Main article: Angular Momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics There is a thing which we call the angular momentum tensor. It's a 3×3 matrix, so it has nine elements which are defined as: L ij = ri · pj - rj · pi. Because of this definition, it's an antisymmetric tensor of the second order in three dimensions, so it's got only three independent components Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar)

A possible explanation I have: the zeroth-components of the Angular Momentum Tensor corresponds to boosts and there's no boosts when dealing with spin, so there's no zeroth-components. Am I correct? quantum-mechanics special-relativity angular-momentum quantum-spin. Share. Cite. Improve this question . Follow edited Dec 6 '19 at 18:13. asked Dec 6 '19 at 7:08. user242977 user242977 $\endgroup. The mathematics allows it: the sum of **angular** momenta is an **angular** **momentum** acting in the appropriate **tensor** product. As we will see below, each **angular** **momentum** lives on a diﬀerent vector space, but the sum ﬁnds a home in the **tensor** product of the vector spaces. What is an **angular** **momentum**? It is a triplet

angular momentum since the multiplicities correspond to the multiplicities of the angular momenta l = 0,1,2. We deﬁne the spherical tensor T(k) q of rank k so that the argument nˆ of the spherical function Ym l (nˆ) = hnˆ|lm Lecture L26 - 3D Rigid Body Dynamics: The Inertia Tensor In this lecture, we will derive an expression for the angular momentum of a 3D rigid body. We shall see that this introduces the concept of the Inertia Tensor. Angular Momentum We start from the expression of the angular momentum of a system of particles about the center of mass, As discussed above, angular momentum cannot be a rank-1 tensor. One approach is to define a rank-2 angular momentum tensor L ab = r a p b − r b p a

2 Deﬁnition of the tensor spherical harmonics In the coordinate representation, the total angular momentum basis consists of simultaneous eigenstates of J~2, J z, ~L2, S~2. These are the tensorsphericalharmonics, which satisfy, J~2Yℓs jm(θ,φ) = ~2j(j +1)Yℓs jm(θ,φ), Jz Y ℓs jm(θ,φ) = ~mYℓs jm(θ,φ), L~2Yℓs jm(θ,φ) = ~ 2ℓ(ℓ+1)Yℓ Moment of Inertia Tensor Consider a rigid body rotating with fixed angular velocity about an axis which passes through the origin--see Figure 28. Let be the position vector of the th mass element, whose mass is. We expect this position vector to precess about the axis of rotation (which is parallel to) with angular velocity

The Angular Momentum Tensor As mentioned previously, there is no such thing as a vector cross product in four dimensions, so the nonrelativistic deﬁnition of angular momentum as L = r × p needs to be modiﬁed to be usable in relativity In the third chapter graphical methods of representing angular-momentum theory are introduced, and this technique is used in the fourth chapter to develop tensor-operator methods further. In later chapters angular-momentum graphs will be used frequently to evaluate the angular part of interaction matrix elements Lecture Series on Dynamics of Machines by Prof. Amitabha GhoshDepartment of Mechanical Engineering IIT KanpurFor more details on NPTEL visit http://nptel.iit.. The generalized moment of momentum tensor γ ij is then defined by time derivation of the generalized spin energy of each particle: (15) γ . ij χ . ij = 1 2 d d t ( I ik χ . ij χ . kj ) Volume double forces c ij may exist in a continuum medium and can be understood as the application of two force vectors acting on the ends of a material segment of small but finite length

In this video I will develop the diagonal component notation of the inertia tensor by rel... Visit http://ilectureonline.com for more math and science lectures 2.3Spin angular momentum tensor S ij k and the Dirac eld 2.4Non-uniqueness of the momentum and spin currents 3.Currents of matter and gravitational theory 3.1Metric momentum current !general relativity (GR) 3.2Canonical momentum and spin currents ! Einstein-Cartan (EC) and Poincar e gauge theory (PG) 2/30 . 1. Momentum current 1.1 (Force) stress ˙ ab in continuum mechanics Classical body: gas. 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 Angular momentum operators can not only be used to classify wavefunctions, but also operators acting on these functions. If T denotes a general operator acting on the space spanned by the eigenfunctions of the angular momentum operator , then a spherical tensor operator of rank k can be defined as a set of 2k+1 operators, , that fulfills the following commutation relations (116) where (117) As.

