Mathematical physics

Mathematical physics is a special area where math and science work together. It uses math to help us understand physics. Scientists and mathematicians work as a team to create new math tools. These tools help solve big questions in physics.

One important part of mathematical physics is creating new math ideas. When physicists find a hard problem, they sometimes need new math to solve it. This helps both math and physics grow.

Math makes complex physics ideas easier to understand. For example, equations can show how planets move or how light behaves. This helps us 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 this field. It shows how closely math and physics are linked.

Scope

Mathematical physics uses 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. Here, math helps us understand how objects move. This uses special methods called Lagrangian mechanics and Hamiltonian mechanics. These ideas also help in other areas of physics.

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

Quantum theory is another big part. Here, 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 mean different things. Some math areas that began with physics ideas are not always part of mathematical physics. For example, ordinary differential equations and symplectic geometry are usually just math, while dynamical systems and Hamiltonian mechanics are part of mathematical physics.

Mathematical physics uses careful math to solve physics problems. This is different from theoretical physics, which often uses simpler ideas to match real experiments. Mathematical physicists make physics theories more exact. Their work helps both physics and math improve.

Prominent mathematical physicists

Before Newton

Mathematical ideas about nature started with ancient Greek thinkers like Euclid, Archimedes, and Ptolemy. Later, scholars from Islamic and Byzantine cultures added to these ideas. These ideas came back to Europe in the 1100s and during the Renaissance.

In the 1500s, Nicolaus Copernicus suggested that Earth orbits the Sun. Johannes Kepler showed that planets move in oval paths called ellipses. Galileo Galilei used experiments to show that all objects fall at the same rate and supported the idea that Earth orbits the Sun.

René Descartes used math to describe nature, creating tools to show positions in space. Christiaan Huygens was one of the first to use math to explain how light behaves.

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.

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. Joseph Fourier found new methods to solve heat problems. Others like Pierre-Simon Laplace and Carl Friedrich Gauss advanced math for electricity, magnetism, and astronomy.

Relativistic

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

Quantum mechanics

The early 1900s brought big changes in understanding tiny particles. Scientists like Max Planck and Albert Einstein started this work. Others like Niels Bohr, Erwin Schrödinger, and Richard Feynman built on it. They used math to describe how tiny particles behave differently 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–)