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76-655: Due to Aavasaksa's distinctive elevation above other nearby hills, it was first used by Pierre Louis Maupertuis in the French Geodesic Mission (1736–1737), and later became part of the Struve Geodetic Arc . As a result of this, UNESCO named Aavasaksa a World Heritage Site , along with the 33 other sites used in the Struve Geodetic Arc. Aavasaksa is often considered the southernmost point in Finland where

152-494: A Lagrangian describing the physical system. The accumulated value of this energy function between two states of the system is called the action . Action principles apply the calculus of variation to the action. The action depends on the energy function, and the energy function depends on the position, motion, and interactions in the system: variation of the action allows the derivation of the equations of motion without vector or forces. Several distinct action principles differ in

228-424: A Lagrangian imply a continuous symmetry and conversely. For examples, a Lagrangian independent of time corresponds to a system with conserved energy; spatial translation independence implies momentum conservation; angular rotation invariance implies angular momentum conservation. These examples are global symmetries, where the independence is itself independent of space or time; more general local symmetries having

304-399: A different Lagrangian: the principle itself is independent of coordinate systems. The explanatory diagrams in force-based mechanics usually focus on a single point, like the center of momentum , and show vectors of forces and velocities. The explanatory diagrams of action-based mechanics have two points with actual and possible paths connecting them. These diagrammatic conventions reiterate

380-440: A functional dependence on space or time lead to gauge theory . The observed conservation of isospin was used by Yang Chen-Ning and Robert Mills in 1953 to construct a gauge theory for mesons , leading some decades later to modern particle physics theory . Action principles apply to a wide variety of physical problems, including all of fundamental physics. The only major exceptions are cases involving friction or when only

456-566: A major role for the inheritor quantity of ‘live force’. For Maupertuis, however, it was important to retain the concept of the hard body. And the beauty of his principle of least action was that it applied just as well to hard and elastic bodies. Since he had shown that the principle also applied to systems of bodies at rest and to light, it seemed that it was truly universal. The final stage of his argument came when Maupertuis set out to interpret his principle in cosmological terms. ‘Least action’ sounds like an economy principle, roughly equivalent to

532-461: A path according to the "action", a continuous sum or integral of the Lagrangian along the path. Introductory study of mechanics, the science of interacting objects, typically begins with Newton's laws based on the concept of force , defined by the acceleration it causes when applied to mass : F = m a . {\displaystyle F=ma.} This approach to mechanics focuses on

608-620: A probability amplitude at a different point later in time: ψ ( x k + 1 , t + ε ) = 1 A ∫ e i S ( x k + 1 , x k ) / ℏ ψ ( x k , t ) d x k , {\displaystyle \psi (x_{k+1},t+\varepsilon )={\frac {1}{A}}\int e^{iS(x_{k+1},x_{k})/\hbar }\psi (x_{k},t)\,dx_{k},} where S ( x k + 1 , x k ) {\displaystyle S(x_{k+1},x_{k})}

684-528: A quantity which he again assimilated to action. Finally, in 1746 he gave a further paper, the Loix du mouvement et du repos ( Laws of movement and rest ), this time to the Berlin Academy of Sciences, which showed that point masses also minimise action. Point masses are bodies that can be treated for the purposes of analysis as being a certain amount of matter (a mass) concentrated at a single point. A major debate in

760-460: A rigid body with no net force, the actions are identical, and the variational principles become equivalent to Fermat's principle of least time: δ ( t 2 − t 1 ) = 0. {\displaystyle \delta (t_{2}-t_{1})=0.} When the physics problem gives the two endpoints as a position and a time, that is as events , Hamilton's action principle applies. For example, imagine planning

836-540: A rocket to the Moon today, how can it land there in 5 days? The Newtonian and action-principle forms are equivalent, and either one can solve the same problems, but selecting the appropriate form will make solutions much easier. The energy function in the action principles is not the total energy ( conserved in an isolated system ), but the Lagrangian , the difference between kinetic and potential energy. The kinetic energy combines

