# Theoretical mechanics

The theoretical mechanics or analytical mechanics deals with the mathematical foundations of classical mechanics and relativistic mechanics . She examines the properties of the basic equations and their solutions and develops methods for the exact or approximate solution of certain problem classes.

## Formalisms

In principle, Newton's or relativistic equations already contain all of classical mechanics. In practice, however, these equations are not ideal for dealing with many problems. Therefore, alternative formulations of mechanics have been developed that are better suited to most problems. In addition, these alternative formulations can usually be used to better examine the relationship between classical mechanics and quantum mechanics .

One of these alternative formulations is the principle of extremal action (often somewhat imprecisely referred to as the "principle of least action", since in most cases the extremum is a minimum). It provides the basis for Noether's theorem , which establishes a connection between the symmetries of a physical system and its conservation quantities. In addition, the stationary phase approximation results as a limit case of quantum mechanics for short wavelengths, which allows a formal derivation of classical mechanics as a limit case of quantum mechanics ( correspondence principle ). However, this principle is usually not favorable for the immediate practical calculation of specific problems.

However, the Lagrange formalism can be derived from the principle of extremal action, which is the method of choice for most specific problems. It provides a consistent formal method to determine the equations of motion for a physical system . In particular, any constraint conditions (for example the condition that the wheel of a bicycle should only roll, but not slip) can be included without having to consider in advance which constraining forces occur; the latter is obtained as a result from the formalism. The Lagrange formalism also provides the basis for the path integral formalism of quantum mechanics.

The Hamilton formalism can be derived from the Lagrange formalism . This is also well suited for solving many specific problems. It is also well suited for the theoretical investigation of the properties of classic trajectories . Since he - unlike the previously presented formalisms - works in phase space , he can use the complete mathematical apparatus of symplectic geometry . The Hamilton formalism is also the starting point for canonical quantization , the easiest way to set up the Schrödinger equation for a physical system.

The Hamilton-Jacobi formalism can in turn be derived from Hamiltonian mechanics . Due to the use of partial differential equations, this is usually not ideal for solving specific problems, but is suitable for theoretical investigations. The Hamilton-Jacobi equation can also be obtained directly as a first approximation of the phase of the quantum mechanical wave function from the Schrödinger equation with formal expansion according to nach. They therefore provide a particularly direct connection between classical mechanics and quantum mechanics.

## Methods

Theoretical mechanics uses various methods to study the behavior of physical systems. The most obvious method, the closed mathematical solution of the equations of motion, is only possible in the rarest of cases. In addition, it only reveals something about the individual system in question; In theoretical physics, however, one is often more interested in properties that whole classes of physical systems have in common.

The methods of perturbation theory form an important class . These describe how the behavior of a system changes if its properties are changed only slightly (for example a pendulum is only slightly deflected from its rest position, or a weak electric field is applied to a system). Perturbation theoretic methods often provide the only way to calculate analytical solutions in a specific case, but they also often allow deeper insights into the behavior of a physical system.

## literature

### Textbooks

• LD Landau, EM Lifschitz: Textbook of theoretical physics. Volume 1: Mechanics. Akademie Verlag, Berlin 1970.
• W. Nolting: Basic course: Theoretical Physics. Volume 2: Analytical Mechanics. 3. Edition. Verlag Zimmermann-Neufang, 1993, ISBN 3-922410-21-9 .