Editing Mathematical physics (section)

=== Newtonian physics and post Newtonian ===

The prevailing framework for science in the 16th and early 17th centuries was one borrowed from [[Greek mathematics|Ancient Greek mathematics]], where geometrical shapes formed the building blocks to describe and think about space, and time was often thought as a separate entity. With the introduction of algebra into geometry, and with it the idea of a coordinate system, time and space could now be thought as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and used in all further mathematical frameworks developed in next centuries. 

By the middle of the 17th century, important concepts such as the [[fundamental theorem of calculus]] (proved in 1668 by Scottish mathematician [[James Gregory (mathematician)|James Gregory]]) and finding extrema and minima of functions via differentiation using Fermat's theorem (by French mathematician [[Pierre de Fermat]]) were already known before Leibniz and Newton.<ref name=geometriae>{{cite book| last=Gregory | first=James | title=Geometriae Pars Universalis | url=https://archive.org/details/gregory_universalis | publisher= Patavii: typis heredum Pauli Frambotti | year=1668 | location=[[Museo Galileo]] }}</ref> [[Isaac Newton]] (1642–1727) developed [[calculus]] (although [[Gottfried Wilhelm Leibniz]] developed similar concepts outside the context of physics) and [[Newton's method]] to solve problems in mathematics and physics. He was extremely successful in his application of [[calculus]] and other methods to the study of motion. Newton's theory of motion, culminating in his ''[[Philosophiæ Naturalis Principia Mathematica]]'' (''Mathematical Principles of Natural Philosophy'') in 1687, modeled three Galilean laws of motion along with Newton's [[law of universal gravitation]] on a framework of [[absolute space]]—hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presuming [[absolute time]], supposedly justifying knowledge of absolute motion, the object's motion with respect to absolute space.<ref>{{citation|contribution=The Mathematical Principles of Natural Philosophy|title=Encyclopædia Britannica|place=London|contribution-url=https://www.britannica.com/EBchecked/topic/369153/The-Mathematical-Principles-of-Natural-Philosophy}}</ref> The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. Having ostensibly reduced the Keplerian celestial laws of motion as well as Galilean terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity.<ref name=Lakatos1980>Imre Lakatos, auth, Worrall J & Currie G, eds, ''The Methodology of Scientific Research Programmes: Volume 1: Philosophical Papers'' (Cambridge: Cambridge University Press, 1980), pp [https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA213 213–214], [https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA220 220]</ref>

In the 18th century, the Swiss [[Daniel Bernoulli]] (1700–1782) made contributions to [[fluid dynamics]], and [[vibrating string]]s. The Swiss [[Leonhard Euler]] (1707–1783) did special work in [[Calculus of variations|variational calculus]], dynamics, fluid dynamics, and other areas. Also notable was the Italian-born Frenchman, [[Joseph-Louis Lagrange]] (1736–1813) for work in [[analytical mechanics]]: he formulated [[Lagrangian mechanics]]) and variational methods. A major contribution to the formulation of Analytical Dynamics called [[Hamiltonian dynamics]] was also made by the Irish physicist, astronomer and mathematician, [[William Rowan Hamilton]] (1805–1865). Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicist [[Joseph Fourier]] (1768 – 1830) introduced the notion of [[Fourier series]] to solve the [[heat equation]], giving rise to a new approach to solving partial differential equations by means of [[integral transforms]].

Into the early 19th century, following mathematicians in France, Germany and England had contributed to mathematical physics. The French [[Pierre-Simon Laplace]] (1749–1827) made paramount contributions to mathematical [[astronomy]], [[potential theory]]. [[Siméon Denis Poisson]] (1781–1840) worked in [[analytical mechanics]] and [[potential theory]]. In Germany, [[Carl Friedrich Gauss]] (1777–1855) made key contributions to the theoretical foundations of [[electricity]], [[magnetism]], [[mechanics]], and [[fluid dynamics]]. In England, [[George Green (mathematician)|George Green]] (1793–1841) published ''[[An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism]]'' in 1828, which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism.

A couple of decades ahead of Newton's publication of a particle theory of light, the Dutch [[Christiaan Huygens]] (1629–1695) developed the wave theory of light, published in 1690. By 1804, [[Thomas Young (scientist)|Thomas Young]]'s double-slit experiment revealed an interference pattern, as though light were a wave, and thus Huygens's wave theory of light, as well as Huygens's inference that light waves were vibrations of the [[luminiferous aether]], was accepted. [[Jean-Augustin Fresnel]] modeled hypothetical behavior of the aether. The English physicist [[Michael Faraday]] introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the Scottish [[James Clerk Maxwell]] (1831–1879) reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four [[Maxwell's equations]]. Initially, optics was found consequent of{{clarify|date=January 2018}} Maxwell's field. Later, radiation and then today's known [[electromagnetic spectrum]] were found also consequent of{{clarify|date=January 2018}} this electromagnetic field.

The English physicist [[Lord Rayleigh]] [1842–1919] worked on [[sound]]. The Irishmen [[William Rowan Hamilton]] (1805–1865), [[George Gabriel Stokes]] (1819–1903) and [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1824–1907) produced several major works: Stokes was a leader in [[optics]] and fluid dynamics; Kelvin made substantial discoveries in [[thermodynamics]]; Hamilton did notable work on [[analytical mechanics]], discovering a new and powerful approach nowadays known as [[Hamiltonian mechanics]]. Very relevant contributions to this approach are due to his German colleague mathematician [[Carl Gustav Jacobi]] (1804–1851) in particular referring to [[canonical transformations]]. The German [[Hermann von Helmholtz]] (1821–1894) made substantial contributions in the fields of [[electromagnetism]], waves, [[fluid]]s, and sound. In the United States, the pioneering work of [[Josiah Willard Gibbs]] (1839–1903) became the basis for [[statistical mechanics]]. Fundamental theoretical results in this area were achieved by the German [[Ludwig Boltzmann]] (1844–1906). Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics.