Angular velocity tensor

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Topic: Angular velocity tensor

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	Moment of inertia - Enpsychlopedia
		However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity.
		For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.
		In the case of the principal axes, the angular momentum is parallel to the angular velocity, so the object can be rotated in free space without an external torque applied.
	www.grohol.com /wiki/Moment_of_inertia

	A Unified Field Theory by N.F.J. Matthews
		A vanishing divergence of the source tensor of the Bianchi-type field equations represents a set of six equations that we interpret locally as conservation laws of angular momentum and velocity of center of gravity.
		In spherical coordinates, for example, the z component of angular momentum density would have to be computed from superposition of two components of angular momentum density, namely, the radial component and the copolar-angle component, each of which would have to be determined separately.
		Clearly, the local vanishing of the divergence of the conformal tensor in (2.38) is a necessary consequence of Einstein's empty-space field equations (2.1) locally.
	www.nofreewill.com /uft (10618 words)

	Inertia Tensor
		The angular momentum of a rigid body rotating about an axis passing through the origin of the local reference frame is in fact the product of the inertia tensor of the object and the angular velocity.
		A 2-dimensional symmetric matrix is not necessarily a tensor of the 2nd rank.
		This property is explained in detail in Transformation of the Inertia Tensor.
	kwon3d.com /theory/moi/iten.html (237 words)

	AZCompuguy - ScottoBobScotto - Subspace Technology
		Once the second layer is activated, the total energy of the warp field is _divided_ among the two stages of the tensor, and thus among the two layers of the subspace field.
		The energy of the ship itself (associated with it's velocity) has only increased by a factor of 2, while the warp engines are now having to output eight times as much power into the warp field because the higher energy warp field is much less efficient (quickly bleeding it's energy back into normal space).
		If these two total angular momentums are to be equal, then we must set the t3 angular momentum to zero.
	www.azcompuguy.com /subspace.htm (15696 words)

	Euclidean Tensors
		If the components of an antisymmetric tensor are the infinitesimal velocities of points of a fluid relative to a certain point, then the associated vector is twice the infinitesimal angle of angular velocity of rotation, the vorticity, and is a pseudovector.
		The 23 component of the antisymmetric tensor is the same as the 1 component of c, for example.
		The pressure tensor in a liquid is a multiple of this tensor, expressing the property that the pressure is independent of direction.
	www.du.edu /~jcalvert/math/eucltens.htm (5403 words)

	mass, forces and angular acceleration - GameDev.Net Discussion Forums
		Meanwhile, the angular inertia is expressed as a constant inertia tensor in OBJECT SPACE.
		i assume angular momentum is roughly the same as linear momentum, p=mv, but where v is now the angular velocity.
		as the force cannot disappear, the force resulting in acceleration of the mass and the force resulting in the angular acceleration of the mass must sum to equal the force being applied at the application point.
	www.gamedev.net /community/forums/viewreply.asp?ID=884456 (4240 words)

	Moment of inertia - Wikipedia, the free encyclopedia
		However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity.
		Rotational versions of Newton's second law and the formulas for momentum and kinetic energy, use moment of inertia in place of the mass of an object (with torque, angular velocity and angular acceleration replacing force, velocity and acceleration, respectively).
		In the case of rotation about a principal axis with constant angular velocity, the angular momentum is, like the angular velocity, along this axis, so it remains constant.
	en.wikipedia.org /wiki/Rotational_inertia (704 words)

	Physics - Rotation - Changing Frame-Of-Reference - Martin Baker
		As covered in kinematics the absolute angular velocity of a solid object is the angular velocity in the local coordinates plus the angular velocity of the frame of reference.
		As covered in kinematics the absolute velocity of a point is the velocity in the local coordinates plus the velocity of the frame of reference.
		Inertia [I] I think the inertial tensor defined in the inertial frame is always used as this has constant terms.
	www.euclideanspace.com /physics/dynamics/rotation/rotationfor (704 words)

	Tensors and Ellipsoids
		Since the angular velocity passes through the point of contact P, there is no relative motion between the energy ellipsoid and the invariant plane: the energy ellipsoid must roll on the invariant plane without slipping.
		If we consider a space in which the components of the angular velocity are the coordinates, then this equation also defines an ellipsoid in this space, which is similar to the inertia ellipsoid.
		The angular velocity must terminate on the energy ellipsoid in any case.
	www.du.edu /~jcalvert/phys/ellipso.htm (704 words)

