Classical mechanics

Newton's Second Law

History of ...

This box: view • talk • edit

Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature A projectile is any object propelled through space by the exertion of a force which ceases after launch A machine is any device that uses Energy to perform some activity s are significant physical entities, associations or structures which current Science has confirmed to exist in Space. A spacecraft is a Vehicle or machine designed for Spaceflight. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter It produces very accurate results within these domains, and is one of the oldest and largest subjects in science and technology. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Technology is a broad concept that deals with a Species ' usage and knowledge of Tools and Crafts and how it affects a species' ability to control and adapt

Besides this, many related specialties exist, dealing with gases, liquids, and solids, and so on. This page is about the physical properties of gas as a state of matter Liquid is one of the principal States of matter. A liquid is a Fluid that has the particles loose and can freely form a distinct surface at the boundaries of A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. Classical mechanics is enhanced by special relativity for objects moving with high velocity, approaching the speed of light. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics, velocity is defined as the rate of change of Position. Furthermore, general relativity is employed to handle gravitation at a deeper level. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitation is a natural Phenomenon by which objects with Mass attract one another In physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements A physical law or scientific law is a Scientific generalization based on empirical Observations of physical behavior (i In Physics, a physical body (sometimes called simply a body or even an object) is a collection of Masses taken to be one The other sub-field is quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century workers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer Tycho Brahe, born Tyge Ottesen Brahe ( December 14 1546 &ndash October 24 1601) was a Danish nobleman Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Therefore, some sources exclude so-called "relativistic physics" from that category. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. However, a number of modern sources do include Einstein's mechanics, which in their view represents classical mechanics in its most developed and most accurate form. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements This is further described in the following sections. More abstract and general methods include Lagrangian mechanics and Hamiltonian mechanics. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. While the terms classical mechanics and Newtonian mechanics are usually considered equivalent (if relativity is excluded), much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements

Contents 1 Description of the theory 1.1 Displacement and its derivatives 1.1.1 Velocity and speed 1.1.2 Acceleration 1.1.3 Frames of reference 1.2 Forces; Newton's Second Law 1.3 Energy 1.4 Beyond Newton's Laws 1.5 Classical transformations 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 See also 6.1 Branches 7 External links

Description of the theory

The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, objects with negligible size. A point particle (or point-like, often spelled pointlike) is an idealized object heavily used in Physics. In Engineering, Mathematics, Physics and similar disciplines the term negligible refers to the quantities so small that they can be ignored (neglected The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it. In Mathematics, Statistics, and the mathematical Sciences a parameter ( G auxiliary measure) is a quantity that defines certain characteristics Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics, a force is whatever can cause an object with Mass to Accelerate. Each of these parameters is discussed in turn.

In reality, the kind of objects which classical mechanics can describe always have a non-zero size. (The physics of very small particles, such as the electron, is more accurately described by quantum mechanics). The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, a baseball can spin while it is moving. For information on degrees of freedom in other sciences see Degrees of freedom. Baseball is a Bat-and-ball Sport played between two teams of nine players each A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.

Displacement and its derivatives

The SI derived units with kg, m and s
displacement	m
speed	m s−1
acceleration	m s−2
jerk	m 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	kg m−1 s−2
surface tension	kg s−2
irradiance	kg s−3
kinematic viscosity	m² s−1
dynamic viscosity	kg m−1 s

The displacement, or position, of a point particle is defined with respect to an arbitrary fixed reference point, O, in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of In Physics, jerk, jolt (especially in British English) surge or lurch, is the rate of change of Acceleration; that is Specific energy is defined as the Energy per unit Mass: J/kg or in basic SI units m2/s2 This article is about the moment of inertia of a rotating object. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, a force is whatever can cause an object with Mass to Accelerate. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, power (symbol P) is the rate at which work is performed or energy is transmitted or the amount of energy required or expended for Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface For the work of fiction see Surface Tension (short story. Surface tension is a property of the surface of a Liquid that causes it to Irradiance, radiant emittance, and radiant exitance are Radiometry terms for the power of Electromagnetic radiation at a surface per unit Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another It is defined as the vector r from O to the particle. In general, the point particle need not be stationary relative to O, so r is a function of t, the time elapsed since an arbitrary initial time. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i. Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial e. , the time interval between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space. In Physics, the concept of absolute time and absolute space are Hypothetical models in which time either runs at the same rate for all the observers in Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. ^[1]

Velocity and speed

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or

. In Physics, velocity is defined as the rate of change of Position. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