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912-409: A single point in space and time, attempting to answer the question: "What happens next?". Mechanics based on action principles begin with the concept of action , an energy tradeoff between kinetic energy and potential energy , defined by the physics of the problem. These approaches answer questions relating starting and ending points: Which trajectory will place a basketball in the hoop? If we launch

988-463: A small number found themselves constructed in such a manner that the parts of the animal were able to satisfy its needs; in another infinitely greater number, there was neither fitness nor order: all of these latter have perished. Animals lacking a mouth could not live; others lacking reproductive organs could not perpetuate themselves; the only ones that remained are those in which order and fitness were found; and these species, which we see today, are but

1064-1134: A trip to the Moon. During your voyage the Moon will continue its orbit around the Earth: it's a moving target. Hamilton's principle for objects at positions q ( t ) {\displaystyle \mathbf {q} (t)} is written mathematically as ( δ S ) Δ t = 0 ,   w h e r e   S [ q ]   = d e f   ∫ t 1 t 2 L ( q ( t ) , q ˙ ( t ) , t ) d t . {\displaystyle (\delta {\mathcal {S}})_{\Delta t}=0,\ \mathrm {where} \ {\mathcal {S}}[\mathbf {q} ]\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt.} The constraint Δ t = t 2 − t 1 {\displaystyle \Delta t=t_{2}-t_{1}} means that we only consider paths taking

1140-412: A universal principle of wisdom provides an undeniable proof of the shaping of the universe by a wise creator. Hence the principle of least action is not just the culmination of Maupertuis's work in several areas of physics, he sees it as his most important achievement in philosophy too, giving an incontrovertible proof of God. The flaws in his reasoning are principally that there is no obvious reason why

1216-490: A year later. Maupertuis's difficult disposition involved him in constant quarrels, of which his controversies with Samuel König and Voltaire during the latter part of his life are examples. Some historians of science point to his work in biology as a significant precursor to the development of evolutionary theory, specifically the theory of natural selection . Other writers contend that his remarks are cursory, vague, or incidental to that particular argument. Mayr's verdict

1292-401: Is a second velocity term corresponding to their actual motion. This was anathema to Cartesians and Newtonians. An inherent tendency towards motion was an ‘occult quality’ of the kind of favoured by mediaeval scholastics and to be resisted at all costs. Today the concept of a ‘hard’ body is rejected; and mass times the square of velocity is just twice kinetic energy so modern mechanics reserves

1368-512: Is a translation from the Essai de cosmologie , followed by the original French passage: But could one not say that, in the fortuitous combinations of the productions of nature, as there were but some where certain relations of fitness were present which be able to subsist, it is not to be wondered at that this fitness is present in all the species that are currently in existence? Chance, it may be said, had produced an innumerable multitude of individuals;

1444-407: Is equivalent of the integral over time of the live force. Leibniz had already shown that this quantity is likely to be either minimised or maximised in natural phenomena. Minimising this quantity could conceivably demonstrate economy, but how could maximising it? (See also the corresponding principles of stationary actions by Lagrange and Hamilton ). In Universal Natural History and Theory of

1520-403: Is fixed during the variation, but not the time, the reverse of the constraints on Hamilton's principle. Consequently, the same path and end points take different times and energies in the two forms. The solutions in the case of this form of Maupertuis's principle are orbits : functions relating coordinates to each other in which time is simply an index or a parameter. For time-invariant system,

1596-469: Is the Einstein gravitational constant . The action principle is so central in modern physics and mathematics that it is widely applied including in thermodynamics , fluid mechanics , the theory of relativity , quantum mechanics , particle physics , and string theory . The action principle is preceded by earlier ideas in optics . In ancient Greece , Euclid wrote in his Catoptrica that, for

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1672-410: Is the Lagrangian . Some textbooks write ( δ W ) E = 0 {\displaystyle (\delta W)_{E}=0} as Δ S 0 {\displaystyle \Delta S_{0}} , to emphasize that the variation used in this form of the action principle differs from Hamilton's variation . Here the total energy E {\displaystyle E}

