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72-468: Heilig, who today would be thought of as a "multimedia" specialist, in the 1950s saw theater as an activity that could encompass all the senses in an effective manner, thus drawing the viewer into the onscreen activity. He dubbed it "Experience Theater", and detailed his vision of multi-sensory theater in a 1955 paper, "The Cinema of the Future" (Robinett 1994). In 1962 he built a prototype of his vision, dubbed

144-432: A {\displaystyle F=ma} , is valid. Non-inertial reference frames accelerate in relation to another inertial frame. A body rotating with respect to an inertial frame is not an inertial frame. When viewed from an inertial frame, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter

216-413: A Legendre transformation on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory , Routhian mechanics , and Appell's equation of motion . All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called

288-514: A baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles. The center of mass of a composite object behaves like a point particle. Classical mechanics assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at

360-412: A configuration space M {\textstyle M} and a smooth function L {\textstyle L} within that space called a Lagrangian. For many systems, L = T − V , {\textstyle L=T-V,} where T {\textstyle T} and V {\displaystyle V} are the kinetic and potential energy of

432-997: A close relationship with geometry (notably, symplectic geometry and Poisson structures ) and serves as a link between classical and quantum mechanics . In this formalism, the dynamics of a system are governed by Hamilton's equations, which express the time derivatives of position and momentum variables in terms of partial derivatives of a function called the Hamiltonian: d q d t = ∂ H ∂ p , d p d t = − ∂ H ∂ q . {\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.} The Hamiltonian

504-416: A decrease in the magnitude of velocity " v " is referred to as deceleration , but generally any change in the velocity over time, including deceleration, is referred to as acceleration. While the position, velocity and acceleration of a particle can be described with respect to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in which

576-447: A distance ). The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O . A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P . In general, the point particle does not need to be stationary relative to O . In cases where P

648-428: A fictitious centrifugal force and Coriolis force . A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within a field , such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others. Newton was the first to mathematically express

720-729: A particular formalism based on Newton's laws of motion . Newtonian mechanics in this sense emphasizes force as a vector quantity. In contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its kinetic energy and potential energy . The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation . Two dominant branches of analytical mechanics are Lagrangian mechanics , which uses generalized coordinates and corresponding generalized velocities in configuration space , and Hamiltonian mechanics , which uses coordinates and corresponding momenta in phase space . Both formulations are equivalent by

792-487: A reformulation of Lagrangian mechanics . Introduced by Sir William Rowan Hamilton , Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta . Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has

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864-493: A separate discipline in physics, formally treated as distinct from mechanics, whether it be classical fields or quantum fields . But in actual practice, subjects belonging to mechanics and fields are closely interwoven. Thus, for instance, forces that act on particles are frequently derived from fields ( electromagnetic or gravitational ), and particles generate fields by acting as sources. In fact, in quantum mechanics, particles themselves are fields, as described theoretically by

936-411: A solid body into a collection of points.) In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The behavior of very small particles, such as the electron , is more accurately described by quantum mechanics .) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom , e.g.,

1008-464: Is a limiting case of the Poincaré group used in special relativity . The limiting case applies when the velocity u is very small compared to c , the speed of light . The transformations have the following consequences: 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

1080-419: Is an accepted version of this page Classical mechanics is a physical theory describing the motion of objects such as projectiles , parts of machinery , spacecraft , planets , stars , and galaxies . The development of classical mechanics involved substantial change in the methods and philosophy of physics. The qualifier classical distinguishes this type of mechanics from physics developed after

1152-633: Is another tradition that goes back to the ancient Greeks where mathematics is used more extensively to analyze bodies statically or dynamically , an approach that may have been stimulated by prior work of the Pythagorean Archytas . Examples of this tradition include pseudo- Euclid ( On the Balance ), Archimedes ( On the Equilibrium of Planes , On Floating Bodies ), Hero ( Mechanica ), and Pappus ( Collection , Book VIII). In

1224-408: Is based on the choice of mathematical formalism. Classical mechanics can be mathematically presented in multiple different ways. The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to

1296-401: Is called the equation of motion . As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example: where λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is This can be integrated to obtain where v 0

1368-412: Is equal to the change in kinetic energy E k of the particle: Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted 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

1440-426: Is moving relative to O , r is defined as a function of t , time . In pre-Einstein relativity (known as Galilean relativity ), time is considered an absolute, i.e., the time interval that is observed to elapse 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. The velocity , or

1512-422: Is non-conservative. The kinetic energy E k of a particle of mass m travelling at speed v is given by For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles. The work–energy theorem states that for a particle of constant mass m , the total work W done on the particle as it moves from position r 1 to r 2