- The helicity current and symmetrized Poincaré currents (energy-momentum and angular momentum tensors, and) are common and dual-symmetric. At the same time the canonical energy-momentum and angular-momentum tensors, Tαβ and Mαβγ, conflict with the helicity and energy conservation laws in the standard electromagnetic theory
- The angular momentum of a finite size rigid body (rigid rotor) will be introduced. The angular momentum is expressed with respect to a laboratory system of axes with origin at the center of mass of the rotor. The following formula will be derived: where the inertia tensor is given b
- 4.4 Moment-of-inertia tensor. The moment-of-inertia tensor relates the angular momentum of a rigid body to its angular velocity, The corresponding rotational kinetic energy has the form The tensor can be written Its components are The moment-of-inertia is symmetric. In other words, its components form a real symmetric matrix, This means that the tensor has only six independent components.
- The vector TAM waves of given total angular momentum can be decomposed further into a set of three basis functions of fixed orbital angular momentum (OAM), a set of fixed helicity, or a basis consisting of a longitudinal (L) and two transverse (E and B) TAM waves. The symmetric traceless rank-2 tensor TAM waves can be similarly decomposed into a basis of fixed OAM or fixed helicity, or a basis that consists of a longitudinal (L), two vector (VE and VB, of opposite parity), and two.

- §A-2 Spherical Tensor and Rotation Matrix 399 Before the reaction, the total angular momentum J of the r-mesic atom is 1, as the intrinsic spin of the pion is 0 (see also §2-7), the spin of the deuteron is 1 (see §3-1), and the orbital angular momentum of the rd-system is 0 (the r- is h the atomic s-state). Total angular momentum is conserved in the reaction of Eq
- Electromagnetic angular momentum describes the ability of electromagnetic field to impose torque on matter. We show that for an electromagnetic field - such as an optical beam field - in a.
- No headers. In this entry I work through section 4.4.3 of David Griffithsʹ Introduction to Quantum Mechanics (2nd ed.) in which he treats the addition of angular momentum for two identical spin‐1/2 particles. The tensor algebra approach is illustrated. The four spin states of two spin‐1/2 particles are written below in the spin‐z basis in tensor format
- angular momentum tensor. angular momentum tensor: translation. judesio kiekio momento tenzorius statusas T sritis fizika atitikmenys: angl. angular momentum tensor vok. Drehimpulstensor, m rus. тензор момента количества движения, m pranc..

We may compute the angular momentum for a rigid body rotating about an axis going through its center of mass in the same way. Now use the vector identity we computed earlier. Now lets write this for the components of . The angular momentum can be written in terms of the same inertia tensor. Now we notice an important feature of rotations of rigid bodies. The angular moment will not be parallel. In this section, we generalize Newton's laws of motion (conservation of linear and angular momentum); mass conservation; and the laws of thermodynamics for a continuum. 5.1 Mass Conservation The total mass of any subregion within a deformable solid must be conserved

Advanced answer: it is neither. In reality angular momentum is an anti-symmetric tensor of degree 2. This means it is naturally an object with two indices, [math]L^{ij}[/math] — where [math]i,j[/math] range over the number of spatial dimensions —. and the angular momentum tensor (17.86) which is therefore conserved. This form of the stress tensor can also be directly coupled to source terms, resulting in the covariant form of the work energy theorem for the combined system of particles and fields. This is about all we will say about this at this time. I realize that it is unsatisfactory and apologize. If we had one more semester. * But Did You Check eBay? Check Out Angular Momentum On eBay*. Looking For Great Deals On Angular Momentum? From Everything To The Very Thing. All On eBay The problem of labelling the representations ofSO(3) contained in each representation ofU(3) is solved by using for this purpose an operatorZ with integral eigenvalues. A description is given of the decomposition of any tensor representation ofU(3) intoSO(3)-irreducible tensors labelled by these eigenvalues. An implicit definition ofZ is given in terms of theU(3) generators, and the relationship ofZ to the labelling operators (with irrational eigenvalues) proposed by Racah and others is made.

