Integral of movement

An integral of the movement or first integral ( English first integral ) is for a given dynamic system a feature that a longitudinal trajectory of the system is constant. For example, according to the law of conservation of energy , the total energy of an isolated system is invariable and a significant integral of motion.

The constants of motion , as the integrals are also called, have fundamental representatives in the conserved quantities, but unlike these do not have to restrict the motion, but are rather classification features of a type of motion. In the case of the Kovalevskaya top, the Kovalevskaya constant determines the stability of the top and the specific orbital energy of a celestial body, whether its Kepler orbit is elliptical, hyperbolic or parabolic. Often the integrals allow conclusions to be drawn about the further course of the path and help in solving the equations of motion .

Definitions

In the literature there are differently formulated definitions: (t is the independent variable (time), x ∈ V ⊆ ℝⁿ the solution function (location) and v the time derivative of x)

An integral of the movement of a movement type is a function F (x, v), which is constant on any path of the movement type and only depends on the path as a whole and therefore solely on the initial conditions.
The integral of the movement is a function of the coordinates along a phase space - trajectory remains constant.
For a given dynamic system, an integral of the motion is any real-valued, infinitely often differentiable function (∈ C ^∞ ) that is constant along the integral curves of the vector field on which the system is based .
A first integral of an ordinary differential equation D (t, x, v) = 0 is a (non-constant) continuously differentiable function F (t, x), which is locally constant on a solution x (t) of D = 0.
First integrals of Newton's second law, force equals mass times acceleration, are called equations of the form F (x, v, t) = const. on the nature that the time derivative dF / dt vanishes identically by virtue of Newton's law.

General

The point mechanism considers the movement of mass points in which a first integral depends only on the location and the velocity of the point, but along a curved path is fixed. The value of the constant is therefore fixed with the initial conditions , i.e. the initial position and the initial speed. If six independent integrals can be found for such a system, then the location can be determined from them as a function of time and the initial conditions, so that the trajectory is completely known.

In the theory of the heavy gyro there are always three first integrals (the Euler-Poisson equations ) for six unknowns. If a fourth integral is found, then a fifth integral can be constructed using a method devised by Carl Gustav Jacob Jacobi , with which the equations of motion are solved. Because one of the six unknowns takes on the role of the independent variable, since the time does not appear explicitly in the equations.

In physical laws , equations of motion are usually systems of differential equations of the second order, such as Newton's law of gravitation or Coulomb's law . A constant that only depends on the position and the velocity can be developed in such systems into a Taylor series by continuing the time derivative of the equation of motion , see solution of the N-body problem with a Taylor series . In most cases, however, a first integral is understood to be a function that is built up in a simple manner from elementary functions of its arguments, with a quadrature also occasionally having to be carried out.

In general, the quantities only remain unchangeable under special, idealized conditions - in the mathematical model - as in the example of the total energy of an isolated system mentioned above, because the suppression of any interaction between the system and its environment can only be ensured temporarily and approximately in reality, see irreversible process .

An explicit dependence of the integrals on time, as in the last of the following examples, is permitted or not, depending on the source, and the integrals are also called motion constants or differentiated from them.

Examples

At constant acceleration is where c is a constant and the superpoints form the second derivative of time . The function ${\ displaystyle {\ ddot {x}} = c}$

{\ displaystyle F (x, {\ dot {x}}) = {\ frac {{\ dot {x}} ^ {2}} {2}} - cx = {\ text {const.}}}

is then an integral of the movement, which can be verified by deriving it from time. A harmonic oscillator is an oscillatory system that obeys the differential equation. Here is ${\ displaystyle {\ ddot {x}} + c ^ {2} x = 0}$

{\ displaystyle F (x, {\ dot {x}}) = {\ frac {{\ dot {x}} ^ {2}} {2}} + {\ frac {c ^ {2}} {2} } x ^ {2} = {\ text {const.}}}

Motion constant. If there is a potential where the dash 'indicates the derivative with respect to x and the minus sign is a convention, the integral is found ${\ displaystyle {\ ddot {x}} = - V '(x)}$

{\ displaystyle F (x, {\ dot {x}}) = {\ frac {{\ dot {x}} ^ {2}} {2}} + V (x) = {\ text {const.}} }

In these examples, x can also stand for an n-dimensional vector, where the line 'then indicates the formation of the gradient and the square (.) ^{2 is formed} with the scalar product .

The last form arises when a mass point moves in a potential field , such as the gravitational field , where the first term in the motion integral represents the kinetic energy , the second the potential energy and the integral the total energy of the mass point. If the potential field is radially symmetric, i.e. the gradient is proportional to the position vector, with scalar c and components , then the differences are ${\ displaystyle {\ ddot {x}} = - \ operatorname {grad} (V) = c (| x |) x}$ ${\ displaystyle {\ ddot {x}} _ {i} = c (| x |) x_ {i}}$

{\ displaystyle F_ {ij} = {\ dot {x}} _ {i} x_ {j} -x_ {i} {\ dot {x}} _ {j}}

Constants of motion. In the three-dimensional space of our perception, these are the components of the angular momentum , which is therefore an integral of the movement in a central force field .