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1584-483: Is the extensive use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them. Quantum mechanics is of a bigger scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the correspondence principle , there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that

1656-416: Is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the conservation of energy ), and the particle is slowing down. This expression can be further integrated to obtain the position r of

1728-515: Is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is a = d v /d t , the second law can be written in the simplified and more familiar form: So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation , which

1800-511: The forces applied to it. Classical mechanics also describes the more complex motions of extended non-pointlike objects. Euler's laws provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. The rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass". (These generalizations/extensions are derived from Newton's laws, say, by decomposing

1872-429: The forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics . Dynamics goes beyond merely describing objects' behavior and also considers the forces which explain it. Some authors (for example, Taylor (2005) and Greenwood (1997) ) include special relativity within classical dynamics. Another division

1944-572: The kinetic energy of a free particle is E = ⁠ 1 / 2 ⁠ mv , whereas in relativistic mechanics, it is E = ( γ − 1) mc (where γ is the Lorentz factor ; this formula reduces to the Newtonian expression in the low energy limit). For high-energy processes, quantum mechanics must be adjusted to account for special relativity; this has led to the development of quantum field theory . Dynamics (mechanics) This

2016-451: The principle of least action . One result is Noether's theorem , a statement which connects conservation laws to their associated symmetries . Alternatively, a division can be made by region of application: For simplicity, classical mechanics often models real-world objects as point particles , that is, objects with negligible size. The motion of a point particle is determined by a small number of parameters : its position, mass , and

2088-413: The rate of change of displacement with time, is defined as the derivative of the position with respect to time: In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at 60 − 50 = 10 km/h . However, from

2160-434: The revolutions in physics of the early 20th century , all of which revealed limitations in classical mechanics. The earliest formulation of classical mechanics is often referred to as Newtonian mechanics . It consists of the physical concepts based on the 17th century foundational works of Sir Isaac Newton , and the mathematical methods invented by Newton, Gottfried Wilhelm Leibniz , Leonhard Euler and others to describe

2232-463: The speed of light . With objects about the size of an atom's diameter, it becomes necessary to use quantum mechanics . To describe velocities approaching the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity becomes applicable. Some modern sources include relativistic mechanics in classical physics, as representing the field in its most developed and accurate form. Classical mechanics

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2304-565: The stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique . Lagrangian mechanics describes a mechanical system as a pair ( M , L ) {\textstyle (M,L)} consisting of

2376-482: The theory of impetus , which later developed into the modern theories of inertia , velocity , acceleration and momentum . This work and others was developed in 14th-century England by the Oxford Calculators such as Thomas Bradwardine , who studied and formulated various laws regarding falling bodies. The concept that the main properties of a body are uniformly accelerated motion (as of falling bodies)

2448-509: The wave function . The following are described as forming classical mechanics: The following are categorized as being part of quantum mechanics: Historically, classical mechanics had been around for nearly a quarter millennium before quantum mechanics developed. Classical mechanics originated with Isaac Newton 's laws of motion in Philosophiæ Naturalis Principia Mathematica , developed over

2520-545: The 20th century based in part on earlier 19th-century ideas. The development in the modern continuum mechanics, particularly in the areas of elasticity, plasticity, fluid dynamics, electrodynamics, and thermodynamics of deformable media, started in the second half of the 20th century. The often-used term body needs to stand for a wide assortment of objects, including particles , projectiles , spacecraft , stars , parts of machinery , parts of solids , parts of fluids ( gases and liquids ), etc. Other distinctions between

2592-519: The Middle Ages, Aristotle's theories were criticized and modified by a number of figures, beginning with John Philoponus in the 6th century. A central problem was that of projectile motion , which was discussed by Hipparchus and Philoponus. Persian Islamic polymath Ibn Sīnā published his theory of motion in The Book of Healing (1020). He said that an impetus is imparted to a projectile by

2664-531: The Sensorama using a short film piece that detailed a bicycle ride through Brooklyn, created in the 1950s, and still seemed quite impressed by what it could do more than 40 years later. The Sensorama was able to display stereoscopic 3-D images in a wide-angle view, provide body tilting, supply stereo sound , and also had tracks for wind and aromas to be triggered during the film. Heilig was unable to obtain financial backing for his visions and patents , and so

2736-666: The Sensorama work was halted. This technology-related article is a stub . You can help Misplaced Pages by expanding it . Mechanics Mechanics (from Ancient Greek μηχανική ( mēkhanikḗ )  'of machines ') is the area of physics concerned with the relationships between force , matter , and motion among physical objects . Forces applied to objects may result in displacements , which are changes of an object's position relative to its environment. Theoretical expositions of this branch of physics has its origins in Ancient Greece , for instance, in