127: Addition of Spin Angular Momentum: A Tensor Algebra Approach. In this entry I work through section 4.4.3 of David Griffithsʹ Introduction to Quantum Mechanics (2nd ed.) in which he treats the addition of angular momentum for two identical spin‐1/2 particles. The tensor algebra approach is illustrated * In summary, the angular momentum vector is given by the mass moment of inertia tensor times the angular-velocity vector representing the axis of rotation*. Note that the angular momentum vector does not in general point in the same direction as the angular-velocity vector 6 Addition of Angular Momentum Back in section2.3.1, we understood how to describe composite systems using the tensor product of the Hilbert spaces of the individual subsystems. However, in many circum- stances, the basis of the tensor product formed by taking all possible pairs of basis ele-ments from the individual subspaces is not the most convenient. A key case is when the subspaces. The moment-of-inertia tensor relates the angular momentum of a rigid body to its angular velocity, The corresponding rotational kinetic energy has the form The tensor can be writte 2.3 Spin angular momentum tensor S ij k and the Dirac eld I Invariance under 3+3 in nitesimal Lorentz transformations: x0 i= x + !ijx j, with !(ij) = 0, yields, via the Noether theorem and L= = 0, angular-momentum conservation, @ k S ij k |{z} spin + x [iT j] k |{z} orb. angular mom. = 0; S ij k |{z} 6 4:= @L @@ k f ij |{z} Lor. gen. = S ji k: The canonical (Noether) spin S i

- In reality angular momentum is an anti-symmetric tensor of degree 2. This means it is naturally an object with two indices, — where range over the number of spatial dimensions — and satisfying . (By comparison, a vector is a tensor of degree 1, having just one index, , and a scalar is a tensor of degree 0, having no spatial indices.) Why so? Angular momentum is a measure of how something is rotating. In order to specify the rotation of an object you need to specify the plane in which it.
- So, in the most general case, we need to apply two rotations to the diagonal inertia tensor in order to find the angular momentum via Unity's angular velocity vector: rigidbody.inertiaTensorRotaion defines the rotation for the tensor into the body's local frame, and rigidbody.rotation defines the rotation for that transformed tensor into the world's frame
- Firstly I will give an example of a second rank tensor, and then state the transformation properties of tensors. Consider an object of mass m at position ~x moving with velocity ~v. The angular velocity, ω~, is related to these in the usual way: ~v = ω~ ×~x. (22) The angular momentum6 is given by ~L = m~x×~v = m~x×(ω~ ×~x) = m x2ω~ −(~x·ω~)~

Angular momentum operators can not only be used to classify wavefunctions, but also operators acting on these functions. If T denotes a general operator acting on the space spanned by the eigenfunctions of the angular momentum operator , then a spherical tensor operator of rank k can be defined as a set of 2k+1 operators, , that fulfills the following commutation relation this anti-symmetric divergence, we obtain the conventional angular momentum tensor J µλρ λ µρ ρ µλ =− x T x T (3.9) It defines the same Lorentz generators as M µλρ See also: Moment of Inertia . © 1996-2007 Eric W. Weisstei The angular momentum, ~L, will depend on the distribution of mass within the body and upon the angular velocity. Write this as ~L= I(!~); where the function Iis called the tensor of inertia. For a second example, take a system consisting of a mass suspended by six springs. At equilibrium the springs are perpendicular to each other. If now a (small) force F~is applied to the mass it will. In the third section we introduce abstract angular momentum operators acting on ab-stractspaces,andinthefourthweconsiderthedecompositionoftensorproductsofthese spaces. Then we will introduce irreducible tensor operators (angular momentum operators are anexampleofsuchoperators)anddiscussthecelebratedWigner-Eckarttheorem

