Describing and solving these equations has become an important mathematical problem in Newtonian mechanics. This kind of research continues to this day. For example, the three-body and various classical dynamical systems in celestial mechanics are long-term research objects. /kloc-in the 8th century, the foundation of Newtonian mechanics began to be described by variational principles, which promoted the development of variational methods. Later, many physical theories took variational principles as their own foundation. /kloc-since the 0/8th century, many partial differential equations have been summarized in the theory of continuum mechanics, heat transfer and electromagnetic field, which are generally called mathematical physical equations (including integral equations, differential integral equations and ordinary differential equations with physical significance). It was not until the beginning of the twentieth century that the study of mathematical physical equations became the main content of mathematical physics.
Since then, many new partial differential equation problems, such as isolated wavelet, discontinuous solution, bifurcation solution and inverse problem, have appeared in connection with the needs of plasma physics, solid state physics, nonlinear optics, space technology and nuclear technology. They further enrich the contents of mathematical and physical equations. Complex variable function, integral transformation, special function, variational method, harmonic analysis, functional analysis, differential geometry and algebraic geometry are all effective tools to study mathematical and physical equations.
Since the 20th century, due to the update of physics content, mathematical physics has taken on a new look. With the in-depth study of electromagnetic theory and gravitational field, people's concept of time and space has changed fundamentally, making the geometry of Minkowski space and Riemann space become the necessary mathematical theory of Einstein's special relativity and general relativity. Many physical quantities are expressed as vectors, tensors and spinors, and global differential geometry is also needed to discuss large-scale spatio-temporal structure.
With the development of electronic computers, many problems in mathematical physics can be solved by numerical calculation, and the developed "computational mechanics" and "computational physics" are playing an increasingly important role. It has also become an important method to directly simulate physical models with computers. In addition, various asymptotic methods continue to develop. The emergence of quantum mechanics and quantum field theory enriches mathematical physics. In quantum mechanics, the state of matter is described by wave function, physical quantity becomes operator, and the measured physical quantity is the spectrum of operator. In quantum field theory, the wave function is re-quantized into an operator, and the generation and elimination of particles are described by electromagnetic interaction, weak interaction and strong interaction. Therefore, it is necessary to study the spectrum of operators, the spectral analysis of functions and the algebra formed by operators in various function spaces. At the same time, the mathematical basis of perturbation expansion and renormalization (dealing with divergence difficulties) should be studied. In addition, it is also a noteworthy topic to study nonlinear field theory by non-perturbation method.