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8,700 | In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for unique factorization domains. | https://en.wikipedia.org/wiki?curid=23666 |
8,701 | The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers formula_181, the ring of complex numbers of the form formula_182 where formula_183 denotes the imaginary unit and formula_36 and formula_41 are arbitrary integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes formula_186 and formula_187. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not. This is a consequence of Fermat's theorem on sums of two squares, | https://en.wikipedia.org/wiki?curid=23666 |
8,702 | which states that an odd prime formula_19 is expressible as the sum of two squares, formula_189, and therefore factorable as formula_190, exactly when formula_19 is 1 mod 4. | https://en.wikipedia.org/wiki?curid=23666 |
8,703 | Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_192 (for integers formula_36 and formula_41) the number formula_195 has two factorizations formula_196, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an ideal, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements. | https://en.wikipedia.org/wiki?curid=23666 |
8,704 | "Prime ideals", which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers. | https://en.wikipedia.org/wiki?curid=23666 |
8,705 | The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. | https://en.wikipedia.org/wiki?curid=23666 |
8,706 | Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers. | https://en.wikipedia.org/wiki?curid=23666 |
8,707 | The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case. | https://en.wikipedia.org/wiki?curid=23666 |
8,708 | In the theory of finite groups the Sylow theorems imply that, if a power of a prime number formula_197 divides the order of a group, then the group has a subgroup of order formula_197. By Lagrange's theorem, any group of prime order is a cyclic group, | https://en.wikipedia.org/wiki?curid=23666 |
8,709 | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics other than the use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. | https://en.wikipedia.org/wiki?curid=23666 |
8,710 | This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public-key cryptography algorithms. | https://en.wikipedia.org/wiki?curid=23666 |
8,711 | These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,712 | The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types. Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments (or all of its prime factors) are called factorization algorithms. | https://en.wikipedia.org/wiki?curid=23666 |
8,713 | Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators. | https://en.wikipedia.org/wiki?curid=23666 |
8,714 | The most basic method of checking the primality of a given integer formula_1 is called "trial division". This method divides formula_1 by each integer from 2 up to the square root of formula_1. Any such integer dividing formula_1 evenly establishes formula_1 as composite; otherwise it is prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,715 | Integers larger than the square root do not need to be checked because, whenever formula_204, one of the two factors formula_36 and formula_41 is less than or equal to the square root of formula_1. Another optimization is to check only primes as factors in this range. | https://en.wikipedia.org/wiki?curid=23666 |
8,716 | For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to formula_208, which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,717 | Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers. However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter. | https://en.wikipedia.org/wiki?curid=23666 |
8,718 | Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed. The oldest method for generating a list of primes is called the sieve of Eratosthenes. The animation shows an optimized variant of this method. | https://en.wikipedia.org/wiki?curid=23666 |
8,719 | Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin. In advanced mathematics, sieve theory applies similar methods to other problems. | https://en.wikipedia.org/wiki?curid=23666 |
8,720 | Some of the fastest modern tests for whether an arbitrary given number formula_1 is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer. | https://en.wikipedia.org/wiki?curid=23666 |
8,721 | For instance the Solovay–Strassen primality test on a given number formula_19 chooses a number formula_36 randomly from formula_136 through formula_213 and uses modular exponentiation to check | https://en.wikipedia.org/wiki?curid=23666 |
8,722 | whether formula_214 is divisible by formula_19. If so, it answers yes and otherwise it answers no. If formula_19 really is prime, it will always answer yes, but if formula_19 is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2. | https://en.wikipedia.org/wiki?curid=23666 |
8,723 | the probability that a composite number could pass the test every time is at most formula_219. Because this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite. | https://en.wikipedia.org/wiki?curid=23666 |
8,724 | In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. | https://en.wikipedia.org/wiki?curid=23666 |
8,725 | The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms, such as the AKS primality test, | https://en.wikipedia.org/wiki?curid=23666 |
8,726 | and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving. | https://en.wikipedia.org/wiki?curid=23666 |
8,727 | When the elliptic curve method concludes that a number is prime, it provides primality certificate that can be verified quickly. | https://en.wikipedia.org/wiki?curid=23666 |
8,728 | The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice. These methods can be used to generate large random prime numbers, by generating and testing random numbers until finding one that is prime; | https://en.wikipedia.org/wiki?curid=23666 |