tesseral tensor angular momentum operator as applied to spin Hamiltonians is due to Scherz [4]. It was reintroduced by Tuszynski et al. [5] and is the generalization of the concept of a real tesseral spherical harmonic formulation for CEFs of various symme-tries due to Prather [6]. Daniels [7] introduzed polarization harmonic operators Cm l and Sm l which were the operator equivalents of the. angular momentum tensor; spin tensor; موتر الإجهاد لكوشي ; موتر إجهاد-طاقة; EM tensor; gluon field strength tensor; Einstein tensor; موتر متري; رياضياتيون: ليونهارت أويلر; كارل فريدريش غاوس; أوغستين لوي كوشي; هيرمان غراسمان; غريغوريو ريتشي; توليو ليفي تشيفيتا; ج Angular Momentum Each component of the angular Momentum is 1= ( 2− 2) 1− 2− 3 2=− 1+ ( 2− 2) 2− 3 3=− 1− 2+ ( 2− 2) 3 It can be written in the tensor formula. = ** Action of Operators ∇, n and Angular Momentum Operators**. Sums of Tensor Spherical Harmonics. Orthogonality, Normalization and Completeness. Expansion in a Series of Tensor Spherical Harmonics. SPINOR SPHERICAL HARMONICS. Definition. Components of Spinor Spherical Harmonics. Complex Conjugation. Time Reversal

Here we extend this framework by localizing the angular momentum of the linearized gravitational field, deriving a gravitational spin tensor which possesses similarly desirable properties. By examining the local exchange of angular momentum (between matter and gravity) we find that gravitational intrinsic spin is localized, separately from orbital angular momentum, in terms of a gravitational spin tensor. This spin tensor is then uniquely determined by requiring that it obey two. Angular Momentum and Killing-Yano Tensors 613 generates symmetry transformations on the phase space linear in momentum {x µ,K} = Kµνp ν. The Poisson brackets satisfy {H,K} =0, where H = 1 2 gµνp µp ν. Thus, in the phase space there is a reciprocal model with constant of motion H and the Hamil-tonian K. 4 Extended Lagrangians and their corresponding geometries Let us assume that a. the components of angular momentum. As an example of a tensor operator, let V and W be vector operators, and write Tij = ViWj. (25) Then Tij is a tensor operator (it is the tensor product of V with W). This is just an example; in general, a tensor operator cannot be written as the product of two vector operators as in Eq. (25)

**ANGULAR** **MOMENTUM** 8.1 Introduction Now that we have introduced three-dimensional systems, we need to introduce into our quantum-mechanical framework the concept of **angular** **momentum**. Recall that in classical mechanics **angular** **momentum** is deﬁned as the vector product of position and **momentum**: L ≡ r ×p = i j k xy z p x p y p z . (8.1) Note that. In this lecture, we will derive expressions for the angular momentum and kinetic energy of a 3D rigid body. We shall see that this introduces the concept of the Inertia Tensor. Angular Momentum We start form the expression of the angular momentum of a system of particles about the center of mass, HG, derived in lecture D17, HG = Z m r′ ×v′ dm . Here, r′ is the position vector relative.

The vector TAM waves of given total angular momentum can be decomposed further into a set of three basis functions of fixed orbital angular momentum, a set of fixed helicity, or a basis consisting of a longitudinal (L) and two transverse (E and B) TAM waves. The symmetric traceless rank-2 tensor TAM waves can be similarly decomposed into a basis of fixed orbital angular momentum or fixed helicity, or a basis that consists of a longitudinal (L), two vector (VE and VB, of opposite parity), and. * The commutation relations of an arbitrary angular momentum vector can be reduced to those of the harmonic oscillator*. This provides a powerful method for constructing and developing the properties of angular momentum eigenvectors. In this paper many known theorems are derived in this way, and some new results obtained. Among the topics treated are the properties of the rotation matrices; the addition of two, three, and four angular momenta; and the theory of tensor operators

