Analytical mechanics

Theoretical mechanics deals with the mathematical foundations of classical Newtonian and relativistic mechanics. It examines the characteristics of the basic equations and their solutions and methods developed for exact or approximate solution of certain classes of problems.

Formalisms

In principle, Newtonian or relativistic equations already contain all sorts of classical mechanics. In practice, however, these equations are not ideal for the treatment of many problems. Therefore, alternative formulations of the mechanism have been developed that are better suited for most problems. In addition, can be combined with these alternative formulations usually the connection between classical and quantum mechanics better investigate.

One of these alternative formulations is the principle of extremal action (often somewhat inaccurately as a " principle of least action ", since in most cases the extremum is a minimum ). It provides a basis for the Noether theorem, which establishes a relationship between the symmetries of a physical system and its conservation values ​​. Moreover, it follows by means of the stationary -phase approximation as a limiting case of quantum mechanics for short wavelengths, which allows a formal derivation of classical mechanics as a limiting case of quantum mechanics (correspondence principle). For the immediate practical calculation of specific problems, however, this principle is not generally favorable.

From the principle of extremal action, however, the Lagrangian formalism can be derived, which is the method of choice for most practical problems. It delivers a consistent formal method to determine the equations of motion of a physical system. This may, in particular any constraints ( for example, the condition that a bicycle wheel only roll, but that does not slip ) are included, without the need to think in advance what potential constraint forces there; latter is obtained as the result of the formalism. The Lagrange formalism provides the basis for the path integral formalism of quantum mechanics.

From the Lagrangian formalism the Hamiltonian formalism can be derived. Again, this is well suited for the solution of many practical problems. In addition, it is well suited for the theoretical investigation of the properties of classical trajectories. Because he - unlike the previously presented formalisms - works in phase space, it can use the full mathematical apparatus of symplectic geometry. The Hamiltonian formalism is also the starting point for the canonical quantization, the easiest way to set up the Schrödinger equation for a physical system.

From the Hamiltonian mechanics, the Hamilton -Jacobi formalism can in turn be derived. This is because of the use of partial differential equations are generally not ideal for solving specific problems, but is suitable for theoretical studies. The Hamilton -Jacobi equation can be directly as a first approximation of the phase of the quantum mechanical wave function of the Schrödinger equation with gain formal development by ħ. They therefore provide a particularly direct relationship between classical mechanics and quantum mechanics.

Methods

The theoretical mechanics used different methods to study the behavior of physical systems. The most obvious method, the closed mathematical solution of the equations of motion is possible only in the rarest cases. In addition, it reveals something about the corresponding single system; in theoretical physics but one is often interested more for properties that whole classes of physical systems have in common.

An important class are the methods of perturbation theory. These describe how the behavior of a system is changed, if only slightly changed its properties ( for example, a pendulum only slightly deflects from the rest position, or applies it to a system, a weak electric field ). Perturbative methods often provide the specific case, the only way to calculate analytical solutions, but they also often allow a deeper insight into the behavior of a physical system.