Mathematical physics

Mathematical physics is the study of mathematical methods used to understand the physical world. It involves two main directions of discovery. First, researchers develop new mathematical tools specifically to solve complex physics problems. Second, the laws of physics often inspire the creation of entirely new branches of mathematics. This second process is known as physical mathematics. While it is closely related to theoretical physics, mathematical physics emphasizes mathematical rigor. This means it seeks to prove that physical theories are logically sound and complete.

In classical mechanics, mathematical physics reformulates Newtonian laws into more advanced frameworks. These include Lagrangian mechanics and Hamiltonian mechanics. These approaches allow scientists to study mechanical systems even when they face complex constraints. One major result of these methods is the understanding of symmetry. Through Noether's theorem, researchers see how symmetry relates to conserved quantities during a system's evolution. These classical ideas have since expanded into other fields like statistical mechanics and classical field theory. They also provided the foundation for modern differential geometry, including symplectic geometry and vector bundles.

Other branches of the field rely on specific mathematical structures to explain natural phenomena. For example, the theory of partial differential equations is central to mathematical physics. This area includes variational calculus, Fourier analysis, and vector analysis. These tools were developed intensely between the late 18th century and the 1930s by thinkers like Euler and Lagrange. They allow us to model many real-world systems. These include hydrodynamics, which studies fluids, and acoustics, which studies sound. They also explain electricity, magnetism, and the way air moves in aerodynamics.

Quantum theory represents another major pillar of the field. The study of atomic spectra and quantum mechanics grew alongside mathematical fields like linear algebra and functional analysis. Nonrelativistic quantum mechanics uses Schrödinger operators to connect math to atomic and molecular physics.

A modern subspecialty is quantum information theory. On a much larger scale, the theories of relativity require different mathematical tools. Group theory was essential for special and general relativity. Later, researchers added topology and functional analysis to describe cosmological phenomena. Today, concepts from category theory and homological algebra also help describe these vast physical systems.

The history of this field stretches back to ancient Greece. Scholars like Euclid and Archimedes used mathematical analysis to study optics and equilibrium. During the Renaissance, the field began to transform. In 1543, Nicolaus Copernicus proposed a heliocentric model. Later, Johannes Kepler used equations to formalize the laws of planetary motion, showing that orbits are ellipses rather than perfect circles.

Galileo Galilei famously asserted that the book of nature is written in mathematics. He used experimentation to establish the laws of free fall and inertial motion, which are central to classical mechanics.

As math and physics merged, new frameworks emerged to describe the universe. René Descartes developed analytic geometry, which allowed for plotting locations in 3D space using Cartesian coordinates. This helped bridge the gap between geometry and algebra. Christiaan Huygens was a vital forerunner of modern mathematical physics. He was among the first to use mathematical parameters to idealize physical problems. He also used math to explain unobservable phenomena, such as the nature of light.

By the mid-17th century, the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz provided the essential language for studying change.

Mathematical physics also plays a critical role in statistical mechanics. This field includes the study of phase transitions and relies heavily on Hamiltonian mechanics. It is closely linked to ergodic theory and probability theory. There are even increasing interactions between physics and combinatorics. While theoretical physicists might use intuitive or approximate arguments to link theory to observation, mathematical physicists strive for absolute precision. They often re-examine problems that others believe are solved. For instance, they have investigated the subtleties of synchronization in relativity and the mathematical foundations of the second law of thermodynamics.