however, the square of the angular momentum vector commutes with all the components. This will give us the operators we need to label states in 3D central potentials. Lets just compute the commutator. Since there is no difference between , and , we can generalize this to where is the completely antisymmetric tensor and we assume a sum over repeated indices. The tensor is equal to 1 for cyclic. carries an intrinsic orbital angular momentum and include also spherical tensor operators such as the electric charge multipole operator. As an application of this method, we study 9) a two-cluster model 3) of 6 Li which includes nucleons in 1p-wave orbits. The aim of this report is to provide the necessary formulas concerning the projection of angular momentum The angular momentum can be written in terms of the same inertia tensor. Now we notice an important feature of rotations of rigid bodies. The angular moment will not be parallel to the angular velocityif the inertia tensor has off diagonal components. Jim Branson2012-10-21 Killing-Yano Tensors and Angular Momentum. February 2004 · Czechoslovak Journal of Physics. Dumitru Baleanu; Ozlem Defterli; New geometries were obtained by adding a suitable term involving the. If the angular velocity vector is directed along one of the three coordinate axes that would get rid of the non-diagonal inertia tensor elements, we expect to see the following relation between the angular velocity vector and the angular momentum

The angular momentum of a rigid body with angular velocity is given by , where is the inertia tensor. This Demonstration shows the rotation of an axially symmetric ellipsoid rotating about a fixed angular velocity vector .The body axes , , (indicated by the red, green, and blue spheres) and the angular momentum rotate as functions of time. The space axes , , are indicated by the red, green. The angular momentum operators define a set of irreducible tensors that are unique except for a normalization constant. The normalization is conveniently defined in terms of statistical tensors that describe oriented states. The properties of the tensors discussed include: the trace of products of components of such tensors; symmetry properties of the traces; and the expansion of products of. Inertia Tensor. We have already seen that that linear momentum is conserved for a closed system but momentum can be exchanged between objects within the closed system by means of equal and opposite forces. In the same way angular momentum can be exchanged between object within a closed system by means of equal and opposite torques. The relationship between torque and the acceleration can be. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3- j , 6- j , and 9- j symbols I is called the moment of inertia tensor. I xx, I yy, I zz are the MI about the x,y and z-axis of the body frame respectively. Off-diagonal terms are the product of inertia. Since I yx = I xy, I zx = I xz, and I yz = I zy, out of the nine components, only six at most are different and the matrix is symmetric. The angular momentum and the kinetic energy can be expressed as 11 and 22 L I T L I Z.

It is customary to assume that the law of conservation of the angular momentum is violated for an asymmetric energy-momentum tensors. This is the reason for criticizing the Minkowski tensor and other asymmetric energy-momentum tensors. In this paper, it is shown that the laws of conservation of energy and momentum following from an asymmetric tensor in the form of its total divergence are. Because of this we'll switch to angular momentum as the primary quantity in our physics simulation and we'll derive angular velocity from it. For consistency we'll also switch from linear velocity to linear momentum. Calculating The Inertia Tensor. Now we need a way to calculate the inertia tensor of our go stone. The general case is quite complicated because inertia tensors are capable. - Angular momentum, torque, angular velocity - Moment of inertia, moment of inertia tensor - Rotation around fixed and free axes - Gyroscope, precession, nutation - Euler's equations . 2 Remarks Warning. The rotational speed meter works with a red laser (630-670 nm) of laser class 2 (maximum power < 1 mW). The beam window is indicated by a triangular laser warning sign. Do not look directly.

- Keywords: angular momentum ﬂux, symmetrized AM tensor, spin AM tensor (Some ﬁgures may appear in colour only in the online journal) 1. Introduction In addition to linear momentum, light can also carry angular momentum (AM) [1-4] that can result in observable mechanic effects when interacting with small objects [5-7].TheAM density of light J is deﬁned as the cross-product of position.
- Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity about a particular axis. In the simple case of revolution of a particle in a circle about a center of rotation, the particle remaining always in the same plane and having always the same distance from the center, it is sufficient to discard.
- Energy Momentum Tensor Momentum Angular Momentum Energy density, Pressure Observables Deeply Virtual Compton Scattering, moments of GPDs etc. QCD Energy Momentum Tensor Energy Momentum Pressure Shear stress Energy Momentum Pressure Shear stress. GPD based definition of Angular Momentum To access OAM, we take the difference between total angular momentum and spin Xiangdong Ji, PRL 78.610,1997.
- ing the local exchange.