1748-556: Is the classical action. Instead of single path with stationary action, all possible paths add (the integral over x k {\displaystyle x_{k}} ), weighted by a complex probability amplitude e i S / ℏ {\displaystyle e^{iS/\hbar }} . The phase of the amplitude is given by the action divided by the Planck constant or quantum of action: S / ℏ {\displaystyle S/\hbar } . When

1824-545: The Académie des Sciences . His early mathematical work revolved around the vis viva controversy, for which Maupertuis developed and extended the work of Isaac Newton (whose theories were not yet widely accepted outside England) and argued against the waning Cartesian mechanics. In the 1730s, the shape of the Earth became a flashpoint in the battle among rival systems of mechanics. Maupertuis, based on his exposition of Newton (with

1900-664: The Battle of Mollwitz on a donkey , where he was taken prisoner by the Austrians. On his release he returned to Berlin, and thence to Paris , where he was elected director of the Academy of Sciences in 1742, and in the following year was admitted into the Académie française . Returning to Berlin in 1744, again at the desire of Frederick II, he was chosen president of the Royal Prussian Academy of Sciences in 1746, which he controlled with

1976-620: The abbreviated action W [ q ]   = def   ∫ q 1 q 2 p ⋅ d q , {\displaystyle W[\mathbf {q} ]\ {\stackrel {\text{def}}{=}}\ \int _{q_{1}}^{q_{2}}\mathbf {p} \cdot \mathbf {dq} ,} (sometimes written S 0 {\displaystyle S_{0}} ), where p = ( p 1 , p 2 , … , p N ) {\displaystyle \mathbf {p} =(p_{1},p_{2},\ldots ,p_{N})} are

2052-493: The midnight sun is literally visible. The hill is surrounded by rivers running next to it: Torne River to the west and the smaller Tengeliö river  [ fi ] to the east and north. Asteroid 2678 Aavasaksa is named after the hill. [REDACTED] Media related to Aavasaksa at Wikimedia Commons Pierre Louis Maupertuis Pierre Louis Moreau de Maupertuis ( / ˌ m oʊ p ər ˈ t w iː / ; French: [mopɛʁtɥi] ; 1698 – 27 July 1759)

2128-540: The Heavens , Immanuel Kant quotes Maupertuis' 1745 discussion of nebula -like objects, which Maupertuis notes are actually collections of stars, including Andromeda . Arthur Schopenhauer suggested that Immanuel Kant 's "most important and brilliant doctrine"—contained in the Critique of Pure Reason (1781)—was asserted by Maupertuis: But what are we to say when we find Kant's most important and brilliant doctrine, that of

2204-455: The Paris Academy, he gave his Accord de plusieurs lois naturelles qui avaient paru jusqu'ici incompatibles ( Agreement of several natural laws that had hitherto seemed to be incompatible ) to show that the behaviour of light during refraction – when it bends on entering a new medium – was such that the total path it followed, from a point in the first medium to a point in the second, minimised

2280-528: The action S {\displaystyle S} relates simply to the abbreviated action W {\displaystyle W} on the stationary path as Δ S = Δ W − E Δ t {\displaystyle \Delta S=\Delta W-E\Delta t} for energy E {\displaystyle E} and time difference Δ t = t 2 − t 1 {\displaystyle \Delta t=t_{2}-t_{1}} . For

2356-534: The action integral builds in value from zero at the starting point to its final value at the end. Any nearby path has similar values at similar distances from the starting point. Lines or surfaces of constant partial action value can be drawn across the paths, creating a wave-like view of the action. Analysis like this connects particle-like rays of geometrical optics with the wavefronts of Huygens–Fresnel principle . [Maupertuis] ... thus pointed to that remarkable analogy between optical and mechanical phenomena which