8,729 | when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,730 | The following table lists some of these tests. Their running time is given in terms of formula_1, the number to be tested and, for probabilistic algorithms, the number formula_72 of tests performed. Moreover, formula_222 is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters formula_1 and formula_72. | https://en.wikipedia.org/wiki?curid=23666 |
8,731 | In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. | https://en.wikipedia.org/wiki?curid=23666 |
8,732 | For example, the Lucas–Lehmer primality test can determine whether a Mersenne number (one less than a power of two) is prime, deterministically, | https://en.wikipedia.org/wiki?curid=23666 |
8,733 | in the same time as a single iteration of the Miller–Rabin test. This is why since 1992 () the largest "known" prime has always been a Mersenne prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,734 | The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. | https://en.wikipedia.org/wiki?curid=23666 |
8,735 | Given a composite integer formula_1, the task of providing one (or all) prime factors is referred to as "factorization" of formula_1. It is significantly more difficult than primality testing, and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and Pollard's rho algorithm can be used to find very small factors of formula_1, and elliptic curve factorization can be effective when formula_1 has factors of moderate size. Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve. the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,736 | Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer. However, current technology can only run this algorithm for very small numbers. the largest number that has been factored by a quantum computer running Shor's algorithm is 21. | https://en.wikipedia.org/wiki?curid=23666 |
8,737 | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (2048-bit primes are common). RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers formula_21 and formula_123 than to calculate formula_21 and formula_123 (assumed coprime) if only the product formula_176 is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing formula_234), while the reverse operation (the discrete logarithm) is thought to be a hard problem. | https://en.wikipedia.org/wiki?curid=23666 |
8,738 | Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to formula_72-independent hashing by using higher-degree polynomials, again modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table. | https://en.wikipedia.org/wiki?curid=23666 |
8,739 | Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than formula_236. | https://en.wikipedia.org/wiki?curid=23666 |
8,740 | Prime numbers are also used in pseudorandom number generators including linear congruential generators and the Mersenne Twister. | https://en.wikipedia.org/wiki?curid=23666 |
8,741 | Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area. Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square. | https://en.wikipedia.org/wiki?curid=23666 |
8,742 | The concept of a prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The prime decomposition of 3-manifolds is another example of this type. | https://en.wikipedia.org/wiki?curid=23666 |
8,743 | Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas. | https://en.wikipedia.org/wiki?curid=23666 |
8,744 | with formula_72 a nonnegative integer. They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but formula_239 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular formula_1-gon is constructible using straightedge and compass if and only if the odd prime factors of formula_1 (if any) are distinct Fermat primes. Likewise, a regular formula_1-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of formula_1 are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form formula_244. | https://en.wikipedia.org/wiki?curid=23666 |
8,745 | It is possible to partition any convex polygon into formula_1 smaller convex polygons of equal area and equal perimeter, when formula_1 is a power of a prime number, but this is not known for other values of formula_1. | https://en.wikipedia.org/wiki?curid=23666 |
8,746 | Beginning with the work of Hugh Montgomery and Freeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems. Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures. | https://en.wikipedia.org/wiki?curid=23666 |
8,747 | The evolutionary strategy used by cicadas of the genus "Magicicada" makes use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles. | https://en.wikipedia.org/wiki?curid=23666 |
8,748 | In contrast, the multi-year periods between flowering in bamboo plants are hypothesized to be smooth numbers, having only small prime numbers in their factorizations. | https://en.wikipedia.org/wiki?curid=23666 |
8,749 | The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as "La Nativité du Seigneur" (1935) and "Quatre études de rythme" (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations". | https://en.wikipedia.org/wiki?curid=23666 |
8,750 | In his science fiction novel "Contact", scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975. In the novel "The Curious Incident of the Dog in the Night-Time" by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel "The Solitude of Prime Numbers", in which they are portrayed as "outsiders" among integers. | https://en.wikipedia.org/wiki?curid=23666 |
8,751 | Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: | https://en.wikipedia.org/wiki?curid=55212 |
8,752 | The three laws of motion were first stated by Isaac Newton in his "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), originally published in 1687. Newton used them to investigate and explain the motion of many physical objects and systems, which laid the foundation for classical mechanics. In the time since Newton, the conceptual content of classical physics has been reformulated in alternative ways, involving different mathematical approaches that have yielded insights which were obscured in the original, Newtonian formulation. Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds (special relativity), are very massive (general relativity), or are very small (quantum mechanics). | https://en.wikipedia.org/wiki?curid=55212 |
