Mathematical physics

Mathematical physics is a special area where math and science come together. It uses math to understand and explain the world of physics. This means scientists and mathematicians work side by side to create new math tools that help solve big questions in physics.

One important part of mathematical physics is developing new math ideas because of problems in physics. When physicists find a tricky problem, sometimes they need new math to solve it. This back-and-forth helps both fields grow and learn.

Math helps make complex ideas in physics easier to understand. For example, equations and formulas can show how planets move or how light behaves. This makes it possible to predict and explain many things in nature.

Mathematical physics is important because it leads to new discoveries. Many of today's technologies, from computers to space travel, depend on the work done in this field. It shows how closely linked math and physics really are.

Scope

Mathematical physics is about using math to understand and solve problems in physics. It has several main areas, each linked to different parts of our history.

One area is classical mechanics, where math helps us understand how objects move. This uses special methods called Lagrangian mechanics and Hamiltonian mechanics, which show how symmetry and conserved quantities work together in moving systems. These ideas also help in other areas of physics.

Another area involves partial differential equations. These are math equations that describe how things change, and they are very important in physics. They help us understand movement in fluids, sound, electricity, and more.

Quantum theory is another big part, where math helps explain the tiny particles that make up everything. This area connects with linear algebra and other math fields.

Finally, relativity and quantum relativistic theories use math like group theory and topology to describe the universe on large and small scales. These help us understand space, time, and tiny particles.

Usage

The term "mathematical physics" can be used in different ways. Some areas of mathematics that started from physics ideas are not always considered part of mathematical physics. For instance, ordinary differential equations and symplectic geometry are usually seen as pure math, while dynamical systems and Hamiltonian mechanics are part of mathematical physics.

Mathematical physics focuses on solving physics problems using very careful mathematical methods. This makes it different from theoretical physics, which often uses simpler ideas to connect with real experiments. Mathematical physicists work to make physics theories more exact and sometimes find that earlier solutions were not complete or correct. Their work helps both physics and mathematics grow.

Prominent mathematical physicists

Before Newton

Mathematical ideas about nature go back to ancient Greek thinkers like Euclid, Archimedes, and Ptolemy. Later, scholars from Islamic and Byzantine cultures added to these ideas, which were later brought back to Europe in the 12th century and during the Renaissance.

In the 1500s, Nicolaus Copernicus suggested that Earth orbits the Sun, though he used complex ideas to explain planetary motion. Johannes Kepler improved this by showing planets move in oval paths called ellipses. Galileo Galilei used experiments to show that all objects fall at the same rate and helped support the idea that Earth orbits the Sun.

René Descartes used math to describe nature, creating tools to plot positions in space. Christiaan Huygens was one of the first to use math to explain how light behaves, laying groundwork for modern physics.

Newtonian and post-Newtonian physics

In the 1600s, new math ideas like calculus helped scientists understand motion better. Isaac Newton used these ideas to explain how objects move and how gravity works. His work brought together many earlier ideas into one system.

Later scientists like Daniel Bernoulli, Leonhard Euler, and Joseph-Louis Lagrange used math to study fluids, strings, and motion. William Rowan Hamilton created new ways to describe motion that influenced modern physics. Joseph Fourier found new methods to solve heat problems, and others like Pierre-Simon Laplace and Carl Friedrich Gauss advanced math for electricity, magnetism, and astronomy.

Relativistic

In the late 1800s, scientists noticed puzzles about light and motion. Albert Einstein solved these with his theory of relativity, which changed how we think about space and time. He showed that space and time are linked and can bend, especially around massive objects.

Quantum mechanics

The early 1900s brought big changes in understanding tiny particles. Scientists like Max Planck and Albert Einstein started this work, and others like Niels Bohr, Erwin Schrödinger, and Richard Feynman built on it. They used math to describe how tiny particles behave in ways very different from everyday objects.

List of prominent contributors to mathematical physics in the 20th century

Prominent contributors to the 20th century's mathematical physics include (ordered by birth date):

• William Thomson (Lord Kelvin) (1824–1907)
• Oliver Heaviside (1850–1925)
• Jules Henri Poincaré (1854–1912)
• David Hilbert (1862–1943)
• Arnold Sommerfeld (1868–1951)
• Constantin Carathéodory (1873–1950)
• Albert Einstein (1879–1955)
• Emmy Noether (1882–1935)
• Max Born (1882–1970)
• George David Birkhoff (1884–1944)
• Hermann Weyl (1885–1955)
• Satyendra Nath Bose (1894–1974)
• Louis de Broglie (1892–1987)
• Norbert Wiener (1894–1964)
• John Lighton Synge (1897–1995)
• Mário Schenberg (1914–1990)
• Wolfgang Pauli (1900–1958)
• Paul Dirac (1902–1984)
• Eugene Wigner (1902–1995)
• Andrey Kolmogorov (1903–1987)
• Lars Onsager (1903–1976)
• John von Neumann (1903–1957)
• Sin-Itiro Tomonaga (1906–1979)
• Hideki Yukawa (1907–1981)
• Nikolay Nikolayevich Bogolyubov (1909–1992)
• Subrahmanyan Chandrasekhar (1910–1995)
• Mark Kac (1914–1984)
• Julian Schwinger (1918–1994)
• Richard Phillips Feynman (1918–1988)
• Irving Ezra Segal (1918–1998)
• Ryogo Kubo (1920–1995)
• Arthur Strong Wightman (1922–2013)
• Chen-Ning Yang (1922–2025)
• Rudolf Haag (1922–2016)
• Freeman John Dyson (1923–2020)
• Martin Gutzwiller (1925–2014)
• Abdus Salam (1926–1996)
• Jürgen Moser (1928–1999)
• Michael Francis Atiyah (1929–2019)
• Joel Louis Lebowitz (1930–)
• Roger Penrose (1931–)
• Elliott Hershel Lieb (1932–)
• Yakir Aharonov (1932–)
• Sheldon Glashow (1932–)
• Steven Weinberg (1933–2021)
• Ludvig Dmitrievich Faddeev (1934–2017)
• David Ruelle (1935–)
• Yakov Grigorevich Sinai (1935–)
• Vladimir Igorevich Arnold (1937–2010)
• Arthur Michael Jaffe (1937–)
• Roman Wladimir Jackiw (1939–)
• Leonard Susskind (1940–)
• Rodney James Baxter (1940–)
• Michael Victor Berry (1941–)
• Giovanni Gallavotti (1941–)
• Stephen William Hawking (1942–2018)
• Jerrold Eldon Marsden (1942–2010)
• Michael C. Reed (1942–)
• John Michael Kosterlitz (1943–)
• Israel Michael Sigal (1945–)
• Alexander Markovich Polyakov (1945–)
• Barry Simon (1946–)
• Herbert Spohn (1946–)
• John Lawrence Cardy (1947–)
• Giorgio Parisi (1948-)
• Abhay Ashtekar (1949-)
• Edward Witten (1951–)
• F. Duncan Haldane (1951–)
• Ashoke Sen (1956–)
• Juan Martín Maldacena (1968–)