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2432-455: The action of a particle is much larger than ℏ {\displaystyle \hbar } , S / ℏ ≫ 1 {\displaystyle S/\hbar \gg 1} , the phase changes rapidly along the path: the amplitude averages to a small number. Thus the Planck constant sets the boundary between classical and quantum mechanics. All of the paths contribute in

2508-640: The constraints on their initial and final conditions. The names of action principles have evolved over time and differ in details of the endpoints of the paths and the nature of the variation. Quantum action principles generalize and justify the older classical principles. Action principles are the basis for Feynman's version of quantum mechanics , general relativity and quantum field theory . The action principles have applications as broad as physics, including many problems in classical mechanics but especially in modern problems of quantum mechanics and general relativity. These applications built up over two centuries as

2584-588: The controversy in his favour. The book included an adventure narrative of the expedition, and an account of the Käymäjärvi Inscriptions in Sweden. On his return home he became a member of almost all the scientific societies of Europe. After the Lapland expedition, Maupertuis set about generalising his earlier mathematical work, proposing the principle of least action as a metaphysical principle that underlies all

2660-409: The different strong points of each method. Depending on the action principle, the two points connected by paths in a diagram may represent two particle positions at different times, or the two points may represent values in a configuration space or in a phase space . The mathematical technology and terminology of action principles can be learned by thinking in terms of physical space, then applied in

2736-416: The distance it had travelled and the velocity at which it was travelling. In 1741, he gave a paper to the Paris Academy of Sciences, Loi du repos des corps , ( Law of bodies at rest ). In it he showed that a system of bodies at rest tends to reach a position in which any change would create the smallest possible change in a quantity that he argued could be assimilated to action. In 1744, in another paper to

2812-521: The early part of the eighteenth century concerned the behaviour of such bodies in collisions. Cartesian and Newtonian physicists argued that in their collisions, point masses conserved both momentum and relative velocity. Leibnizians, on the other hand, argued that they also conserved what was called live force or vis viva . This was unacceptable for their opponents for two reasons: the first that live force conservation did not apply to so-called ‘hard’ bodies, bodies that were totally incompressible, whereas

2888-424: The endpoints are fixed, Maupertuis's least action principle applies. For example, to score points in basketball the ball must leave the shooters hand and go through the hoop, but the time of the flight is not constrained. Maupertuis's least action principle is written mathematically as the stationary condition ( δ W ) E = 0 {\displaystyle (\delta W)_{E}=0} on

2964-502: The energy of motion for all the objects in the system; the potential energy depends upon the instantaneous position of the objects and drives the motion of the objects. The motion of the objects places them in new positions with new potential energy values, giving a new value for the Lagrangian. Using energy rather than force gives immediate advantages as a basis for mechanics. Force mechanics involves 3-dimensional vector calculus , with 3 space and 3 momentum coordinates for each object in

3040-452: The help of Leonhard Euler until his death. His position became extremely awkward with the outbreak of the Seven Years' War between his home country and his patron's, and his reputation suffered in both Paris and Berlin . Finding his health declining, he retired in 1757 to the south of France with a young girl, leaving his wife and children behind and went in 1758 to Basel , where he died

3116-416: The help of his mentor Johan Bernoulli ) predicted that the Earth should be oblate , while his rival Jacques Cassini measured it astronomically to be prolate . In 1736 Maupertuis acted as chief of the French Geodesic Mission sent by King Louis XV to Lapland to measure the length of a degree of arc of the meridian . His results, which he published in a book detailing his procedures, essentially settled

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3192-490: The idea of economy of effort in daily life. A universal principle of economy of effort would seem to display the working of wisdom in the very construction of the universe. This seems, in Maupertuis's view, a more powerful argument for the existence of an infinitely wise creator than any other that can be advanced. He published his thinking on these matters in his Essai de cosmologie ( Essay on cosmology ) of 1750. He shows that

