Pseudo vector

The angular momentum L as an example of a pseudo-vector: while the position vector r and momentum m · v reverse their direction in a point reflection, that of the angular momentum L = m · r × v remains unchanged.

A pseudo-vector , and rotation vector , axial vector or axial vector called, is in the physics a vector quantity , which at a point mirroring maintains its direction of the observed physical system. In contrast, polar or shear vectors reverse their direction when a point is mirrored.

The picture shows a body with a rotary movement and its mirror image. The angular momentum does not change with the point reflection, because the rotational speed is described by an axial vector. The web speed is after the point of reflection as the pulse in the opposite direction and is therefore a polar vector.

The direction of an axial vector is defined in terms of an orientation of space , usually right-handed . Axial vectors typically occur when a physical relationship is expressed by the cross product (which uses the right-hand rule in right-handed coordinate systems.)

definition

Transformation behavior under a movement of the system

Let one physical system and a second one be given, which emerges from the first one at every point in time through always the same spatial movement χ (i.e. through an image that is true to length and angle, no movement in the kinematic sense !). For χ, movements in the opposite direction (reversing orientation) are allowed. In the first system, at a fixed point in time t , a particle that is at location P (t) is mapped onto a particle at location P '(t) in the moving system. The mass and charge of the particle remain unchanged. For continuous distributions this means that a density of a density with being imaged. It is said that a physical quantity has a certain transformation behavior under movement if this transformation maps the physical quantity to the corresponding quantity in the moving system. For example, the moving particle has at the location P ' the transformed speed represented by the speed of the original particle at the location P is determined. ${\ displaystyle \ rho}$ ${\ displaystyle \ rho '}$ ${\ displaystyle \ rho '(t, P') = \ rho (t, P)}$ ${\ displaystyle {\ vec {v}} '}$ ${\ displaystyle {\ vec {v}}}$

Axial and polar vectors

If the movement χ is made up of only shifts and rotations , the transformation behavior is the same for all vector quantities. If, on the other hand, one considers the case of point reflection in space at the center , ie , where and are the position vectors of a particle and its mirror image, then two cases must be distinguished. A polar vector , such as the speed of the particle is characterized in that it transforms as the position vectors: . An axial vector , such as the angular velocity of the particle on the other hand displayed below the point reflection on themselves: . The property of a vector quantity to be axial or polar already determines the transformation behavior under any movement χ. Because every movement can be represented by executing translations, rotations and point reflections one after the other. ${\ displaystyle P_ {Z}}$ ${\ displaystyle {\ vec {r}} '= - {\ vec {r}}}$ ${\ displaystyle {\ vec {r}} = {\ overrightarrow {P_ {Z} \, P}}}$ ${\ displaystyle {\ vec {r}} '= {\ overrightarrow {P_ {Z} \, P'}}}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {v}} '= - {\ vec {v}}}$ ${\ displaystyle {\ vec {\ omega}}}$ ${\ displaystyle {\ vec {\ omega}} '= {\ vec {\ omega}}}$

Active and passive transformation

These considerations regarding the transformation behavior of a vector quantity under an active movement χ of the system has nothing to do with the transformation behavior of the components of the vector under an ordinary coordinate transformation . The latter is the same for axial and polar vectors, namely the coordinates of a tensor of rank one. So they are real vectors in the sense of tensor calculus, which is why the term pseudo vector is misleading in this context. In fact, there are authors who do not clearly separate these different terms. Many authors describe an inconsistent movement of the system as a coordinate transformation with a simultaneous change in the orientation with respect to which the cross product is to be calculated. This corresponds to a passive transformation, whereby the observer experiences the same transformation as the coordinate system. This clearly means that the right hand becomes a left hand when the coordinate system is mirrored. In terms of calculation, this is achieved by introducing a pseudotensor , the components of which are given by the Levi-Civita symbol , regardless of the orientation of an orthonormal coordinate system . This completely antisymmetric pseudotensor (also called tensor density with weight -1) is not a tensor. In this sense, the term pseudo vector is to be understood, which in this view changes its direction in the event of a point reflection of the coordinate system (whose components, however, remain unchanged). This passive view yields the same results as the active one with regard to the distinction between axial and polar vectors.

Calculation rules

The cross product of two polar or two axial vectors is an axial vector.
The cross product of a polar and an axial vector is a polar vector.
The scalar product of two polar or two axial vectors is a scalar (ie it retains its sign under any motion).
The scalar product of a polar and an axial vector is a pseudoscalar (ie changes its sign under a point reflection).