8,753 | Newton's laws are often stated in terms of "point" or "particle" masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface. | https://en.wikipedia.org/wiki?curid=55212 |
8,754 | The mathematical description of motion, or kinematics, is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body's trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by a single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or origin, with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body's location as a function of time is formula_1, then its average velocity over the time interval from formula_2 to formula_3 is formula_4Here, the Greek letter formula_5 (delta) is used, per tradition, to mean "change in". A positive average velocity means that the position coordinate formula_6 increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends the time interval in the same place as it began. Calculus gives the means to define an "instantaneous" velocity, a measure of a body's speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace formula_5 with the symbol formula_8, for example,formula_9This denotes that the instantaneous velocity is the derivative of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position formula_10 to the infinitesimally small time interval formula_11 over which it occurs. More carefully, the velocity and all other derivatives can be defined using the concept of a limit. A function formula_12 has a limit of formula_13 at a given input value formula_2 if the difference between formula_15 and formula_13 can be made arbitrarily small by choosing an input sufficiently close to formula_2. One writes, formula_18Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero:formula_19 "Acceleration" is to velocity as velocity is to position: it is the derivative of the velocity with respect to time. Acceleration can likewise be defined as a limit:formula_20Consequently, the acceleration is the "second derivative" of position, often written formula_21. | https://en.wikipedia.org/wiki?curid=55212 |
8,755 | Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction. Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in formula_22, or in bold typeface, such as formula_23. Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and the magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body's velocity vector might be formula_24, indicating that it is moving at 3 metres per second along a horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different coordinate system will be represented by different numbers, and vector algebra can be used to translate between these alternatives. | https://en.wikipedia.org/wiki?curid=55212 |
8,756 | The physics concept of "force" makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity. | https://en.wikipedia.org/wiki?curid=55212 |
8,757 | Newton's first law expresses the principle of inertia: the natural behavior of a body is to move in a straight line at constant speed. In the absence of outside influences, a body's motion preserves the status quo. | https://en.wikipedia.org/wiki?curid=55212 |
8,758 | The modern understanding of Newton's first law is that no inertial observer is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example, a person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger "feels" no motion. The principle expressed by Newton's first law is that there is no way to say which inertial observer is "really" moving and which is "really" standing still. One observer's state of rest is another observer's state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest. | https://en.wikipedia.org/wiki?curid=55212 |
8,759 | By "motion", Newton meant the quantity now called momentum, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving. In modern notation, the momentum of a body is the product of its mass and its velocity: | https://en.wikipedia.org/wiki?curid=55212 |
8,760 | If the mass formula_27 does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of the velocity, which is the acceleration: | https://en.wikipedia.org/wiki?curid=55212 |
8,761 | As the acceleration is the second derivative of position with respect to time, this can also be written | https://en.wikipedia.org/wiki?curid=55212 |
8,762 | A common visual representation of forces acting in concert is the free body diagram, which schematically portrays a body of interest and the forces applied to it by outside influences. For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, "normal" force, friction, and string tension. | https://en.wikipedia.org/wiki?curid=55212 |
8,763 | Newton's second law is sometimes presented as a "definition" of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. In order for this to be more than a tautology — acceleration implies force, force implies acceleration — some other statement about force must also be made. For example, an equation detailing the force might be specified, like Newton's law of universal gravitation. By inserting such an expression for formula_30 into Newton's second law, an equation with predictive power can be written. Newton's second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter. | https://en.wikipedia.org/wiki?curid=55212 |
8,764 | Overly brief paraphrases of the third law, like "action equals reaction" might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is "not" the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth. | https://en.wikipedia.org/wiki?curid=55212 |