In classical mechanics, velocities are directly additive and subtractive. For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, then from the perspective of the slower car, the faster car is traveling east at 60 − 50 = 10 km/h. Whereas, from the perspective of the faster car, the slower car is moving 10 km/h to the West. Velocities are directly additive as vector quantities; they must be dealt with using vector analysis. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector and the velocity of the second object by the vector where u is the speed of the first object, v is the speed of the second object, and and are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is:

Similarly:

When both objects are moving in the same direction, this equation can be simplified to:

Or, by ignoring direction, the difference can be given in terms of speed only:

Acceleration

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time) or

. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both. If only the magnitude, v, of the velocity decreases, this is sometimes referred to as deceleration, but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.

Frames of reference

While the position and velocity and acceleration of a particle can be referred to any arbitrary point of reference and accompanying coordinate system (reference frame), Classical Mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. They are characterized by the absence of accelerated motion between any two of them and the requirement of forces to produce accelerated motion of particles relative to any one of them. Any non-inertial reference frame would be accelerated with respect to an inertial one and relative to such a non-inertial frame a particle would, nevertheless, display accelerated motion. A weakness in the concept of inertial frames is the absence of any guaranteed method for identifying them. For practical purposes, reference frames that are unaccelerated with respect to the distant stars are regarded as good approximations to inertial frames.

The following consequences can be derived about the perspective of an event in two inertial reference frames, S and S', where S' is traveling at a relative velocity of to S.

• (the velocity of a particle from the perspective of S' is slowed by than its velocity from the perspective of S)
• = (the acceleration of a particle remains the same regardless of reference frame)
• = (the force on a particle remains the same regardless of reference frame)
• F' = F (since F = ma, as long as the mass m stays constant) (the force on a particle remains the same regardless of reference frame; see Newton's law)
• 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism.
• the form of Maxwell's equations is not preserved across such inertial reference frames. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric However, in Einstein's theory of special relativity, the assumed constancy (invariance) of the vacuum speed of light alters the relationships between inertial reference frames so as to render Maxwell's equations invariant. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

Forces; Newton's Second Law

Newton was the first to mathematically express the relationship between force and momentum. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Physics, a force is whatever can cause an object with Mass to Accelerate. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it to be a fundamental postulate, a law of nature. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":

.

The quantity is called the (canonical) momentum. In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The net force on a particle is, thus, equal to rate change of momentum of the particle with time. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Typically, the mass m is constant in time, and Newton's law can be written in the simplified form

where is the acceleration. It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. A rocket or rocket vehicle is a Missile, Aircraft or other Vehicle which obtains Thrust by the reaction of the Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used.

Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for , obtained by considering the particular physical entities with which the particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, for example:

with λ a positive constant (although this relation is known to be incorrect for drag in dense air, for example, it is accurate enough for elementary work). In physics a resistive force is a Force that acts on a body due to its motion relative to other bodies with which it is in contact whose direction is opposite to the Velocity In Physics, mechanical work is the amount of Energy transferred by a Force. 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. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its Continuing the example, assume that friction is the only force acting on the particle. Then the equation of motion is

.

This can be integrated to obtain

where is the initial velocity. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative This means that the velocity of this particle decays exponentially to zero as time progresses. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value This expression can be further integrated to obtain the position of the particle as a function of time.

Important forces include the gravitational force and the Lorentz force for electromagnetism. Gravitation is a natural Phenomenon by which objects with Mass attract one another In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force on another particle B, it follows that B must exert an equal and opposite reaction force, -, on A. The strong form of Newton's third law requires that and - act along the line connecting A and B, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.

Energy

If a force is applied to a particle that achieves a displacement , the work done by the force is defined as the scalar product of force and displacement vectors:

.

If the mass of the particle is constant, and W_total is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:

,

where E_k is called the kinetic energy. The kinetic energy of an object is the extra Energy which it possesses due to its motion 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:

. In Physics, mechanical work is the amount of Energy transferred by a Force.

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:

. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Potential energy can be thought of as Energy stored within a physical system

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

. Potential energy can be thought of as Energy stored within a physical system

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

is constant in time. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός It is often useful, because many commonly encountered forces are conservative.

Beyond Newton's Laws

Classical mechanics also includes descriptions of the complex motions of extended non-pointlike objects. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems.

Classical transformations

Consider two reference frames S and S' . See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to 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). In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. The Galilean transformation is used to transform between the coordinates of two Reference frames which differ only by constant relative motion within the constructs of Newtonian 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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

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. In physics the Coriolis effect is an apparent deflection of moving objects when they are viewed from a Rotating frame of reference.

