Perturbation theory (classical physics)

The perturbation calculation is a branch of applied mathematics . It is mainly used in physics and celestial mechanics and deals with the effects of small disturbances on a system.

Calculation methodology

Perturbation calculation is a solution approach for physical systems based on series development . Similarly, the error calculation , for example, a mass as m = 5 is represented kg +/- 0.2 in which, when only 0.2 kg is accurately known, a slightly varying size of the mold also in the perturbation theory used, the Disturbance parameter represents a disturbance of the output data assumed to be small. In contrast to the calculation of errors, is a variable. It is believed that the problem, e.g. B. a continuum mechanical problem, a trajectory , a differential equation , or a linear system of equations , in a known developable form, depends on the disturbance. As a rule, a representation is sought so that the solution in holomorphic form depends on the disturbance. Now you make a perturbation approach and represent the solution you are looking for as a series expansion in the perturbation parameter. You get an analytical, graduated system of equations for the expansion coefficients of the solution and can thereby determine them. ${\ displaystyle x = x_ {0} + \ varepsilon}$ ${\ displaystyle \ varepsilon}$ ${\ displaystyle \ varepsilon}$

It should be noted that certain solution components can be of a singular nature in the development parameter, and that multi-dimensional solution spaces can be divided into different solution branches due to the interference parameter . Example: In elasticity / acoustics, rigid body displacements are divided into longitudinal and transverse waves when considering small wave numbers . In the mathematical theory required for this, relatively compact operator properties are particularly relevant.

Perturbation calculation analytically continues a solution in the vicinity of a solution point in any order as a function of an interference parameter. By successively applying this continuation one obtains the so-called homotopy method , with which solutions can be pursued beyond the actual convergence interval of the developments.

principle

The differential equation of the nth order is given

{\ displaystyle F (t, y, {\ dot {y}}, {\ ddot {y}}, \ cdots, y ^ {(n)}, \ varepsilon) = 0}

with as a small parameter. ${\ displaystyle 0 <\ varepsilon \ ll 1}$

The following series of functions is used for an approximate solution :

{\ displaystyle y = y_ {0} + \ varepsilon \ cdot y_ {1} + \ varepsilon ^ {2} \ cdot y_ {2} + \ cdots + \ varepsilon ^ {n} \ cdot y_ {n}}

Insertion into the differential equation and comparison of coefficients with respect to ε results in a system of differential equations for the functions . For the undisturbed system with ε = 0 the solution is. If the undisturbed system can be solved analytically, at least the first approximation of the disturbance can often also be solved analytically. ${\ displaystyle y_ {i}}$ ${\ displaystyle y = y_ {0}}$

example

The differential equation of a vibratory system with Newtonian friction

{\ displaystyle {\ ddot {x}} + \ varepsilon {\ dot {x}} ^ {2} + x = 0}

the initial conditions

{\ displaystyle x (0) = 1}

{\ displaystyle {\ dot {x}} (0) = 0}

and the small coefficient of friction ε can be solved approximately analytically by means of the first order perturbation calculation using the approach

{\ displaystyle x = x_ {0} + \ varepsilon x_ {1}}

Inserting into the differential equation and sorting according to powers of ε, whereby only terms of the first order are taken into account, since ε is very small according to the assumption, the differential equation system yields:

{\ displaystyle {\ ddot {x}} _ {0} + x_ {0} = 0}

{\ displaystyle {\ ddot {x}} _ {1} + x_ {1} = - {\ dot {x}} _ {0} ^ {2}}

with the initial conditions

{\ displaystyle x_ {0} (0) = 1}

{\ displaystyle {\ dot {x}} _ {0} (0) = 0}

{\ displaystyle x_ {1} (0) = 0}

{\ displaystyle {\ dot {x}} _ {1} (0) = 0}

.

The solutions taking into account the initial conditions are

{\ displaystyle x_ {0} (t) = \ cos (t)}

{\ displaystyle x_ {1} (t) = - {\ tfrac {1} {3}} (\ cos (t) -1) ^ {2}}

and thus the solution is in the 1st fault order

{\ displaystyle x (t) = x_ {0} (t) + \ varepsilon x_ {1} (t) = \ cos (t) - {\ tfrac {1} {3}} \ varepsilon \ left (\ cos ( t) -1 \ right) ^ {2}}

.

The solution in the 2nd fault order is obtained with the approach

{\ displaystyle x = x_ {0} + \ varepsilon x_ {1} + \ varepsilon ^ {2} x_ {2}}

.

Insertion into the differential equation yields the same equations for and . For one finds: ${\ displaystyle x_ {0}}$ ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$

{\ displaystyle {\ ddot {x}} _ {2} + x_ {2} = - 2 {\ dot {x}} _ {0} {\ dot {x}} _ {1} = {\ tfrac {4 } {3}} \ sin ^ {2} (t) (\ cos (t) -1)}

This procedure can be continued for any order of ε.

Development history

The trigger for research in the area of perturbation calculus was the discovery in the first 1820s that the orbit of the planet Uranus deviates from the previous calculations. In 1844, the French mathematician and astronomer Urbain Le Verrier began using perturbation calculations to calculate part of the orbit of an imaginary planet in order to explain the deviations in the orbit of Uranus. As a result, the German astronomer Johann Gottfried Galle observed in 1846 only a deviation of one degree from the calculated orbit. A few days later he was able to determine a movement of a newly discovered celestial body, whereupon this planet was named Neptune . Due to the orbital anomalies of Uranus and Neptune, the US astronomers Percival Lowell and William Henry Pickering used perturbation calculations to calculate the orbit of Pluto , which could only be discovered decades later at the Lowell Observatory in Arizona.

The perturbation theory of celestial mechanics had less success in the field of atomic theory. Niels Bohr and Arnold Sommerfeld attempted to describe complex atoms using mechanical perturbation theory using the angle variable method . Only quantum mechanics was able to provide corresponding results.

literature

Kato Tosio : Perturbation Theory for linear operators . Springer Science + Business, New York 1966, ISBN 978-3-662-12680-6 (standard work on perturbation calculation).
Richard Bellman : Perturbation techniques in mathematics, physics and engineering . Holt, Rinehart & Winston, New York 1964.
Fault calculation. In: Lexicon of Physics. Spectrum of Science, accessed August 17, 2015 .