2808-418: The Sensorama, along with five short films for it to display. The Sensorama was a mechanical device, which includes a stereoscopic color display, fans, odor emitters, stereo‐sound system, and a motional chair. It simulated a motorcycle ride through New York and created the experience by having the spectator sit in an imaginary motorcycle while experiencing the street through the screen, fan-generated wind, and

2880-455: The basis of Newtonian mechanics . There is some dispute over priority of various ideas: Newton's Principia is certainly the seminal work and has been tremendously influential, and many of the mathematics results therein could not have been stated earlier without the development of the calculus. However, many of the ideas, particularly as pertain to inertia and falling bodies, had been developed by prior scholars such as Christiaan Huygens and

2952-426: The behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers , i.e. if quantum mechanics is applied to large systems (for e.g. a baseball), the result would almost be the same if classical mechanics had been applied. Quantum mechanics has superseded classical mechanics at the foundation level and is indispensable for the explanation and prediction of processes at

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3024-456: The distinction between quantum and classical mechanics, Albert Einstein 's general and special theories of relativity have expanded the scope of Newton and Galileo 's formulation of mechanics. The differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a body approaches the speed of light . For instance, in Newtonian mechanics ,

3096-402: The equations of motion solely as a result of the relative acceleration. These forces are referred to as fictitious forces , inertia forces, or pseudo-forces. Consider two reference frames S and S' . For observers in each of the reference frames an event has space-time coordinates of ( x , y , z , t ) in frame S and ( x' , y' , z' , t' ) in frame S' . Assuming time is measured

3168-558: The ideas of other great thinkers of his time and began to calculate motion in terms of distance travelled from some starting position and the time that it took. He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects traveling at speeds close to

3240-447: The less-known medieval predecessors. Precise credit is at times difficult or contentious because scientific language and standards of proof changed, so whether medieval statements are equivalent to modern statements or sufficient proof, or instead similar to modern statements and hypotheses is often debatable. Two main modern developments in mechanics are general relativity of Einstein , and quantum mechanics , both developed in

3312-407: 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. If a constant force F is applied to a particle that makes a displacement Δ r , the work done by the force is defined as the scalar product of the force and displacement vectors: More generally, if the force varies as a function of position as

3384-451: The mayl is spent. He also claimed that a projectile in a vacuum would not stop unless it is acted upon, consistent with Newton's first law of motion. On the question of a body subject to a constant (uniform) force, the 12th-century Jewish-Arab scholar Hibat Allah Abu'l-Barakat al-Baghdaadi (born Nathanel, Iraqi, of Baghdad) stated that constant force imparts constant acceleration. According to Shlomo Pines , al-Baghdaadi's theory of motion

3456-405: The mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames . An inertial frame is an idealized frame of reference within which an object with zero net force acting upon it moves with a constant velocity; that is, it is either at rest or moving uniformly in a straight line. In an inertial frame Newton's law of motion, F = m

3528-468: The molecular, atomic, and sub-atomic level. However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult (mainly due to computational limits) in quantum mechanics and hence remains useful and well used. Modern descriptions of such behavior begin with a careful definition of such quantities as displacement (distance moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however, motion

3600-471: The motion of bodies under the influence of forces . Later, methods based on energy were developed by Euler, Joseph-Louis Lagrange , William Rowan Hamilton and others, leading to the development of analytical mechanics (which includes Lagrangian mechanics and Hamiltonian mechanics ). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics. If

3672-408: The motion of a spacecraft, regarding its orbit and attitude ( rotation ), is described by the relativistic theory of classical mechanics, while the analogous movements of an atomic nucleus are described by quantum mechanics. The following are the three main designations consisting of various subjects that are studied in mechanics. Note that there is also the " theory of fields " which constitutes

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3744-472: The particle as a function of time. Important forces include the gravitational force and the Lorentz force for electromagnetism . In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force F on another particle B , it follows that B must exert an equal and opposite reaction force , − F , on A . The strong form of Newton's third law requires that F and − F act along

3816-409: The particle moves from r 1 to r 2 along a path C , the work done on the particle is given by the line integral If the work done in moving the particle from r 1 to r 2 is the same no matter what path is taken, the force is said to be conservative . Gravity is a conservative force, as is the force due to an idealized spring , as given by Hooke's law . The force due to friction

3888-406: The perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as −10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities ; they must be dealt with using vector analysis . Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = u d and the velocity of

