The Lagrangian mechanics and the Hamiltonian mechanics are embodied in the so-called analytical mechanics. It leads, for instance, to discover the deep interplay of the notion of symmetry and that of conserved quantities during the dynamical evolution, stated within the most elementary formulation of Noether's theorem. These approaches and ideas have been extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory and quantum field theory. Moreover they have provided several examples and basic ideas in differential geometry (e.g. the theory of vector bundles and several notions in symplectic geometry).
Quantum theory
This theory developed almost concurrently with the mathematical fields of linear algebra, the spectral theory of operators, operator algebras and more broadly, functional analysis. It has connections to atomic and molecular physics. Quantum information theory is another subspecialty.
Partial differential equations
The special and general theories 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 cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays.
Geometrically advanced formulations of classical mechanics
This theory forms a separate field, which includes the theory of phase transitions. 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.
Statistical mechanics
This theory (and the related areas of variational calculus, Fourier analysis, potential theory, and vector analysis) is perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticitytheory, acoustics, thermodynamics,electricity, magnetism, and aerodynamics.