3268-569: The ideality of space and of the merely phenomenal existence of the corporeal world, expressed already thirty years previously by Maupertuis? ... Maupertuis expresses this paradoxical doctrine so decidedly, and yet without the addition of proof, that it must be supposed that he also obtained it from somewhere else. Principle of least action Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called

3344-434: The initial position and velocities are given. Different action principles have different meaning for the variations; each specific application of an action principle requires a specific Lagrangian describing the physics. A common name for any or all of these principles is "the principle of least action". For a discussion of the names and historical origin of these principles see action principle names . When total energy and

3420-404: The laws of mechanics. He also expanded into the biological realm, anonymously publishing a book that was part popular science, part philosophy, and part erotica: Vénus physique . In that work, Maupertuis proposed a theory of generation (i.e., reproduction) in which organic matter possessed a self-organizing “intelligence” that was analogous to the contemporary chemical concept of affinities , which

3496-423: The major arguments advanced to prove God, from the wonders of nature or the apparent regularity of the universe, are all open to objection (what wonder is there in the existence of certain particularly repulsive insects, what regularity is there in the observation that all the planets turn in nearly the same plane – exactly the same plane might have been striking but 'nearly the same plane' is far less convincing). But

3572-627: The methods useful for particle mechanics also apply to continuous fields. The action integral runs over a Lagrangian density, but the concepts are so close that the density is often simply called the Lagrangian. For quantum mechanics, the action principles have significant advantages: only one mechanical postulate is needed, if a covariant Lagrangian is used in the action, the result is relativistically correct, and they transition clearly to classical equivalents. Both Richard Feynman and Julian Schwinger developed quantum action principles based on early work by Paul Dirac . Feynman's integral method

3648-502: The minimum or "least action". The path variation implied by δ {\displaystyle \delta } is not the same as a differential like d t {\displaystyle dt} . The action integral depends on the coordinates of the objects, and these coordinates depend upon the path taken. Thus the action integral is a functional , a function of a function. An important result from geometry known as Noether's theorem states that any conserved quantities in

3724-471: The more powerful and general abstract spaces. Action principles assign a number—the action—to each possible path between two points. This number is computed by adding an energy value for each small section of the path multiplied by the time spent in that section: where the form of the kinetic ( KE {\displaystyle {\text{KE}}} ) and potential ( PE {\displaystyle {\text{PE}}} ) energy expressions depend upon

3800-404: The other two conservation principles did; the second was that live force was defined by the product of mass and square of velocity. Why did the velocity appear twice in this quantity, as squaring it suggests? The Leibnizians argued this was simple enough: there was a natural tendency in all matter towards motion, so even at rest, there is an inherent velocity in bodies; when they begin to move, there

3876-536: The particle momenta or the conjugate momenta of generalized coordinates , defined by the equation p k   = def   ∂ L ∂ q ˙ k , {\displaystyle p_{k}\ {\stackrel {\text{def}}{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}},} where L ( q , q ˙ , t ) {\displaystyle L(\mathbf {q} ,{\dot {\mathbf {q} }},t)}

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3952-412: The particular path taken has a stationary value for the system's action: similar paths near the one taken have very similar action value. This variation in the action value is key to the action principles. The symbol δ {\displaystyle \delta } is used to indicate the path variations so an action principle appears mathematically as meaning that at the stationary point ,

4028-441: The path followed by a physical system. His work in natural history is interesting in relation to modern science, since he touched on aspects of heredity and the struggle for life . Maupertuis was born at Saint-Malo , France, to a moderately wealthy family of merchant- corsairs . His father, Renė, had been involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. The son

4104-427: The path of light reflecting from a mirror, the angle of incidence equals the angle of reflection . Hero of Alexandria later showed that this path has the shortest length and least time. Building on the early work of Pierre Louis Maupertuis , Leonhard Euler , and Joseph-Louis Lagrange defining versions of principle of least action , William Rowan Hamilton and in tandem Carl Gustav Jacob Jacobi developed