	Moment of inertia - Wikipedia, the free encyclopedia
		However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity.
		For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.
		Alternatively the elements of the inertia tensor can be expressed as:
	www.wikipedia.org /wiki/Moment_of_inertia (936 words)

	Moment of inertia - Wikipedia, the free encyclopedia
		However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity.
		For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.
		Alternatively the elements of the inertia tensor can be expressed as:
	en.wikipedia.org /wiki/Moment_of_inertia (806 words)

	Moment of inertia
		However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity.
		Rotational versions of Newton's second law and the formulas for momentum and kinetic energy, use the moment of inertia of an object (with torque, angular velocity and angular acceleration replacing force, velocity and acceleration, respectively).
		Moment of inertia is to rotational motion as mass is to linear motion.
	www.omniknow.com /common/wiki.php?in=en&term=Rotational_inertia (924 words)

	Ivanova_Elena
		Darboux problem is the problem of determination of the turn-tensor (or other quantities describing position of a rigid body) by the known angular velocity vector.
		That is why the left Darboux problem can be reformulated as the problem of determination of the right angular velocity vector by the known left one and the right Darboux problem can be reformulated as the problem of determination of the left angular velocity vector by the known right one.
		This approach allows to reduce Darboux problem to solution of the 3-order linear differential equation in a real-valued unknown.
	www.eng.abdn.ac.uk /~eng580/apm/abstracts/Ivanova_Elena (924 words)

	A960907.txt
		We find furthermore that the zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector.
		By translating from Clifford into tensor algebra, we also propose a new (non-relativistic) velocity operator for a spin 1/2 particle.
		In presence of a non-constant spin vector (Pauli case) we have, besides the component normal to spin present even in the Schroedinger theory, also a component of the local velocity which is parallel to the rotor of the spin vector.
	www.clifford.org /anonftp/clf-alg/abstracts/1996/A960907.txt (924 words)

	My Personal Reading List
		A fairly large family of old and new metrics are thus included as special cases: Schwarzschild, NUT, and Kerr exterior solutions, and new interior solutions for a stationary body of fluid, either spherical and nonrotating, or rigidly rotating with arbitrary angular velocity.
		All Petrov type-D stationary (non-static) axisymmetric (without a higher symmetry) rigidly rotating perfect-fluid metrics which have the velocity of the fluid in the 2-space generated by the principal null directions of the Weyl tensor are explicitly obtained.
		The formalism of the previous article is used to obtain solutions of Einstein's field equations for the interior of a rigidly rotating perfect fluid, with zero magnetic Weyl tensor.
	members.localnet.com /~atheneum/bib/fluids.html (5872 words)

	Andrew Zaferakis: Rigid Body Dynamics
		During the simulation we want to know how fast the rigid body is spinning, this is called the angular velocity, represented by "omega".
		In rigid body dynamics, we must compute the intertia tensor of the rigid body.
		Once we have the inertia tensor of the body, we can easily compute the inverse, which is what is needed in the simulation.
	www.cs.unc.edu /~andrewz/dynamics/inertia_tensor.html (237 words)

	Electrodynamics Syllabus
		Energy in electromagnetism, Poynting's theorem and conservation of energy, momentum density, radiation pressure, Maxwell stress tensor, source of radiation pressure for a conductor, conservation of linear and angular momentum in electromagnetic fields.
		Mathematical background - reminder of 3-tensors, dyadic products, illustration by momentum tensor, covariant and contravariant descriptions, four-vectors, properties to make consistent with SR defining the interval and the Lorentz transformation, four-vector calculus, illustration of four vectors for displacement, momentum, velocity and acceleration, four-tensors.
		Understand conservation of energy and momentum to include the contribution of the electromagnetic fields.
	physweb.spec.warwick.ac.uk /teach/syllabi/year4/px421.html (444 words)

	Inertia Tensor
		The angular momentum of a rigid body rotating about an axis passing through the origin of the local reference frame is in fact the product of the inertia tensor of the object and the angular velocity.
		This property is explained in detail in Transformation of the Inertia Tensor.
		The diagonal elements in the inertia tensor shown in
	kwon3d.com /theory/moi/iten.html (237 words)

