introduction page

The physical framework established by Sir Isaac Newton’s mechanics was a revolutionary step in our understanding of how the universe works. Newton’s laws allowed, for the first time, precise and accurate predictions to be made about the motion of objects. The development of differential and integral calculus by Newton and Leibniz allowed for analysis of how dynamical systems evolve, given the initial conditions of the system. Newton’s mechanics is useful in describing a broad range of situations by identifying force vectors and plugging them into Newton’s second law, which allows prediction of subsequent motion of the system. This is a powerful method, but it has its limitations, especially in complicated situations where it becomes impractical to write down all the force vectors acting in a system.

In 1788, an Italian mathematician named Joseph-Louis Lagrange introduced a reformulated version of the laws of dynamics, now known as Lagrangian Mechanics, which allowed for a wider variety of problems to be solved, using an equivalent underlying physical framework to Newton’s. This new system of mechanics was developed by Lagrange over several years with the assistance of other mathematicians and physicists, notably Euler and d’Alembert. The most important feature of Lagrange’s approach is that it deals with the scalar quantities of kinetic and potential energy, which are employed together in an important quantity called the Lagrangian, L:

The Lagrangian is a scalar quantity that is defined as the total kinetic energy K minus the total potential energy U of a system. The Lagrangian is chosen in this particular way to allow for the equations of motion governing the system’s behavior to be derived from the Principle of Stationary Action, also known as Hamilton's Principal.

The Lagrangian method allows for use of convenient types of coordinates, called Generalized Coordinates, to be chosen. In simple problems, Newton’s approach may be easier to employ than Lagrange’s, however, in more complicated problems and when the possible motion of a system is constrained, the Lagrangian approach often prevails. Since its development and application to mechanics, Lagrangian techniques have been utilized in electromagnetic theory, the theory of relativity, quantum mechanics and quantum field theory, and a variety of other applications.