4180-470: The path of stationary action. Schwinger's approach relates variations in the transition amplitudes ( q f | q i ) {\displaystyle (q_{\text{f}}|q_{\text{i}})} to variations in an action matrix element: where the action operator is The Schwinger form makes analysis of variation of the Lagrangian itself, for example, variation in potential source strength, especially transparent. For every path,

4256-563: The path. Solution of the resulting equations gives the world line q ( t ) {\displaystyle \mathbf {q} (t)} . Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy. The action S {\displaystyle S} in Hamilton's principle is the Legendre transformation of the action in Maupertuis' principle. The concepts and many of

4332-449: The physics problem, and their value at each point on the path depends upon relative coordinates corresponding to that point. The energy function is called a Lagrangian; in simple problems it is the kinetic energy minus the potential energy of the system. A system moving between two points takes one particular path; other similar paths are not taken. Each path corresponds to a value of the action. An action principle predicts or explains that

4408-443: The power of the method and its further mathematical development rose. This article introduces the action principle concepts and summarizes other articles with more details on concepts and specific principles. Action principles are " integral " approaches rather than the " differential " approach of Newtonian mechanics . The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of

4484-401: The product of mass, velocity and distance should be particularly viewed as corresponding to action, and even less reason why its minimisation should be an 'economy' principle like a minimisation of effort. Indeed, the product of mass, velocity and distance is mathematically the equivalent of the product of the live force and time; thus the integral over distance of the product of mass and velocity

4560-425: The quantum action principle. At the end point, where the paths meet, the paths with similar phases add, and those with phases differing by π {\displaystyle \pi } subtract. Close to the path expected from classical physics, phases tend to align; the tendency is stronger for more massive objects that have larger values of action. In the classical limit, one path dominates –

4636-629: The renowned mathematician David Hilbert applied the principle of least action to derive the field equations of general relativity. His action, now known as the Einstein–Hilbert action , contained a relativistically invariant volume element − g d 4 x {\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x} and the Ricci scalar curvature R {\displaystyle R} . The scale factor κ {\displaystyle \kappa }

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4712-427: The same time, as well as connecting the same two points q ( t 1 ) {\displaystyle \mathbf {q} (t_{1})} and q ( t 2 ) {\displaystyle \mathbf {q} (t_{2})} . The Lagrangian L = T − V {\displaystyle L=T-V} is the difference between kinetic energy and potential energy at each point on

4788-456: The scenario; energy is a scalar magnitude combining information from all objects, giving an immediate simplification in many cases. The components of force vary with coordinate systems; the energy value is the same in all coordinate systems. Force requires an inertial frame of reference; once velocities approach the speed of light , special relativity profoundly affects mechanics based on forces. In action principles, relativity merely requires

4864-444: The shape of elastic rods under load, the shape of a liquid between two vertical plates (a capillary ), or the motion of a pendulum when its support is in motion. Quantum action principles are used in the quantum theory of atoms in molecules ( QTAIM ), a way of decomposing the computed electron density of molecules in to atoms as a way of gaining insight into chemical bonding. Inspired by Einstein's work on general relativity ,

4940-1006: The smallest part of what blind destiny had produced. Mais ne pourroit-on pas dire, que dans la combinaison fortuite des productions de la Nature, comme il n'y avoit que celles où se trouvoient certains rapports de convenance, qui pussent subsister, il n'est pas merveilleux que cette convenance se trouve dans toutes les especes qui actuellement existent? Le hasard, diroit-on, avoit produit une multitude innombrable d'Individus; un petit nombre se trouvoit construit de maniere que les parties de l'Animal pouvoient satisfaire à ses besoins; dans un autre infiniment plus grand, il n'y avoit ni convenance, ni ordre: tous ces derniers ont péri; des Animaux sans bouche ne pouvoient pas vivre, d'autres qui manquoient d'organes pour la génération ne pouvoient pas se perpétuer; les feuls qui soient restés, sont ceux où se trouvoient l'ordre & la convenance: & ces especes que nous voyons aujourd'hui, ne sont que la plus petite partie de ce qu'un destin aveugle avoit produit. The same text