History

Main article: 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. Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Ancient Greek philosophy focused on the role of Reason and Inquiry. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. 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. The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or 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,^[2] and concepts relating to Newton's laws of motion were enunciated by several Muslim physicists during the Middle Ages. Scientific method refers to bodies of Techniques for investigating phenomena Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the 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)^[3][4] and Avicenna. The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized TemplateInfobox Muslim scholars --> ( Persian /ابو علی الحسین ابن عبدالله ابن سینا (born ^[5][6] The proportionality between force and acceleration, an important principle in classical mechanics, was first stated by Hibat Allah Abu'l-Barakat al-Baghdaadi,^[7] Ibn al-Haytham,^[8] and al-Khazini. In Physics, a force is whatever can cause an object with Mass to Accelerate. Hibat Allah Abu'l-Barakat al-Baghdaadi (c 1080-1165 was a Muslim physicist, philosopher, psychologist and scientist of Jewish-Arab TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized Abd al-Rahman al-Khazini ( عبدالرحمن الخزيني) (flourished 1115–1130 was a Muslim scientist, physicist, astronomer, biologist ^[9] It is known that Galileo Galilei's mathematical treatment of acceleration and his concept of impetus^[10] grew out of earlier medieval analyses of motion, especially those of Avicenna,^[5] Ibn Bajjah,^[11] and Jean Buridan. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving In Physics, motion means a constant change in the location of a body TemplateInfobox Muslim scholars --> ( Persian /ابو علی الحسین ابن عبدالله ابن سینا (born Abū-Bakr Muhammad ibn Yahya ibn al-Sāyigh ( Arabic أبو بكر محمد بن يحيى بن الصائغ known as Ibn Bājjah (ابن باجة was an Andalusian Jean Buridan (in Latin, Johannes Buridanus; ca 1295 &ndash 1358 was a French Priest who sowed the seeds of the Copernican revolution

The first published causal explanation of the motions of planets was Johannes Kepler's Astronomia nova published in 1609. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is Johannes Kepler 's Astronomia nova, published in 1609 contains the results of the astronomer's ten-year long investigation of the motion of Mars He concluded, based on Tycho Brahe's observations of the orbit of Mars, that the orbits were ellipses. Tycho Brahe, born Tyge Ottesen Brahe ( December 14 1546 &ndash October 24 1601) was a Danish nobleman This break with ancient thought was happening around the same time that Galilei was proposing abstract mathematical laws for the motion of objects. This page lists some links to ancient philosophy. In Western philosophy, the spread of Christianity through the Roman Empire marked the end of Hellenistic Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher 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 Leaning Tower of Pisa (Torre pendente di Pisa or simply The Tower of Pisa (it La Torre di Pisa is the Campanile, or freestanding bell tower of the The reality of this experiment is disputed, but, more importantly, he did carry out quantitative experiments by rolling balls on an inclined plane. This article deals with the physical structure For related terms see Canal inclined plane, Cable railway, Funicular, or Fixed-wing 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Classical mechanics, Newton's third law states that Forces occur in pairs one called the Action and the other the Reaction ( actio et 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 phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. The Philosophiæ Naturalis Principia Mathematica ( Latin: "mathematical principles of natural philosophy" often Principia Newton also enunciated the principles of conservation of momentum and angular momentum. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position 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. Gravitation is a natural Phenomenon by which objects with Mass attract one another Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass 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. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System.

Newton previously invented the calculus, of mathematics, and used it to perform the mathematical calculations. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives 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. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

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. Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Even when discovering the so-called Newton's rings (a wave interference phenomenon) his explanation remained with his own corpuscular theory of light. The phenomenon of Newton's rings, named after Isaac Newton, is an Interference pattern caused by the reflection of Light between two surfaces In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern In Optics, the corpuscular theory of light, set forward by Sir Isaac Newton, says that light is made up of small discrete particles called "corpuscles" (little

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. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Originally considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances, the Gibbs Paradox (Gibbs' paradox or Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms. A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The effort at resolving these problems led to the development of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism.

Since the end of the 20th century, the place of classical mechanics in physics has been no longer that of an independent theory. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Along with classical electromagnetism, it has become embedded in relativistic quantum mechanics or quantum field theory. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In quantum field theory (QFT the forces between particles are mediated by other particles ^[12] It is the non-relativistic, non-quantum mechanical limit for massive particles.

