Editing Mathematical physics (section)

== Scope ==
There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world.

=== Classical mechanics ===
{{Main|Lagrangian mechanics|Hamiltonian mechanics}}
Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of [[Lagrangian mechanics]] and [[Hamiltonian mechanics]] (including both approaches in the presence of constraints). Both formulations are embodied in [[analytical mechanics]] and lead to an understanding of the deep interplay between the notions of [[symmetry (physics)|symmetry]] and [[conservation law|conserved quantities]] during the dynamical evolution of mechanical systems, as embodied within the most elementary formulation of [[Noether's theorem]]. These approaches and ideas have been extended to other areas of physics, such as [[statistical mechanics]], [[continuum mechanics]], [[classical field theory]], and [[quantum field theory]]. Moreover, they have provided multiple examples and ideas in [[differential geometry]] (e.g., several notions in [[symplectic geometry]] and [[vector bundle]]s).

=== Partial differential equations ===
{{Main|Partial differential equations}}
Within mathematics proper, the theory of [[partial differential equation]], [[variational calculus]], [[Fourier analysis]], [[potential theory]], and [[vector analysis]] are perhaps most closely associated with mathematical physics. These fields were developed intensively from the second half of the 18th century (by, for example, [[Jean le Rond d'Alembert|D'Alembert]], [[Leonhard Euler|Euler]], and [[Joseph-Louis Lagrange|Lagrange]]) until the 1930s. Physical applications of these developments include [[hydrodynamics]], [[celestial mechanics]], [[continuum mechanics]], [[elasticity theory]], [[acoustics]], [[thermodynamics]], [[electricity]], [[magnetism]], and [[aerodynamics]].

=== Quantum theory ===
{{Main|Quantum mechanics}}
The theory of [[atomic spectra]] (and, later, [[quantum mechanics]]) developed almost concurrently with some parts of the mathematical fields of [[linear algebra]], the [[spectral theory]] of [[Operator (mathematics)|operators]], [[operator algebra]]s and, more broadly, [[functional analysis]]. Nonrelativistic quantum mechanics includes [[Schrödinger]] operators, and it has connections to [[atomic, molecular, and optical physics|atomic and molecular physics]]. [[Quantum information]] theory is another subspecialty.

=== Relativity and quantum relativistic theories ===
{{main|Theory of relativity|Quantum field theory}}
The [[Special relativity|special]] and [[general relativity|general]] theories of relativity require a rather different type of mathematics. This was [[group theory]], which played an important role in both [[quantum field theory]] and [[differential geometry]]. This was, however, gradually supplemented by [[topology]] and [[functional analysis]]  in the mathematical description of [[physical cosmology|cosmological]] as well as [[quantum field theory]] phenomena. In the mathematical description of these physical areas, some concepts in [[homological algebra]] and [[category theory]]<ref>{{cite web|title=quantum field theory|url=https://ncatlab.org/nlab/show/quantum+field+theory|website=nLab}}</ref> are also important.

=== Statistical mechanics ===
{{main|Statistical mechanics}}
[[Statistical mechanics]] forms a separate field, which includes the theory of [[phase transition]]s. It relies upon the [[Hamiltonian mechanics]] (or its quantum version) and it is closely related with the more mathematical [[ergodic theory]] and some parts of [[probability theory]]. There are increasing interactions between [[combinatorics and physics]], in particular statistical physics.