Angular velocity tensor

Angular velocity tensor

In physics, the angular velocity tensor is defined as a matrix T such that:

:oldsymbolomega(t) imes mathbf{r}(t) = T(t) mathbf{r}(t)

It allows us to express the cross product:oldsymbolomega(t) imes mathbf{r}(t) as a matrix multiplication. It is, by definition, a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements::T(t) =egin{pmatrix}0 & -omega_z(t) & omega_y(t) \omega_z(t) & 0 & -omega_x(t) \-omega_y(t) & omega_x(t) & 0 \end{pmatrix}

Coordinate-free description

At any instant, t, the angular velocity tensor is a linear map between the position vectors mathbf{r}(t) and their velocity vectors mathbf{v}(t) of a rigid body rotating around the origin:

: mathbf{v} = Tmathbf{r}

where we omitted the t parameter, and regard mathbf{v} and mathbf{r} as elements of the same 3-dimensional Euclidean vector space V.

The relation between this linear map and the angular velocity pseudovector omega is the following.

Because of "T" is the derivative of an orthogonal transformation, the

:B(mathbf{r},mathbf{s}) = (Tmathbf{r}) cdot mathbf{s}

bilinear form is skew-symmetric. (Here cdot stands for the scalar product). So we can apply the fact of exterior algebra that there is a unique linear form L on Lambda^2 V that

:L(mathbf{r}wedge mathbf{s}) = B(mathbf{r},mathbf{s}) ,

where mathbf{r}wedge mathbf{s} in Lambda^2 V is the wedge product of mathbf{r} and mathbf{s}.

Taking the dual vector "L"* of "L" we get

: (Tmathbf{r})cdot mathbf{s} = L^* cdot (mathbf{r}wedge mathbf{s})

Introducing omega := *L^* , as the Hodge dual of "L"* , and apply further Hodge dual identities we arrive at

: (Tmathbf{r}) cdot mathbf{s} = * ( *L^* wedge mathbf{r} wedge mathbf{s}) = * (omega wedge mathbf{r} wedge mathbf{s}) = *(omega wedge mathbf{r}) cdot mathbf{s} = (omega imes mathbf{r}) cdot mathbf{s}

where :omega imes mathbf{r} := *(omega wedge mathbf{r})

by definition.

Because mathbf{s} is an arbitrary vector, from nondegeneracy of scalar product follows

: Tmathbf{r} = omega imes mathbf{r}

See also

* Angular velocity
* Rigid body dynamics


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Angular velocity — Do not confuse with angular frequency The unit for angular velocity is rad/s.In physics, the angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed, and axis about which an object is rotating. The …   Wikipedia

  • Angular momentum — For a generally accessible and less technical introduction to the topic, see Introduction to angular momentum. Classical mechanics Newton s Second Law …   Wikipedia

  • Stress-energy tensor — The stress energy tensor (sometimes stress energy momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • Moment of inertia — This article is about the moment of inertia of a rotating object, also termed the mass moment of inertia. For the moment of inertia dealing with the bending of a beam, also termed the area moment of inertia, see second moment of area. In… …   Wikipedia

  • solids, mechanics of — ▪ physics Introduction       science concerned with the stressing (stress), deformation (deformation and flow), and failure of solid materials and structures.       What, then, is a solid? Any material, fluid or solid, can support normal forces.… …   Universalium

  • Continuum mechanics — Continuum mechanics …   Wikipedia

  • Rigid rotor — The rigid rotor is a mechanical model that is used to explain rotating systems. An arbitrary rigid rotor is a 3 dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the… …   Wikipedia

  • Force — For other uses, see Force (disambiguation). See also: Forcing (disambiguation) Forces are also described as a push or pull on an object. They can be due to phenomena such as gravity, magnetism, or anything that might cause a mass to accelerate …   Wikipedia

  • Rigid body — Classical mechanics Newton s Second Law History of classical mechanics  …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”