1-v^2/c^2}}$ with $m 0 v$ , 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 $f=f_c\frac{m_0}{m_0+T/c^2}$ , where $f c$ is the classical frequency of an electron (or other charged particle) with kinetic energy $T$ and (rest) mass $m 0$ circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV. direct current accelerating voltage.

[вироиш] 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

$\lambda=\frac{h}{p}$

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 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.

[вироиш] Notes

↑ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 614-642 [642]. Routledge, London and New York.
↑ Abdus Salam (1984), "Islam and Science". In C. H. Lai (1987), Ideals and Realities: Selected Essays of Abdus Salam, 2nd ed., World Scientific, Singapore, p. 179-213.
↑ Seyyed Hossein Nasr, "The achievements of Ibn Sina in the field of science and his contributions to its philosophy", Islam & Science, December 2003.
↑ ^4.0 ^4.1 Fernando Espinoza (2005). "An analysis of the historical development of ideas about motion and its implications for teaching", Physics Education 40 (2), p. 141.
↑ Seyyed Hossein Nasr, "Islamic Conception Of Intellectual Life", in Philip P. Wiener (ed.), Dictionary of the History of Ideas, Vol. 2, p. 65, Charles Scribner's Sons, New York, 1973-1974.
↑ Шаблон:Cite encyclopedia
(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [528].)
↑ Shlomo Pines (1964), "La dynamique d’Ibn Bajja", in Mélanges Alexandre Koyré, I, 442-468 [462, 468], Paris.
(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [543].)
↑ Robert Briffault (1938). The Making of Humanity, p. 191.
↑ Nader El-Bizri (2006), "Ibn al-Haytham or Alhazen", in Josef W. Meri (2006), Medieval Islamic Civilization: An Encyclopaedia, Vol. II, p. 343-345, Routledge, New York, London.
↑ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., Encyclopaedia of the History of Arabic Science, Vol. 2, p. 622. London and New York: Routledge.
↑ Galileo Galilei, Two New Sciences, trans. Stillman Drake, (Madison: Univ. of Wisconsin Pr., 1974), pp 217, 225, 296-7.
↑ Ernest A. Moody (1951). "Galileo and Avempace: The Dynamics of the Leaning Tower Experiment (I)", Journal of the History of Ideas 12 (2), p. 163-193.
↑ 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.

[вироиш] References

Feynman, Richard (1996). Six Easy Pieces

. Perseus Publishing. ISBN 0-201-40825-2.

Feynman, Richard; Phillips, Richard (1998). Six Easy Pieces

. Perseus Publishing. ISBN 0-201-32841-0.

Feynman, Richard (1999). Lectures on Physics

. Perseus Publishing. ISBN 0-7382-0092-1.

Landau, L. D.; Lifshitz, E. M. (1972). Mechanics Course of Theoretical Physics , Vol. 1

. Franklin Book Company, Inc.. 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

[вироиш] Ин ҷоро низ нигаред

[вироиш] Шохаҳо

[вироиш] External links

Шаблон:Commonscat

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)

Шаблон:Physics-footer

[0] Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 614-642 [642]. Routledge, London and New York.

[1] Abdus Salam (1984), "Islam and Science". In C. H. Lai (1987), Ideals and Realities: Selected Essays of Abdus Salam, 2nd ed., World Scientific, Singapore, p. 179-213.

[2] Seyyed Hossein Nasr, "The achievements of Ibn Sina in the field of science and his contributions to its philosophy", Islam & Science, December 2003.

[Espinoza-3] 4.0 ^4.1 Fernando Espinoza (2005). "An analysis of the historical development of ideas about motion and its implications for teaching", Physics Education 40 (2), p. 141.

[4] Seyyed Hossein Nasr, "Islamic Conception Of Intellectual Life", in Philip P. Wiener (ed.), Dictionary of the History of Ideas, Vol. 2, p. 65, Charles Scribner's Sons, New York, 1973-1974.

[5] Шаблон:Cite encyclopedia
(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [528].)

[6] Shlomo Pines (1964), "La dynamique d’Ibn Bajja", in Mélanges Alexandre Koyré, I, 442-468 [462, 468], Paris.
(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [543].)

[7] Robert Briffault (1938). The Making of Humanity, p. 191.

[8] Nader El-Bizri (2006), "Ibn al-Haytham or Alhazen", in Josef W. Meri (2006), Medieval Islamic Civilization: An Encyclopaedia, Vol. II, p. 343-345, Routledge, New York, London.

[9] Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., Encyclopaedia of the History of Arabic Science, Vol. 2, p. 622. London and New York: Routledge.

[10] Galileo Galilei, Two New Sciences, trans. Stillman Drake, (Madison: Univ. of Wisconsin Pr., 1974), pp 217, 225, 296-7.

[11] Ernest A. Moody (1951). "Galileo and Avempace: The Dynamics of the Leaning Tower Experiment (I)", Journal of the History of Ideas 12 (2), p. 163-193.

[12] 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.

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