An example with an explicit dependence of the integral on time is given by the uniform motion . Is with her ${\ displaystyle {\ dot {x}} = v}$

{\ displaystyle F (x, t) = x-vt}

constant.

Methods for obtaining the integrals

The following methods are used to obtain the integrals:

In the more or less systematic search for relationships in experimental or numerically simulated data, constants can be noticed and subsequently mathematically proven as such using the equations of motion.
In the gyro theory , general approaches provided with parameters were made with success and those parameters were sought based on the equations of motion that lead to constants.
In the Lagrange formalism , cyclic coordinates indicate first integrals.
With the Hamilton-Jacobi formalism , cyclic coordinates are systematically constructed, with the finding of an integral shifting to the solution of the Hamilton-Jacobi differential equation .

Footnotes

↑ ^a ^b ^c ^d ^e ^f Falk (1966), p. 18 ff.
↑ ^a ^b ^c ^d ^e Stäckel (1908), p. 462 ff.
↑ ^a ^b Guido Walz (Ed.): Lexicon of Mathematics . 2nd edition volume 3 (Inp to Mon). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53501-1 , p. 2 , doi : 10.1007 / 978-3-662-53502-8 .
↑ ^a ^b integral of motion. In: Lexicon of Physics. Spektrum Akademischer Verlag, 1998, accessed March 4, 2020 .
↑ ^a ^b ^c N. N. Ladis: First integral. In: Encyclopedia of Mathematics. Springer Nature in cooperation with the European Mathematical Society , January 15, 2015, accessed on March 6, 2020 .
↑ The method of the last multiplier ( English last multiplier ) see Carl Gustav Jacob Jacobi : Lectures on dynamics . Ed .: A. Clebsch. G. Reimer Verlag, Berlin 1884, p. 73 ff . ( archive.org [accessed March 7, 2020]).
↑ Eugene Leimanis: The general problem of the motion of coupled rigid bodies around a fixed point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 978-3-642-88414-6 , p. 10 , doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google book search - original title: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point .).
↑ ^a ^b Constant of motion. Wikipedia , November 5, 2019, accessed March 6, 2020 .
↑ constant of movement. In: Lexicon of Physics. Spektrum Akademischer Verlag, 1998, accessed March 4, 2020 .

literature

Gottfried Falk : Theoretical physics based on general dynamics . Elementary point mechanics. 1st volume. Springer-Verlag, Berlin, Heidelberg 1966, DNB 456597212 , p. 18th ff ., doi : 10.1007 / 978-3-642-94958-6 .
Paul Stäckel , edited by Felix Klein and Conrad Müller : Encyclopedia of Mathematical Sciences including its applications . Mechanics. Ed .: Academies of Sciences in Göttingen , Leipzig, Munich and Vienna. Fourth volume, 1st part volume, Art. 6.1: Point dynamics. BG Teubner Verlag , 1908, ISBN 978-3-663-16021-2 , p. 462 ff ., doi : 10.1007 / 978-3-663-16021-2 ( wikisource.org [accessed January 24, 2020]).

[falk18f-1] ↑ ^a ^b ^c ^d ^e ^f Falk (1966), p. 18 ff.

[staeckel462-2] Stäckel (1908), p. 462 ff.

[lexmath-3] Guido Walz (Ed.): Lexicon of Mathematics . 2nd edition volume 3 (Inp to Mon). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53501-1 , p. 2 , doi : 10.1007 / 978-3-662-53502-8 .

[spekint-4] integral of motion. In: Lexicon of Physics. Spektrum Akademischer Verlag, 1998, accessed March 4, 2020 .

[ladis-5] N. N. Ladis: First integral. In: Encyclopedia of Mathematics. Springer Nature in cooperation with the European Mathematical Society , January 15, 2015, accessed on March 6, 2020 .

[6] The method of the last multiplier ( English last multiplier ) see Carl Gustav Jacob Jacobi : Lectures on dynamics . Ed .: A. Clebsch. G. Reimer Verlag, Berlin 1884, p. 73 ff . ( archive.org [accessed March 7, 2020]).

[7] Eugene Leimanis: The general problem of the motion of coupled rigid bodies around a fixed point . Springer Verlag, Berlin, Heidelberg 1965, ISBN 978-3-642-88414-6 , p. 10 , doi : 10.1007 / 978-3-642-88412-2 (English, limited preview in Google book search - original title: The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point .).

[enCoM-8] Constant of motion. Wikipedia , November 5, 2019, accessed March 6, 2020 .

[specbew-9] constant of movement. In: Lexicon of Physics. Spektrum Akademischer Verlag, 1998, accessed March 4, 2020 .