# Classic mechanics

The mathematical pendulum - a typical application of classical mechanics

The classical mechanics or Newtonian mechanics is the branch of physics that the movement of solid liquid or gaseous bodies under the influence of forces describes. This also includes the case of inertial motion in the absence of a force and the case of static equilibrium , i.e. H. of remaining in the resting position, although forces are at work. Typical areas of application of classical mechanics are celestial mechanics , technical mechanics , hydrodynamics , aerodynamics , statics and biomechanics .

Classical mechanics are based on the foundations laid by Isaac Newton at the end of the 17th century and were largely fully developed by the end of the 19th century. She served as an important role model in the development of physics and the other natural sciences . Classical mechanics enables very precise predictions and descriptions of all mechanical processes in science, technology and nature, provided that the speed of the bodies compared to the speed of light and their De Broglie wavelength compared to the dimensions of the system under consideration can be neglected.

The physical theories such as the theory of relativity and quantum mechanics , with which these limitations were overcome in the 20th century, are based on the one hand on classical mechanics, but are also based essentially on concepts that are no longer compatible with classical mechanics.

## history

Classical mechanics, developed in the 17th century, became the first natural science in today's sense. The method of knowledge of nature founded by Galileo Galilei , in which experimental observations are made and the results are analyzed using mathematical methods, led to a scientific breakthrough for the first time. Isaac Newton's book Mathematical Principles of Natural Philosophy from 1687 is regarded as the beginning of Classical Mechanics . In it, movements of bodies, especially accelerated movements, are comprehensively analyzed with the help of a new concept of force specially created for this purpose . Newton proved that all observations and measurements on the movements of bodies can be explained by a framework of a few basic assumptions. He demonstrated this, using the mathematical technique of calculus, which is also new, with mathematical rigor for the observation results of Galileo on free fall and those of Johannes Kepler on planetary movements, as well as for numerous own observations and measurements on moving bodies.

Until the middle of the 19th century Christiaan Huygens , Gottfried Wilhelm Leibniz , Johann I Bernoulli , Daniel Bernoulli , Leonhard Euler , Jean-Baptiste le Rond d'Alembert , Joseph-Louis de Lagrange , Pierre-Simon Laplace , Augustin Louis Cauchy , William Rowan Hamilton , (and others) the necessary clarification of some of the Newtonian terms and the introduction of further terms (e.g. angular momentum , work , energy , stress tensor ) and techniques (e.g. d'Alembert's inertial force , Lagrange formalism ). In doing so, they considerably expanded the field of application of Newtonian mechanics. This doctrine of mechanics was so successful in the interpretation of innumerable processes that it was made the basis of a mechanistic world view , which, however, met with severe criticism from traditional philosophy.

From the 19th century Newtonian mechanics gradually found application in construction and mechanical engineering, but the latter only increased from the beginning of the 20th century. While the resulting technical mechanics is based entirely on Newton's concept of force, this was criticized in theoretical mechanics by Ernst Mach , Gustav Kirchhoff , Heinrich Hertz as not really fundamental and subsequently stepped back in its meaning compared to the terms momentum and energy .

It was discovered at the beginning of the 20th century that the validity of classical mechanics has its limits. Findings in electrodynamics led to problems that Albert Einstein solved in the context of his special theory of relativity and general relativity theory with a revision of the classical assumptions about space, time and mass. According to this, Newtonian mechanics remains approximately valid for the movement of bodies whose speeds can be neglected compared to the speed of light and whose gravitational energy can be neglected compared to their rest energy . Another limit of validity of classical mechanics resulted from the knowledge of atomic physics , which - after the first successes of Niels Bohr and Arnold Sommerfeld - could only be explained in the quantum mechanics developed by Werner Heisenberg and Erwin Schrödinger . From quantum mechanics it follows that classical mechanics is approximately valid for processes in which the De Broglie wavelength of the body is negligibly small compared to the relevant spatial distances.

## Formulations

In classical mechanics there are various principles for setting up equations of motion that are used to describe the movement of bodies. These each represent a further development or generalization of Newton's second law. Equations of motion are differential equations of the second order, which can be solved after acceleration and whose solution determines the location and speed of a mass at any time.