Limits of validity

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics Geometric optics is an approximation to the quantum theory of light, and does not have a superior "classical" form. Quantum optics is a field of research in Physics, dealing with the application of Quantum mechanics to phenomena involving Light and its interactions

The Newtonian approximation to special relativity

Newtonian, or non-relativistic classical momentum

p = m₀v

is the result of the first order Taylor approximation of the relativistic expression:

when expanded about

so it is only valid when the velocity is much less than the speed of light. Orders of approximation have been used not only in Science, Engineering, and other quantitative disciplines to make Approximations with various degrees In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives Quantitatively speaking, the approximation is good so long as

For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by , where f_c is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m₀ circling in a magnetic field. A cyclotron is a type of Particle accelerator. Cyclotrons accelerate Charged particles using a high- Frequency, alternating Voltage (potential Gyrotrons are high powered Vacuum tubes which emit Millimeter Wavelength beams by bunching Electrons with Cyclotron motion A cavity magnetron is a high-powered Vacuum tube that generates coherent Microwaves They are commonly found in Microwave ovens as well as various 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. In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave For non-relativistic particles, this wavelength is

where h is Planck's constant and p is the momentum. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta.

Again, this happens with electrons before it happens with heavier particles. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 volts, had a wave length of 0. Clinton Joseph Davisson ( 22 October 1881 &ndash 1 February 1958) was an American physicist who won the 1937 Lester Halbert Germer (1896 &ndash 1971 was an American physicist. 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. Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle In antenna engineering side lobes are the lobes of the far field Radiation pattern that are not the main beam, where the terms "beam" In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating 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. vacuum chamber is a rigid enclosure from which air and other gases are removed by a Vacuum pump. Angular resolution describes the resolving power of any image forming device such as an optical or Radio telescope, a Microscope, a Camera Microchipsjpg|right|thumb|200px|Microchips ( EPROM memory with a transparent window showing the integrated circuit inside

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. In Quantum mechanics, quantum tunnelling is a nanoscopic phenomenon in which a particle violates the principles of Classical mechanics by penetrating a A tunnel Diode or Esaki diode is a type of Semiconductor diode which is capable of very fast operation well into the Microwave frequency In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals The field-effect transistor (FET is a type of Transistor that relies on an Electric field to control the shape and hence the conductivity of a 'channel' Microchipsjpg|right|thumb|200px|Microchips ( EPROM memory with a transparent window showing the integrated circuit inside

Classical mechanics is the same extreme high frequency approximation as geometric optics. A high frequency approximation (or "high energy approximation" for Scattering or other Wave propagation problems in Physics or Engineering It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.

Notes

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). Gerald Jay Sussman is the Panasonic Professor of Electrical Engineering at the Massachusetts Institute of Technology (MIT Jack Wisdom is a Professor of Planetary Sciences at the Massachusetts Institute of Technology. Structure and Interpretation of Classical Mechanics ( SICM) is a Classical mechanics Textbook written by Gerald Jay Sussman and ISBN 0-262-19455-4}
• Herbert Goldstein, Charles P. Herbert Goldstein ( June 26, 1922 &ndash January 12, 2005) was an American Physicist and the author of the standard graduate Poole, John L. Safko, Classical Mechanics (3rd Edition), Addison Wesley; ISBN 0-201-65702-3
• Robert Martin Eisberg, Fundamentals of Modern Physics, John Wiley and Sons, 1961
• M. Alonso, J. Finn, "Fundamental university physics", Addison-Wesley

See also

• History of classical mechanics
• Dynamical systems
• List of equations in classical mechanics
• List of publications in classical mechanics
• Molecular dynamics
• Newton's laws of motion
• Special theory of relativity

Branches

External links

• Binney, James. Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential Nomenclature a = acceleration (m/s² g Optics Book of Optics Molecular dynamics ( MD) is a form of Computer simulation in which atoms and molecules are allowed to interact for a period of time by approximations of Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Celestial mechanics is the branch of Astrophysics that deals with the motions of Celestial objects The field applies principles of Physics, historically Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " 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; see also Motion_Mountain)
• Sussman, Gerald Jay & Wisdom, Jack & Mayer,Meinhard E. Motion Mountain is a free Physics textbook on the Internet written by Christoph Schiller (*1960 (2001). Structure and Interpretation of Classical Mechanics
• Tong, David. Classical Dynamics (Cambridge lecture notes on Lagrangian and Hamiltonian formalism)

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic