Moment (integration)

In the natural sciences and technology, moments are parameters of a distribution that describe the position and form of this distribution. They are integrating on a potentiated calculated weighted distance distribution. The task of determining the position and form of the distribution from given moments is called the moment problem .

Moments of various kinds play important roles in stochastics , technical mechanics and image processing .

Forms and characteristics

History and Development

The concept of moments has its origin in the consideration of the equilibrium of forces in scales. Franciscus Maurolicus (1494–1575) used the term "momentum" explicitly to describe the strength of the rotating force with which weights attached to a lever arm act on a scale. Galileo Galilei then showed in 1638 that the strength of such a “moment of weight” corresponds to the area of ​​the rectangle formed from distance and weight. The current, more abstract term was developed on the basis of this concept.

Continuous and discrete distributions

For the definition of a moment in discrete distributions, a distribution of mass points on a line can be considered as an example. Denotes the distance from a reference point and the mass of the i th point mass, that is n -th moment of the i th point mass is the product of mass and the n -th power of the distance: . The exponent is a natural number and is called the order or degree of the moment. To get the moment of the entire mass distribution, the moments of all point masses are added: ${\displaystyle x_{i}}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle x_{i}^{n}\cdot \alpha _{i}}$ ${\displaystyle n}$ ${\displaystyle m}$

${\displaystyle m_{n}=\sum _{i}x_{i}^{n}\alpha _{i}.}$

The zeroth moment is the total mass. The first moment describes the position of the distribution. If the first moment is divided by the total mass, which corresponds to a normalization of the distribution to one, the distance between the center of mass and the reference point is obtained. The second order moment is the mass moment of inertia (see below).

A moment for continuous distributions can be defined in the same way. Here the distribution does not consist of individual mass points, but a body with continuous mass distribution. This distribution is characterized by its density function (mass per unit length) . The moments of distribution are obtained through integration: ${\displaystyle \rho (x)}$

${\displaystyle m_{n}=\int _{\mathbb {R} }x^{n}\rho (x)\,\mathrm {d} x.}$

With the help of the Lebesgue integral , both definitions can be combined in order to define moments for more general distributions given by a measure : ${\displaystyle \mu }$

${\displaystyle m_{n}=\int _{\mathbb {R} }x^{n}\mathrm {d} \mu (x).}$

Instead of mass distributions, distributions of any other size, for example probabilities, can be considered. If a random variable with probability distribution , then the -th moment is the expected value of . The centered second moment (see below) is the variance . The variable that can be interpreted as a deviation or distance can also be selected from or instead of from . ${\displaystyle X}$ ${\displaystyle \mu }$ ${\displaystyle n}$ ${\displaystyle X^{n}}$ ${\displaystyle x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \mathbb {C} }$

Moments in several dimensions

For moments in several dimensions, the components have to be raised to the power of the basis vectors individually. This results in two dimensions for the moment p + q th order:

${\displaystyle m_{p+q}=\int x^{p}y^{q}\mathrm {d} \mu (x,y)}$

Such a moment is therefore dependent on the choice of the basis and the individual powers p and q . For example, when calculating moments of area ( ), a distinction is made between axial and mixed moments in Cartesian coordinates . For axial moments, the powers are zero up to one direction (e.g. p  = 2, q  = 0). In the case of mixed moments, also known as cross or compound moments, factors in different directions contribute (e.g. p  = 1, q  = 1). Mixed moments are the moments of deviation of an inertia tensor or the covariance of random variables. ${\displaystyle \rho =1}$

With the polar moments, it is not the center distances, but the distance to the origin, i.e. the radial component, that is exponentiated in polar coordinates . ${\displaystyle r}$

Change of the reference point and centered moments

Moments of degree greater than zero generally depend on the position of the reference point. Two moments can only be added meaningfully if they refer to the same point.

From the moment of the first degree, which refers to the origin of the coordinate system, a moment , related to , can be calculated as follows , where the zeroth moment is: ${\displaystyle m_{1}}$ ${\displaystyle m_{1}'}$ ${\displaystyle x'}$ ${\displaystyle m_{0}}$

${\displaystyle m_{1}'=\int _{\mathbb {R} }(x-x')\rho (x)\,\mathrm {d} x=\int _{\mathbb {R} }x\rho (x)\,\mathrm {d} x-x'\int _{\mathbb {R} }\rho (x)\,\mathrm {d} x=m_{1}-x'm_{0}}$

The additional term is also known as the offset moment . In general it can be shown with the binomial theorem for the conversion of a moment of degree n into a moment related to the origin shifted by ${\displaystyle x'm_{0}}$ ${\displaystyle m_{n}}$ ${\displaystyle m_{n}'}$ ${\displaystyle x'}$

${\displaystyle m_{n}'=m_{n}+\sum _{k=0}^{n-1}{\binom {n}{k}}(-x')^{n-k}m_{k}}$

For a moment of the second degree this relation is known as Steiner's theorem and in stochastics as the displacement law . If all moments are of degree zero, the moment is independent of the choice of the reference point. For example, a torque of a couple of forces is independent of the choice of the reference point, since the sum of all forces is zero. ${\displaystyle m_{m}}$ ${\displaystyle m<n}$ ${\displaystyle m_{n}}$

In order to establish comparability, the reference point is often chosen so that the first moment is zero. Such a moment is called central or centered. It then relates to the center of the distribution, for example the expected value or center of gravity. The nth centered moment is calculated through

${\displaystyle {\bar {m}}_{n}=\int _{\mathbb {R} }\left(x-{\frac {m_{1}}{m_{0}}}\right)^{n}\rho (x)\,\mathrm {d} x,}$

where the zeroth moment and the first (not centered) moment means. ${\displaystyle m_{0}}$ ${\displaystyle m_{1}}$

Moments of vector fields

Vector calculation of a moment. The direction of the moment points vertically out of the plane of the paper.

In physics there are often vector-valued quantities . In addition to their amount, they also have a direction. Moments can also be assigned to a distribution of a vector-valued quantity in space, i.e. a vector field . Such a quantity is, for example, the torque ( is the force distribution here), the magnetic moment ( is the current density distribution here) or the angular momentum (formerly also called the momentum momentum , is the momentum distribution here). ${\displaystyle {\vec {\rho }}({\vec {r}})}$ ${\displaystyle {\vec {\rho }}}$ ${\displaystyle {\vec {\rho }}}$ ${\displaystyle {\vec {\rho }}}$

For a vector field , the first order moment is a vector that is given by the integral over the cross product : ${\displaystyle {\vec {\rho }}({\vec {r}})}$ ${\displaystyle {\vec {r}}\times {\vec {\rho }}({\vec {r}})}$

${\displaystyle {\vec {m}}_{1}=\int {\vec {r}}\times {\vec {\rho }}({\vec {r}})\,\mathrm {d} ^{3}r}$

If the component of a moment is to be calculated with respect to a certain direction, only those parts of the vectors of the vector field are to be used which are orthogonal to this direction. If you choose a Cartesian coordinate system , then, for example, the z -component of the moment has to be calculated using the "densities" and . ${\displaystyle \rho _{x}({\vec {r}})}$ ${\displaystyle \rho _{y}({\vec {r}})}$

Trigonometric Moments

If only has an angle dependency, a trigonometric moment can be defined. To do this, one chooses from the complex numbers and obtains ${\displaystyle \mu }$ ${\displaystyle x=e^{\mathrm {i} \theta }}$

${\displaystyle m_{n}=\int _{0}^{2\pi }e^{\mathrm {i} \theta n}\mathrm {d} \mu (\theta )}$

Moment problem

The moment problem is a classic problem of analysis. Instead of calculating the moments from a distribution, conclusions should be drawn about a possible distribution from a given sequence of moments . The term moment problem was introduced by Thomas Jean Stieltjes , who examined the problem in detail for the first time in 1894, taking over the terms and concepts from mechanics. ${\displaystyle (m_{n})}$ ${\displaystyle \mu }$

Depending on the carrier of the distribution (that is the complement of the largest open set of measure zero), different variants of the moment problem are distinguished: In the Hamburg moment problem the carrier is the entire real axis (-∞, ∞), in the Stieltjes moment problem the semi-axis [ 0, ∞) and in the Hausdorff moment problem a bounded interval o. B. d. A. [0.1]. Another variant is the trigonometric moment problem, in which the distribution on a unit circle depending on the angle, i.e. a trigonometric moment, is sought.

If the sequence of moments is limited, the problem is called truncated , if it is unlimited the problem is called infinite .

Examples from mechanics

The force or torque

The torque is the product of the force and the lever arm. It is the most common moment in technology. The word moment is therefore often used as an abbreviation or as a synonym for torque. For special torques compound terms with part of the name -moment , but without rotation- are used. Examples are:

• Drive torque ,
• Load moment ,
• Bending moment and
• Torsional moment .

If several forces act, they can be combined to form a torque or a resulting force with a resulting lever arm. Forces distributed linearly (line force) or area ( surface pressure ) can also be summarized in this way.

Moment of area

Moments also frequently used are the area moments . In order to determine a surface moment of the surface, one chooses for the characteristic function of the surface ${\displaystyle A\subset \mathbb {R} ^{2}}$ ${\displaystyle \rho }$

${\displaystyle \rho \colon \mathbb {R} ^{2}\to \{0,1\},\ r\mapsto {\begin{cases}1,&{\text{falls }}r\in A\\0,&{\text{sonst}}\end{cases}}}$

The zeroth moment of area is the area . If you divide the moments by the area, you get the center of gravity of the area as the first area moment . The centered moment of area of ​​the second degree is the area moment of inertia , which is used as a parameter for cross-sections of beams in their strength and deformation calculations .

The triangle in the xy -plane

As an example, consider a triangle in the xy coordinate plane that is limited by the straight lines x = 4 , y = 0 and y = x / 2 . The area is

${\displaystyle m_{0,0}=\int _{-\infty }^{\infty }\mathrm {d} x\int _{-\infty }^{\infty }\mathrm {d} y\;x^{0}y^{0}\rho (x,y)=\int _{0}^{4}\mathrm {d} x\int _{0}^{x/2}\mathrm {d} y=4}$

The x coordinate of the center of gravity is

${\displaystyle S_{x}={\frac {m_{1,0}}{m_{0,0}}}={\frac {1}{4}}\int _{-\infty }^{\infty }\mathrm {d} x\int _{-\infty }^{\infty }\mathrm {d} y\;x^{1}y^{0}\rho (x,y)={\frac {1}{4}}\int _{0}^{4}\mathrm {d} x\int _{0}^{x/2}\mathrm {d} y\;x={\frac {8}{3}}}$

The axial geometrical moment of inertia around the y- axis is calculated from the square of the x -distance to the center of gravity:

${\displaystyle I_{y}={\bar {m}}_{2,0}=\int _{0}^{4}\mathrm {d} x\int _{0}^{x/2}\mathrm {d} y\;(x-S_{x})^{2}={\frac {32}{9}}}$

Mass moment of inertia

cylinder

The (mass) moment of inertia of a body is related to a specific axis of rotation. It indicates how strongly the body resists a spin. The moment of inertia is a second straight line moment in cylinder coordinates at which the distance to the axis of rotation is squared. It is calculated through integration over a mass distribution , where the mass density (mass per volume) is the volume element . ${\displaystyle J}$ ${\displaystyle r}$ ${\displaystyle \mathrm {d} \mu =\rho \ \mathrm {d} V}$ ${\displaystyle \rho }$ ${\displaystyle \mathrm {d} V}$

As an example, consider a homogeneous cylinder with constant density , radius , height and mass . The moment of inertia of this cylinder for a rotation around the z -axis is then given by the integral: ${\displaystyle \rho _{0}}$ ${\displaystyle R}$ ${\displaystyle h}$ ${\displaystyle m_{Z}=\rho _{0}\pi R^{2}h}$

${\displaystyle J=\int _{\mathbb {R} ^{3}}\mathrm {d} V\;r^{2}\rho (r,\phi ,z)=\rho _{0}\int _{0}^{2\pi }\mathrm {d} \phi \int _{0}^{h}\mathrm {d} z\int _{0}^{R}\mathrm {d} r\;r^{3}={\frac {\rho _{0}\pi hR^{4}}{2}}={\frac {1}{2}}m_{Z}R^{2}}$

Individual evidence

1. ^ John J. Roche: The Mathematics of Measurement: A Critical History . Springer, 1998, ISBN 0-387-91581-8 , pp. 98 ff . ( limited preview in Google Book search).
2. ↑ Vladimir I. Smirnov: Course in Higher Mathematics Part 2 . Harri Deutsch Verlag, 1990, ISBN 3-8171-1298-X , p. 198 ( limited preview in Google Book Search).
3. ^ Palle ET Jørgensen, Keri A. Kornelson, Karen L. Shuman: Iterated Function Systems, Moments, and Transformations of Infinite Matrices Memoirs of the American Mathematical Society . American Mathematical Society, 2011, ISBN 0-8218-8248-1 , pp. 2 ( limited preview in Google Book search).
4. ↑ Volker Läpple: Introduction to strength theory . Springer, 2011, ISBN 3-8348-1605-1 , p. 171 ( limited preview in Google Book search).
5. ^ Analysis of Binary Images , University of Edinburgh
6. ^ NI Fisher: Statistical Analysis of Circular Data . Cambridge University Press, 1995, ISBN 0-521-56890-0 , pp. 41 ( limited preview in Google Book search).
7. ^ Thomas Jean Stieltjes: Recherches sur les Fractions continues . 1894 ( numdam.org [PDF]).
8. ^ Gene H. Golub, Gérard Meurant: Matrices, Moments and Quadrature with Applications . Princeton University Press, 2009, ISBN 1-4008-3388-4 , pp. 15 ( limited preview in Google Book search).
9. ↑ James Alexander Shohat, Jacob David Tamarkin: The Problem of Moments . American Mathematical Society, 1943, ISBN 0-8218-1501-6 , pp. vii ( limited preview in Google Book Search).
10. ^ Henry J. Landau: Moments in Mathematics . American Mathematical Society, 1987, ISBN 0-8218-0114-7 , pp. 1 ( limited preview in Google Book search).
11. ↑ Wolfgang Demtröder: Experimentalphysik 1: Mechanics and heat . Springer DE, May 1, 2008, ISBN 978-3-540-79295-6 , p. 67– (accessed on July 20, 2013).
12. ^ Lev D. Landau: Mechanics . Harri Deutsch Verlag, 1997, ISBN 978-3-8171-1326-2 , p. 133– (accessed on July 20, 2013).
13. ↑ Dubbel - pocket book for mechanical engineering , Chapter B "Mechanics, Kinematics", Sections 1.1 and 3.1
14. ↑ Lothar Papula: Mathematics for Engineers and Natural Scientists 1 . 2007, ISBN 978-3-8348-0224-8 , pp. 536 (Static Moment of a Force).
15. ^ Wolfgang Brauch, Hans-Joachim Dreyer, Wolfhart Haacke: Mathematics for engineers . 11th edition. Teubner, 2006, ISBN 3-8351-0073-4 , p. 372 ( limited preview in Google Book search).