5016-418: The variation of the action S {\displaystyle S} with some fixed constraints C {\displaystyle C} is zero. For action principles, the stationary point may be a minimum or a saddle point , but not a maximum. Elliptical planetary orbits provide a simple example of two paths with equal action – one in each direction around the orbit; neither can be

5092-411: The work on genealogy , coupled with the tracing of phenotypic characters through lineages, foreshadows later work done in genetics. The principle of least action states that in all natural phenomena a quantity called 'action' tends to be minimised. Maupertuis developed such a principle over two decades. For him, action could be expressed mathematically as the product of the mass of the body involved,

5168-458: Was "He was neither an evolutionist , nor one of the founders of the theory of natural selection [but] he was one of the pioneers of genetics ". Maupertuis espoused a theory of pangenesis , postulating particles from both mother and father as responsible for the characters of the child. Bowler credits him with studies on heredity, with the natural origin of human races, and with the idea that forms of life may have changed with time. Maupertuis

5244-673: Was a French mathematician , philosopher and man of letters . He became the Director of the Académie des Sciences , and the first President of the Prussian Academy of Science , at the invitation of Frederick the Great . Maupertuis made an expedition to Lapland to determine the shape of the Earth . He is often credited with having invented the principle of least action ; a version is known as Maupertuis's principle – an integral equation that determines

5320-405: Was a strong critic of the natural theologians , pointing to phenomena incompatible with a concept of a good and wise Creator. He was also one of the first to consider animals in terms of variable populations, in opposition to the natural history tradition that emphasised description of individual specimens. The difficulty of interpreting Maupertuis can be gauged by reading the original works. Below

5396-412: Was educated in mathematics by a private tutor, Nicolas Guisnée, and upon completing his formal education his father secured him a largely honorific cavalry commission. After three years in the cavalry, during which time he became acquainted with fashionable social and mathematical circles, he moved to Paris and began building his reputation as a mathematician and literary wit. In 1723 he was admitted to

5472-636: Was not a variational principle but reduces to the classical least action principle; it led to his Feynman diagrams . Schwinger's differential approach relates infinitesimal amplitude changes to infinitesimal action changes. When quantum effects are important, new action principles are needed. Instead of a particle following a path, quantum mechanics defines a probability amplitude ψ ( x k , t ) {\displaystyle \psi (x_{k},t)} at one point x k {\displaystyle x_{k}} and time t {\displaystyle t} related to

5548-514: Was observed much earlier by John Bernoulli and which was later fully developed in Hamilton's ingenious optico-mechanical theory. This analogy played a fundamental role in the development of modern wave-mechanics. Action principles are applied to derive differential equations like the Euler–Lagrange equations or as direct applications to physical problems. Action principles can be directly applied to many problems in classical mechanics , e.g.

5624-474: Was one that treated the competing theories of generation (i.e. preformationism and epigenesis ). His account of life involved spontaneous generation of new kinds of animals and plants, together with massive elimination of deficient forms. These ideas avoid the need for a Creator, but are not part of modern thinking on evolution. The date of these speculations, 1745, is concurrent with Carl Linnaeus 's own work, and so predates any firm notion of species . Also,

5700-413: Was published earlier (1748) as " Les loix du mouvement et du repos déduites d'un principe metaphysique " (translation: " Derivation of the laws of motion and equilibrium from a metaphysical principle "). King-Hele (1963) points to similar, though not identical, ideas of thirty years later by David Hume in his Dialogues Concerning Natural Religion (1777). The chief debate that Maupertuis was engaged in

5776-424: Was widely read and commented upon favourably by Georges-Louis Leclerc, Comte de Buffon . He later developed his views on living things further in a more formal pseudonymous work that explored heredity , collecting evidence that confirmed the contributions of both sexes and treated variations as statistical phenomena. In 1740 Maupertuis went to Berlin at the invitation of Frederick II of Prussia , and took part in

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