8,765 | Newton's third law relates to a more fundamental principle, the conservation of momentum. The latter remains true even in cases where Newton's statement does not, for instance when force fields as well as material bodies carry momentum, and when momentum is defined properly, in quantum mechanics as well. In Newtonian mechanics, if two bodies have momenta formula_31 and formula_32 respectively, then the total momentum of the pair is formula_33, and the rate of change of formula_34 is formula_35 By Newton's second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and formula_34 is constant. Alternatively, if formula_34 is known to be constant, it follows that the forces have equal magnitude and opposite direction. | https://en.wikipedia.org/wiki?curid=55212 |
8,766 | Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies is the sum of their individual masses. Frank Wilczek has suggested calling attention to this assumption by designating it "Newton's Zeroth Law". Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant. Likewise, the idea that forces add like vectors (or in other words obey the superposition principle), and the idea that forces change the energy of a body, have both been described as a "fourth law". | https://en.wikipedia.org/wiki?curid=55212 |
8,767 | Physicists developed the concept of energy after Newton's time, but it has become an inseparable part of what is considered "Newtonian" physics. Energy can broadly be classified into kinetic, due to a body's motion, and potential, due to a body's position relative to others. Thermal energy, the energy carried by heat flow, is a type of kinetic energy not associated with the macroscopic motion of objects but instead with the movements of the atoms and molecules of which they are made. According to the work-energy theorem, when a force acts upon a body while that body moves along the line of the force, the force does "work" upon the body, and the amount of work done is equal to the change in the body's kinetic energy. In many cases of interest, the net work done by a force when a body moves in a closed loop — starting at a point, moving along some trajectory, and returning to the initial point — is zero. If this is the case, then the force can be written in terms of the gradient of a function called a scalar potential: | https://en.wikipedia.org/wiki?curid=55212 |
8,768 | This is true for many forces including that of gravity, but not for friction; indeed, almost any problem in a mechanics textbook that does not involve friction can be expressed in this way. The fact that the force can be written in this way can be understood from the conservation of energy. Without friction to dissipate a body's energy into heat, the body's energy will trade between potential and (non-thermal) kinetic forms while the total amount remains constant. Any gain of kinetic energy, which occurs when the net force on the body accelerates it to a higher speed, must be accompanied by a loss of potential energy. So, the net force upon the body is determined by the manner in which the potential energy decreases. | https://en.wikipedia.org/wiki?curid=55212 |
8,769 | If a body falls from rest near the surface of the Earth, then in the absence of air resistance, it will accelerate at a constant rate. This is known as free fall. The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time. Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his law of universal gravitation. The latter states that the magnitude of the gravitational force from the Earth upon the body is | https://en.wikipedia.org/wiki?curid=55212 |
8,770 | where formula_27 is the mass of the falling body, formula_41 is the mass of the Earth, formula_42 is Newton's constant, and formula_43 is the distance from the center of the Earth to the body's location, which is very nearly the radius of the Earth. Setting this equal to formula_44, the body's mass formula_27 cancels from both sides of the equation, leaving an acceleration that depends upon formula_42, formula_41, and formula_43, and formula_43 can be taken to be constant. This particular value of acceleration is typically denoted formula_50: | https://en.wikipedia.org/wiki?curid=55212 |
8,771 | If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes projectile motion. When air resistance can be neglected, projectiles follow parabola-shaped trajectories, because gravity affects the body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is formula_50 downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students. | https://en.wikipedia.org/wiki?curid=55212 |
8,772 | When a body is in uniform circular motion, the force on it changes the direction of its motion but not its speed. For a body moving in a circle of radius formula_43 at a constant speed formula_54, its acceleration has a magnitudeformula_55and is directed toward the center of the circle. The force required to sustain this acceleration, called the centripetal force, is therefore also directed toward the center of the circle and has magnitude formula_56. Many orbits, such as that of the Moon around the Earth, can be approximated by uniform circular motion. In such cases, the centripetal force is gravity, and by Newton's law of universal gravitation has magnitude formula_57, where formula_41 is the mass of the larger body being orbited. Therefore, the mass of a body can be calculated from observations of another body orbiting around it. | https://en.wikipedia.org/wiki?curid=55212 |
8,773 | Newton's cannonball is a thought experiment that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because the force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then the curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles). | https://en.wikipedia.org/wiki?curid=55212 |
8,774 | Consider a body of mass formula_27 able to move along the formula_60 axis, and suppose an equilibrium point exists at the position formula_61. That is, at formula_61, the net force upon the body is the zero vector, and by Newton's second law, the body will not accelerate. If the force upon the body is proportional to the displacement from the equilibrium point, and directed to the equilibrium point, then the body will perform simple harmonic motion. Writing the force as formula_63, Newton's second law becomes | https://en.wikipedia.org/wiki?curid=55212 |