Relationship with tensors

Every second-order tensor has a vector invariant in three-dimensional space , which as such is an axial vector. Only the skew-symmetric part of the tensor contributes to the vector invariant . The reverse operation produces the skew-symmetric part of the tensor from the axial vector:

{\ displaystyle {\ vec {a}} \ times \ mathbf {1} = {\ vec {a}} \ times \ left (\ sum _ {i = 1} ^ {3} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {i} \ right) = \ sum _ {i = 1} ^ {3} ({\ vec {a}} \ times {\ hat {e}} _ {i}) \ otimes {\ hat {e}} _ {i} = {\ begin {pmatrix} 0 & -a_ {3} & a_ {2} \\ a_ {3} & 0 & -a_ {1} \\ - a_ {2} & a_ {1} & 0 \ end {pmatrix}} = [{\ vec {a}}] _ {\ times}.}

Here a _{1,2,3 are} the coordinates of the vector with respect to the standard basis , 1 is the unit tensor , "×" is the cross product and " " is the dyadic product . The result is the cross product matrix in coordinate space . ${\ displaystyle {\ vec {a}}}$ ${\ displaystyle {\ hat {e}} _ {1,2,3}}$ ${\ displaystyle \ otimes}$ ${\ displaystyle [{\ vec {a}}] _ {\ times}}$

The vortex strength is the negative vector invariant of the velocity gradient and with the above reverse operation, its skewly symmetrical part, the vortex tensor, is created. With a rigid body motion corresponding angular velocity of Winkelgeschwindigkeitstensor who takes over the role of the velocity here. For the magnetic field B is obtained in this way, the spatial components of the electromagnetic Feldstärketensors .

More generally, an axial vector on the Hodge-duality is a skew-symmetric tensor of second order are allocated. Expressed in coordinates, a vector has the 2-form (for positively or negatively oriented orthonormal basis) with the Levi-Civita symbol and using the sum convention . This relationship can be used to generalize quantities such as angular momentum for spaces with dimensions other than three. Only in R ³ does an antisymmetric 2-form have as many independent components as a vector. In R ^4, for example, there are not 4, but 6 independent components. ${\ displaystyle a_ {i}}$ ${\ displaystyle (* a) _ {ij} = \ mp \ epsilon _ {ijk} a_ {k}}$ ${\ displaystyle \ epsilon _ {ijk}}$

Examples

The following applies to the relationship between position vector , velocity and angular velocity of a particle . With a point reflection it is easy to check that . Position and velocity vectors are therefore polar vectors. This applies to the angular velocity of the reflected particle . So it must apply, ie the angular velocity is an axial vector. ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {\ omega}}}$ ${\ displaystyle {\ vec {v}} = {\ vec {\ omega}} \ times {\ vec {r}}}$ ${\ displaystyle {\ vec {r}} '= - {\ vec {r}}}$ ${\ displaystyle {\ vec {v}} '= - {\ vec {v}}}$ ${\ displaystyle {\ vec {\ omega}} '}$ ${\ displaystyle - {\ vec {v}} = {\ vec {\ omega}} '\ times (- {\ vec {r}})}$ ${\ displaystyle {\ vec {\ omega}} '= {\ vec {\ omega}}}$

The angular momentum is defined as . It follows , so the angular momentum is an axial vector, see introduction.

{\ displaystyle {\ vec {L}} = {\ vec {r}} \ times {\ vec {p}}}

{\ displaystyle {\ vec {L}} '= {\ vec {r}}' \ times {\ vec {p}} '= {\ vec {r}}' \ times m {\ vec {v}} ' = (- {\ vec {r}}) \ times (-m {\ vec {v}}) = {\ vec {L}}}

From the formula for the Lorentz force it follows that the magnetic field must be an axial vector, because the force is proportional to the acceleration and therefore a polar vector. ${\ displaystyle {\ vec {F}} = - q {\ vec {B}} \ times {\ vec {v}}}$ ${\ displaystyle {\ vec {B}}}$ ${\ displaystyle {\ vec {F}}}$

The vortex strength with the rotation of a vector field rot is an axial vector. ${\ displaystyle {\ vec {\ omega}} = \ operatorname {red} \; {\ vec {v}}}$

Reflection of a rotating disk on a plane: A rotating horizontal disk is considered that has a red top and a yellow bottom. The rotation is described by the angular velocity vector. The direction of rotation is such that the angular velocity vector points upwards from the red top. In the case of a mirror image of this rotating disk on a horizontal plane, the vertical components of position vectors are reversed, the upper side is yellow in the mirror image and the lower side is red. The edge of the disc facing the observer moves in the same direction in the original as in the mirror image: the direction of rotation is retained. The angular velocity vector has not reversed itself due to the reflection and also points upwards on the mirror image from the yellow side.

Individual evidence

^ Richard P. Feynman , Robert B. Leighton , Matthew Sands : The Feynman Lectures on Physics . Vol. 1, Mainly mechanics, radiation, and heat. Addison-Wesley , 1964, Section 52-5, pp. 52-6-52-7 ( Online Edition , Caltech ).
↑ axial vector. In: Lexicon of Physics. Spectrum Akademischer Verlag, accessed on July 23, 2008 : “The components of axial vectors remain when the coordinate system is mirrored, i. H. with a sign reversal of all three coordinates, unchanged; ... "
↑ Eric W. Weisstein: Pseudovector. In: MathWorld - A Wolfram Web Resource. Retrieved on July 23, 2008 : "A typical vector (...) is transformed to its negative under inversion of its coordinate axes."
^ Arnold Sommerfeld : Mechanics . In: Lectures on Theoretical Physics . 8th edition. tape I . Harri Deutsch, 1994, ISBN 3-87144-374-3 , pp. 105 .
^ Herbert Goldstein , Charles Poole, John Safko: Classical mechanics . 3. Edition. Addison-Wesley, 2000, pp. 169 .
^ H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 , pp. 34 f. and 109 f .