Variational method is an important method in mathematics and physics, which is widely used in many fields. In mathematics, variational method is mainly used to solve optimization problems, such as extreme value problem and least square problem. By transforming the original problem into the extreme value problem of functional, the optimal solution of the problem is obtained by variational method.

Variational method is also widely used in physics. Firstly, the Lagrange equation and Hamilton equation in classical mechanics are solved by variational method. The energy function of a dynamic system is expressed as a functional, and its extreme value is solved by variational method, so that the conserved quantity and motion equation of the system can be obtained. In addition, the variational method is also used to solve the Schrodinger equation in quantum mechanics. The wave function and energy eigenvalue of the system can be obtained by expressing the wave function as a functional and finding its extreme value by variational method.

In addition, the variational method has important applications in statistical physics. For example, in thermodynamics, the expression of entropy and thermodynamic equilibrium conditions can be derived by variational method. In condensed matter physics, variational method is used to solve electronic structure problems, such as energy band theory and density functional theory.

In a word, the variational method is widely used in the fields of mathematics and physics. It is not only used to solve optimization problems, but also used to solve the motion equation of dynamic system, Schrodinger equation in quantum mechanics and thermodynamic equilibrium conditions in statistical physics.