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Concept
Mathematical physics
Summary
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics).
There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world.
Lagrangian mechanics and Hamiltonian mechanics
The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry and conserved quantities during the dynamical evolution, as embodied 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 ideas in differential geometry (e.g. several notions in symplectic geometry and vector bundle).
Partial differential equations
Following mathematics: the theory of partial differential equation, variational calculus, Fourier analysis, potential theory, and vector analysis are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the 18th century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.
Official source
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