8,775 | where the frequency formula_66 is equal to formula_67, and the constants formula_68 and formula_69 can be calculated knowing, for example, the position and velocity the body has at a given time, like formula_70. | https://en.wikipedia.org/wiki?curid=55212 |
8,776 | One reason that the harmonic oscillator is a conceptually important example is that it is good approximation for many systems near a stable mechanical equilibrium. For example, a pendulum has a stable equilibrium in the vertical position: if motionless there, it will remain there, and if pushed slightly, it will swing back and forth. Neglecting air resistance and friction in the pivot, the force upon the pendulum is gravity, and Newton's second law becomes formula_71where formula_13 is the length of the pendulum and formula_73 is its angle from the vertical. When the angle formula_73 is small, the sine of formula_73 is nearly equal to formula_73 (see Taylor series), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency formula_77. | https://en.wikipedia.org/wiki?curid=55212 |
8,777 | A harmonic oscillator can be "damped," often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be "driven" by an applied force, which can lead to the phenomenon of resonance. | https://en.wikipedia.org/wiki?curid=55212 |
8,778 | Newtonian physics treats matter as being neither created nor destroyed, though it may be rearranged. It can be the case that an object of interest gains or loses mass because matter is added to or removed from it. In such a situation, Newton's laws can be applied to the individual pieces of matter, keeping track of which pieces belong to the object of interest over time. For instance, if a rocket of mass formula_78, moving at velocity formula_79, ejects matter at a velocity formula_80 relative to the rocket, then | https://en.wikipedia.org/wiki?curid=55212 |
8,779 | A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of the body's center of mass and movement around the center of mass. | https://en.wikipedia.org/wiki?curid=55212 |
8,780 | Significant aspects of the motion of an extended body can be understood by imagining the mass of that body concentrated to a single point, known as the center of mass. The location of a body's center of mass depends upon how that body's material is distributed. For a collection of pointlike objects with masses formula_83 at positions formula_84, the center of mass is located at formula_85 where formula_41 is the total mass of the collection. In the absence of a net external force, the center of mass moves at a constant speed in a straight line. This applies, for example, to a collision between two bodies. If the total external force is not zero, then the center of mass changes velocity as though it were a point body of mass formula_41. This follows from the fact that the internal forces within the collection, the forces that the objects exert upon each other, occur in balanced pairs by Newton's third law. In a system of two bodies with one much more massive than the other, the center of mass will approximately coincide with the location of the more massive body. | https://en.wikipedia.org/wiki?curid=55212 |
8,781 | When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque. | https://en.wikipedia.org/wiki?curid=55212 |
8,782 | Angular momentum is calculated with respect to a reference point. If the displacement vector from a reference point to a body is formula_88 and the body has momentum formula_34, then the body's angular momentum with respect to that point is, using the vector cross product, formula_90 Taking the time derivative of the angular momentum gives formula_91 The first term vanishes because formula_92 and formula_93 point in the same direction. The remaining term is the torque, formula_94 When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant. The torque can vanish even when the force is non-zero, if the body is located at the reference point (formula_95) or if the force formula_30 and the displacement vector formula_88 are directed along the same line. | https://en.wikipedia.org/wiki?curid=55212 |
8,783 | The angular momentum of a collection of point masses, and thus of an extended body, is found by adding the contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation. | https://en.wikipedia.org/wiki?curid=55212 |
8,784 | Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them. The size of the attracting force is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Finding the shape of the orbits that an inverse-square force law will produce is known as the Kepler problem. The Kepler problem can be solved in multiple ways, including by demonstrating that the Laplace–Runge–Lenz vector is constant, or by applying a duality transformation to a 2-dimensional harmonic oscillator. However it is solved, the result is that orbits will be conic sections, that is, ellipses (including circles), parabolas, or hyperbolas. The eccentricity of the orbit, and thus the type of conic section, is determined by the energy and the angular momentum of the orbiting body. Planets do not have sufficient energy to escape the Sun, and so their orbits are ellipses, to a good approximation; because the planets pull on one another, actual orbits are not exactly conic sections. | https://en.wikipedia.org/wiki?curid=55212 |
8,785 | If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in closed form. That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time. Numerical methods can be applied to obtain useful, albeit approximate, results for the three-body problem. The positions and velocities of the bodies can be stored in variables within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process is looped to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation. | https://en.wikipedia.org/wiki?curid=55212 |