3960-401: The potential energies corresponding to each force The decrease in the potential energy is equal to the increase in the kinetic energy This result is known as conservation of energy and states that the total energy , is constant in time. It is often useful, because many commonly encountered forces are conservative. Lagrangian mechanics is a formulation of classical mechanics founded on

4032-403: The present state of an object that obeys the laws of classical mechanics is known, it is possible to determine how it will move in the future , and how it has moved in the past. Chaos theory shows that the long term predictions of classical mechanics are not reliable. Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching

4104-406: The relationship between force and momentum . Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law": The quantity m v is called the ( canonical ) momentum . The net force on a particle

4176-440: The same direction, this equation can be simplified to: Or, by ignoring direction, the difference can be given in terms of speed only: 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): Acceleration represents the velocity's change over time. Velocity can change in magnitude, direction, or both. Occasionally,

4248-471: The same in all reference frames, 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 u in the x direction, is: This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform ). This group

4320-400: The second object by the vector v = v e , where u is the speed of the first object, v is the speed of the second object, and d and e are unit vectors in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is: Similarly, the first object sees the velocity of the second object as: When both objects are moving in

4392-426: The seventeenth century. Quantum mechanics developed later, over the nineteenth century, precipitated by Planck's postulate and Albert Einstein's explanation of the photoelectric effect . Both fields are commonly held to constitute the most certain knowledge that exists about physical nature. Classical mechanics has especially often been viewed as a model for other so-called exact sciences . Essential in this respect

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4464-458: The simulated noise and smell of the city. These elements are triggered at the appropriate time, such as the release of exhaust chemicals when the rider approached a bus. The petrol fumes and the smell of pizza snack bars were recreated by chemicals. While the machine still functions today, audiences cannot interact with it and it cannot respond based on the user's actions. Howard Rheingold (in his 1991 book Virtual Reality ) spoke of his trial of

4536-427: The speed of light, Newton's laws were superseded by Albert Einstein 's theory of relativity . [A sentence illustrating the computational complication of Einstein's theory of relativity.] For atomic and subatomic particles, Newton's laws were superseded by quantum theory . For everyday phenomena, however, Newton's three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion. Akin to

4608-467: The speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm. The ancient Greek philosophers were among the first to propose that abstract principles govern nature. The main theory of mechanics in antiquity was Aristotelian mechanics , though an alternative theory is exposed in the pseudo-Aristotelian Mechanical Problems , often attributed to one of his successors. There

4680-436: The system, respectively. The stationary action principle requires that the action functional of the system derived from L {\textstyle L} must remain at a stationary point (a maximum , minimum , or saddle ) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Hamiltonian mechanics emerged in 1833 as

4752-435: The thrower, and viewed it as persistent, requiring external forces such as air resistance to dissipate it. Ibn Sina made distinction between 'force' and 'inclination' (called "mayl"), and argued that an object gained mayl when the object is in opposition to its natural motion. So he concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until

4824-541: The various sub-disciplines of mechanics concern the nature of the bodies being described. Particles are bodies with little (known) internal structure, treated as mathematical points in classical mechanics. Rigid bodies have size and shape, but retain a simplicity close to that of the particle, adding just a few so-called degrees of freedom , such as orientation in space. Otherwise, bodies may be semi-rigid, i.e. elastic , or non-rigid, i.e. fluid . These subjects have both classical and quantum divisions of study. For instance,

4896-469: The writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics ). During the early modern period , scientists such as Galileo Galilei , Johannes Kepler , Christiaan Huygens , and Isaac Newton laid the foundation for what is now known as classical mechanics . As a branch of classical physics , mechanics deals with bodies that are either at rest or are moving with velocities significantly less than

4968-415: Was "the oldest negation of Aristotle 's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law of classical mechanics [namely, that a force applied continuously produces acceleration]." Influenced by earlier writers such as Ibn Sina and al-Baghdaadi, the 14th-century French priest Jean Buridan developed

5040-589: Was explained from a very different point of view. For example, following the ideas of Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down because its natural position is in the Earth; the Sun, the Moon, and the stars travel in circles around the Earth because it is the nature of heavenly objects to travel in perfect circles. Often cited as father to modern science, Galileo brought together

5112-413: Was traditionally divided into three main branches. Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment. Kinematics describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering

5184-439: Was worked out by the 14th-century Oxford Calculators . Two central figures in the early modern age are Galileo Galilei and Isaac Newton . Galileo's final statement of his mechanics, particularly of falling bodies, is his Two New Sciences (1638). Newton's 1687 Philosophiæ Naturalis Principia Mathematica provided a detailed mathematical account of mechanics, using the newly developed mathematics of calculus and providing

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