Classical mechanics

Classical mechanics is a branch of physics that was largely fully developed until the end of the 19th century and is principally engaged in the motion of bodies. Classical mechanics served as the starting point for the development of modern physical theories such as the theory of relativity and quantum mechanics, whose development was necessary because of experimental results that were not compatible with the concepts of classical mechanics. Classical mechanics yet allows very accurate predictions and descriptions of those physical processes in which relativistic and quantum mechanical effects can be neglected. Typical modern application areas of classical mechanics are aerodynamics, structural engineering and biophysics.

• 3.1 The relationship with the theory of relativity
• 3.2 The relation to quantum mechanics

History

First mathematical approaches to describe the mechanics of bodies can be traced back to ancient times. Archimedes already knew the law of the lever. Further developments of the theory followed, among others, by celestial observations and new ideas to the world of Nicolaus Copernicus, Johannes Kepler and Galileo Galilei. However, the generalization to general mathematical concepts goes back to Isaac Newton, who has specially developed the calculus ( independently of Gottfried Wilhelm Leibniz, who developed an equivalent formulation for mathematics ). Other important contributions were followed by Joseph -Louis de Lagrange with the Lagrangian formalism and William Rowan Hamilton with the Hamiltonian mechanics. With these theories physical processes, such as the motion of pendulums, planetary orbits, rigid bodies and bodies in free fall could be the first time fully described.

Formulations

In classical mechanics, different principles for preparation of the equations of motion, which are used to describe the motion of bodies exist. These each represent a further development or generalization of Newton's second law dar. equations of motion are second order differential equations that can be solved for the acceleration and whose solution determines the location and velocity of a mass at any time.

Newton's Laws

Newton's laws are considered the basis of classical mechanics, on the basis for all other models. The central concept of this formulation is the introduction of forces that cause an acceleration of a mass. The equation of motion of the mass is determined by the superposition of the forces acting on the mass:

Lagrangian formalism

The Lagrangian formalism describes the laws of classical mechanics by the Lagrangian, which is given for systems with a generalized potential and holonomic constraints as the difference between kinetic energy and potential energy:

The equations of motion are obtained by applying the Euler -Lagrange equations, the time derivatives, the velocities and the generalized coordinates, into relation:

Hamiltonian mechanics

The Hamiltonian mechanics is the most generalized formulation of classical mechanics and starting point for the development of newer theories and models, such as quantum mechanics. Central equation of this formulation is the Hamiltonian function. It is defined as follows:

Here are the generalized velocities and generalized impulses. Is the potential energy independent of the speed and hang the transformation equations that define the generalized coordinates, not on the time, the Hamiltonian in classical mechanics, is given by the sum of the kinetic energy and potential energy:

The equations of motion are obtained by applying the canonical equations:

The Hamilton-Jacobi formalism exists a modified form of this description, which links the Hamilton function with the effect.

Confines

Many everyday phenomena are described by classical mechanics sufficiently accurate. But there are phenomena that no longer can no longer be reconciled explained by classical mechanics or. In these cases, classical mechanics is replaced by more accurate theories, such as the special theory of relativity or quantum mechanics. These theories include classical mechanics as a limiting case. Known classical unexplained effects are photoelectric effect, Compton scattering and blackbody radiator.

The relationship with the theory of relativity

Unlike the theory of relativity in classical mechanics there are no maximum speed at which signals can propagate. So it is possible in a classical universe, to synchronize all clocks with an infinitely fast signal. This is an absolute, valid in all inertial time possible.

In relativity theory, the largest signal velocity is equal to the vacuum speed of light. Assuming that for the measurement of physical processes needed watches can be perfectly synchronized, can now be the scope of classical mechanics with respect to the theory of relativity determine. The assumption about the ability to synchronize applies namely if and only if the measured velocity is small compared to the ( maximum ) signal speed at which the clocks are synchronized, ie.

The relation to quantum mechanics

In contrast to the quantum mechanics, mass points with identical observables (mass, position, momentum ) can be distinguished, while one starts in the quantum mechanics of indistinguishable entities. This requires that classical body in the sense must be macroscopically that they have individual characteristics that make them distinguishable. Thus, for example, a family of elementary particles can not be regarded as classical mass points. The distinctness of a classical particle is because that if it is left to itself remains in its previous inertial system. This is a quantum mechanical particles described not the case, as a self- over -ground particles does not remain forcibly in his inertial system. This fact can be derived in quantum mechanics, in which one solves the Schrödinger initial value problem for the wave function of a particle whose probability is localized at a point exactly in one place ( a so-called peak). The probability starts increasing with time to melt.

Postgraduate

Classical mechanics is treated in many one - and secondary textbooks of physics.

• Ralph Abraham, Jerrold E. Marsden: Foundations of Mechanics. Addison -Wesley, ISBN 0-201-40840-6.
• Torsten Fließbach: mechanics. 5th edition. Spectrum, 2007, ISBN 978-3-8274-1683-4.
• Herbert Goldstein, Charles. P. Poole, John Safko: Classical Mechanics. Wiley -VCH, ISBN 3-527-40589-5.