8,786 | Newton's laws of motion allow the possibility of chaos. That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of a system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the double pendulum, dynamical billiards, and the Fermi–Pasta–Ulam–Tsingou problem. | https://en.wikipedia.org/wiki?curid=55212 |
8,787 | Newton's laws can be applied to fluids by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The Euler momentum equation is an expression of Newton's second law adapted to fluid dynamics. A fluid is described by a velocity field, i.e., a function formula_98 that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because the velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration formula_99 has two terms, a combination known as a "total" or "material" derivative. The mass of an infinitesimal portion depends upon the fluid density, and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, formula_100 becomes | https://en.wikipedia.org/wiki?curid=55212 |
8,788 | where formula_102 is the density, formula_103 is the pressure, and formula_104 stands for an external influence like a gravitational pull. Incorporating the effect of viscosity turns the Euler equation into a Navier–Stokes equation: | https://en.wikipedia.org/wiki?curid=55212 |
8,789 | It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time. This unphysical behavior, known as a "noncollision singularity", depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as the lack of a relativistic speed limit in Newtonian physics. | https://en.wikipedia.org/wiki?curid=55212 |
8,790 | It is not yet known whether or not the Euler and Navier–Stokes equations exhibit the analogous behavior of initially smooth solutions "blowing up" in finite time. The question of existence and smoothness of Navier–Stokes solutions is one of the Millennium Prize Problems. | https://en.wikipedia.org/wiki?curid=55212 |
8,791 | Classical mechanics can be mathematically formulated in multiple different ways, other than the "Newtonian" description (which itself, of course, incorporates contributions from others both before and after Newton). The physical content of these different formulations is the same as the Newtonian, but they provide different insights and facilitate different types of calculations. For example, Lagrangian mechanics helps make apparent the connection between symmetries and conservation laws, and it is useful when calculating the motion of constrained bodies, like a mass restricted to move along a curving track or on the surface of a sphere. Hamiltonian mechanics is convenient for statistical physics, leads to further insight about symmetry, and can be developed into sophisticated techniques for perturbation theory. Due to the breadth of these topics, the discussion here will be confined to concise treatments of how they reformulate Newton's laws of motion. | https://en.wikipedia.org/wiki?curid=55212 |
8,792 | Lagrangian mechanics differs from the Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant. It is traditional in Lagrangian mechanics to denote position with formula_107 and velocity with formula_108. The simplest example is a massive point particle, the Lagrangian for which can be written as the difference between its kinetic and potential energies: | https://en.wikipedia.org/wiki?curid=55212 |
8,793 | and the potential energy is some function of the position, formula_111. The physical path that the particle will take between an initial point formula_112 and a final point formula_113 is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has the property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. Calculus of variations provides the mathematical tools for finding this path. Applying the calculus of variations to the task of finding the path yields the Euler–Lagrange equation for the particle, | https://en.wikipedia.org/wiki?curid=55212 |
8,794 | which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy. | https://en.wikipedia.org/wiki?curid=55212 |
8,795 | Landau and Lifshitz argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws. Lagrangian mechanics provides a convenient framework in which to prove Noether's theorem, which relates symmetries and conservation laws. The conservation of momentum can be derived by applying Noether's theorem to a Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption. | https://en.wikipedia.org/wiki?curid=55212 |
8,796 | In Hamiltonian mechanics, the dynamics of a system are represented by a function called the Hamiltonian, which in many cases of interest is equal to the total energy of the system. The Hamiltonian is a function of the positions and the momenta of all the bodies making up the system, and it may also depend explicitly upon time. The time derivatives of the position and momentum variables are given by partial derivatives of the Hamiltonian, via Hamilton's equations. The simplest example is a point mass formula_27 constrained to move in a straight line, under the effect of a potential. Writing formula_107 for the position coordinate and formula_118 for the body's momentum, the Hamiltonian is | https://en.wikipedia.org/wiki?curid=55212 |
8,797 | which reproduces the familiar statement that a body's momentum is the product of its mass and velocity. The time derivative of the momentum is | https://en.wikipedia.org/wiki?curid=55212 |
8,798 | which, upon identifying the negative derivative of the potential with the force, is just Newton's second law once again. | https://en.wikipedia.org/wiki?curid=55212 |
8,799 | As in the Lagrangian formulation, in Hamiltonian mechanics the conservation of momentum can be derived using Noether's theorem, making Newton's third law an idea that is deduced rather than assumed. | https://en.wikipedia.org/wiki?curid=55212 |
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