### Newton's Laws

Newton's laws are the basis of classical mechanics on which all other models are based. The central concept of this formulation is the introduction of forces that cause a mass to accelerate . The equation of motion of this mass is determined by the superposition of the forces that act on the mass: ${\ displaystyle {\ ddot {\ vec {x}}}}$${\ displaystyle m}$${\ displaystyle {\ vec {F}} _ {i}}$

${\ displaystyle m {\ ddot {\ vec {x}}} = \ sum _ {i = 1} ^ {N} {{\ vec {F}} _ {i}}}$

### Lagrange formalism

The Lagrange formalism describes the laws of classical mechanics through the Lagrange function , which is given for systems with a generalized potential and holonomic constraints as the difference between kinetic energy and potential energy : ${\ displaystyle L}$ ${\ displaystyle T}$ ${\ displaystyle V}$

${\ displaystyle L = TV}$

The equations of motion are obtained by applying the Euler-Lagrange equations , which relate the derivatives with respect to time , velocities and generalized coordinates : ${\ displaystyle t}$${\ displaystyle {\ dot {q}} _ {i}}$ ${\ displaystyle q_ {i}}$

${\ displaystyle {\ frac {\ text {d}} {{\ text {d}} t}} {\ frac {\ partial L} {\ partial {\ dot {q}} _ {i}}} = { \ frac {\ partial {L}} {\ partial q_ {i}}}}$

### Hamiltonian mechanics

Hamiltonian mechanics is the most generalized formulation of classical mechanics and the starting point for the development of new theories and models, such as quantum mechanics. The central equation of this formulation is the Hamilton function . It is defined as follows: ${\ displaystyle H}$

${\ displaystyle H = \ sum \ limits _ {i} {\ dot {q}} _ {i} p_ {i} -L ({\ vec {q}}, {\ dot {\ vec {q}}} , t)}$

Here are the generalized velocities and the generalized impulses . If the potential energy is independent of the velocity and if the transformation equations that define the generalized coordinates do not depend on time, the Hamilton function in classical mechanics is given by the sum of kinetic energy and potential energy : ${\ displaystyle {\ dot {q}} _ {i}}$${\ displaystyle p_ {i}}$ ${\ displaystyle T}$ ${\ displaystyle V}$

${\ displaystyle H = T + V}$

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

${\ displaystyle {\ dot {q}} _ {i} = {\ frac {\ partial H} {\ partial p_ {i}}}}$
${\ displaystyle {\ dot {p}} _ {i} = - {\ frac {\ partial H} {\ partial q_ {i}}}}$

With the Hamilton-Jacobi formalism, there is a modified form of this description that links the Hamilton function with the action .

## Limits

Many everyday phenomena are described in sufficient detail by classical mechanics. But there are phenomena that cannot be explained or reconciled with classical mechanics. In these cases, classical mechanics is replaced by more precise theories, such as B. by the special theory of relativity or quantum mechanics. These theories contain classical mechanics as a limiting case. Well-known, classically inexplicable effects are photo effect , Compton scattering and cavity radiators .

### The relationship to the theory of relativity

In contrast to the theory of relativity, in classical mechanics there is no maximum speed at which signals can propagate. In a classical universe it is possible to synchronize all clocks with an infinitely fast signal. This means that an absolute time that is valid in every inertial system is conceivable.

In the theory of relativity, the greatest signal speed is equal to the speed of light in a vacuum . Assuming that the clocks required to measure physical processes can be perfectly synchronized, the scope of classical mechanics compared to the theory of relativity can now be determined. The assumption that the clocks can be synchronized applies precisely when the speed to be measured is small compared to the (maximum) signal speed with which the clocks are synchronized, i.e. H. . ${\ displaystyle v}$${\ displaystyle c}$${\ displaystyle v \ ll c}$

### The relationship to quantum mechanics

In contrast to quantum mechanics, mass points with identical observables (mass, location, momentum) can be differentiated, while in quantum mechanics one assumes indistinguishable entities . This means that classical bodies must be macroscopic in the sense that they have individual properties that make them distinguishable. Thus, z. B. Do not consider elementary particles of a family as classical mass points. The distinguishability of a classical particle stems from the fact that, when it is left to itself, it remains in its previous inertial system. This is not the case for a particle described by quantum mechanics, since a particle left to itself does not necessarily remain in its inertial system. This fact can be derived in quantum mechanics, in which one the Schrödinger - initial value problem will solve for the wave function of a particle whose probability at a time is accurately located in a place (a so-called peak). The probability of presence begins to dissipate with increasing time. ${\ displaystyle t = 0}$${\ displaystyle \ delta}$