	Euclidean Tensors
		If the components of an antisymmetric tensor are the infinitesimal velocities of points of a fluid relative to a certain point, then the associated vector is twice the infinitesimal angle of angular velocity of rotation, the vorticity, and is a pseudovector.
		It is a pseudotensor because of the presence of ε
		The property of being a pseudotensor has nothing to do with the property of being antisymmetric.
	www.du.edu /~jcalvert/math/eucltens.htm (237 words)

	SFU MATH 252, Vector Calculus: Spring 2005, Week 2
		Permutation tensor, tensor notation form of cross product; mechanical interpretation: work, torque, angular velocity; parametric and non-parametric representation of equations of lines and planes; scalar triple product, 3x3 determinant, volume
		Vector identities, product of permutation tensors; vector functions of a single variable, limits, continuity, derivatives, differentiation rules; space curves, examples: straight line, circle, helix
		Note: Table 1.2, p.56 gives a nice summary of the various vector products and their analytical and geometric interpretations
	www.math.sfu.ca /~ralfw/math252/week2.html (110 words)

	Euclidean Tensors
		If the components of an antisymmetric tensor are the infinitesimal velocities of points of a fluid relative to a certain point, then the associated vector is twice the infinitesimal angle of angular velocity of rotation, the vorticity, and is a pseudovector.
		The 23 component of the antisymmetric tensor is the same as the 1 component of c, for example.
		The matrix that represents it is the unit matrix, of course.
	www.du.edu /~jcalvert/math/eucltens.htm (110 words)

	[4.05] Rotation Theory of Celestial bodies in Angle-action Variables
		The solution of the Chandler problem (Andoyer’s variables, components of angular velocity w.r.t to the body and space reference systems, direction cosines and their different functions) is presented in elliptical and theta-functions, and in the form of Fourier series in the angle-action variables of unperturbed motion.
		On the base of equations in Andoyer variables describing rotational motion of the celestial bodies with a changeable in the time tensor of inertia (Barkin, 1979, 1984) the problem is reduced to the classical Euler-Poinsot problem for a rigid body, but with another set of constant moments of inertia.
		Secular perturbations in the Earth’s rotation due to second harmonic of the force function have been studied (the determination of the constant of precession; constant additives to the angular velocities of the Chandler and axial motions of the Earth).
	www.aas.org /publications/baas/v36n2/dda04/22.htm (620 words)

	PHY553.html
		Minkowski line element and proper time, four-velocity, four acceleration, four-wave vector, reproduction of all results in relativistic kinematics, four-momentum, four-force, four-angular momentum, reproduction of all results in relativistic kinetics, relativistic Lagrangian and Hamiltonian, invariant action, geodesic equation, relation between geometry and dynamics (notional).
		four potential, covariance of Lorentz gauge condition, field-strength tensor, Maxwell's equations, gauge transformations, covariant form of Lorentz force equation, transformations of electric and magnetic fields, fields due to a point charge moving with uniform velocity, Biot-Savart law recovered.
		Need to redefine momentum, relativistic mass and relativistic momentum, force, work and kinetic energy in relativity, relativistic principle of work, energy-momentum relation, relativistic acceleration, longitudinal and transverse mass, motion under a constant acceleration, Lorentz transformations for energy and momentum, mass-energy equivalence.
	home.iitk.ac.in /~sreerup/phy553.html (408 words)

	Real-time Physics for Computer Games
		The hardest part of the Integration is the angular forces, as mentioned earlier this uses the Inertia Tensor [10]:
		Where a is acceleration, f is the linear forces, m is the mass of the body, v is the accumulated velocity and p is the world position of the centre of the body.
		However calculating spring force is a little more complicated which is discussed in section 4.2, but is applied to the Linear Forces vector in the same way as gravity.
	www.topperware.co.uk /Project/Project.htm (408 words)

	Relativity
		The space-time metric generated by a rotating mass M with angular velocity w was found by Roy Kerr in 1963:
		is known as space-time metric, which is a second rank tensor and a function of the space-time.
		The equations of gravitational field can be solved exactly for the case of a centrally symmetric field in vacuum with mass M at the center.
	universe-review.ca /R15-17-relativity.htm (8674 words)

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