there! I am studying tensor analysis and now try to apply it to solving a quantum physics problem. Here I am trying to calculate angular momentum squared written in terms of the spherical coordinate The theory of angular momenta and irreducible tensors represents, in principle, a development of the classical theory of vectors and tensors. In this chapter only the basic definitions and relations of the vector and tensor theory are represented which will be used throughout. For more detailed analysis see corresponding monographs (e.g., Refs. Mass Moment of Inertia Tensor. As derived in the previous section, the moment of inertia tensor, in 3D Cartesian coordinates, is a three-by-three matrix that can be multiplied by any angular-velocity vector to produce the corresponding angular momentum vector for either a point mass or a rigid mass distribution. Note that the origin of the angular-velocity vector is always fixed at in the. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3-j, 6-j, and 9-j symbols. Throughout, the author provides. 3.4 Angular momentum eigenvalues and matrix elements 113 3.4.1 Eigenvalues of J2 and Jt; irreducibility 1 13 3.4.2 Matrix elements in the spherical basis 116 3.4.3 Matrix elements in the Cartesian basis I17 3.4.4 Operator matrices for j = 112, I, and 3/2 119 3.4.5 Angular momentum: geometrical and dynamical 120 3.3 41 95 . CONTENTS vii 3.5 Reference frames: spin and orbital angular momenta 122.

Energy-momentum tensor (EMT) Shear stress Normal stress (pressure) Energy density Energy flux Momentum density Momentum flux Mass, spin and pressure all encoded in • Nucleon mechanical properties • Quark-gluon plasma • Relativistic hydrodynamics • Stellar structure and dynamics • Cosmology • Gravitational waves • Modified theories of gravitation • Key concept for . Quantum. Beginning with basic principles, the development is carried through the introduction of coupling coefficients for vector addition, the transformation properties of the angular momentum wave functions under rotations of the coordinate axes, irreducible tensors and Racah coefficients. Applications include static moments of systems composed schemes in nuclear reactions and more. In this volume.

Become a Pro with these valuable skills. Start Your Course Today. Join Over 50 Million People Learning Online at Udemy The Tensor of Inertia The expression for angular momentum given by equation 1, can be written in matrix form as, HGx HGy HGz = Ixx −Ixy −Ixz −Iyx Iyy −Iyz −Izx −Izy Izz ωx ωy ωz , or, HG = [IG] ω , where [IG] is the tensor of inertia about the center of mass G and with respect to the xyz axes. The tensor o The angular momentum of a rigid body with angular velocity is given by, where is the inertia tensor. This Demonstration shows the rotation of an axially symmetric ellipsoid rotating about a fixed angular velocity vector. The body axes (indicated by the red, green, and blue spheres) and the angular momentum rotate as functions of time

Note that the angular momentum is itself a vector. The three Cartesian components of the angular momentum are: L x = yp z −zp y,L y = zp x −xp z,L z = xp y −yp x. (8.2) 8.2 Angular momentum operator For a quantum system the angular momentum is an observable, we can measure the angular momentum of a particle in a given quantum state. According to the postulates that w tesseral **tensor** **angular** **momentum** operator as applied to spin Hamiltonians is due to Scherz [4]. It was reintroduced by Tuszynski et al. [5] and is the generalization of the concept of a real tesseral spherical harmonic formulation for CEFs of various symme-tries due to Prather [6]. Daniels [7] introduzed polarization harmonic operators Cm l and S •Angular momentum is defined as an angular mass times the angular velocity •The angular mass is called the moment of inertia (or inertia tensor) of the rigid body •If you spin in your chair while extending your legs, and then suddenly pull your legs closer the chair spins faste Angular momentum of a rigid body about its centre of mass or centre of rotation equals moment of inertia tensor times angular velocity: [itex]\mathbf{L}\ =\ I\,\mathbf{\omega}[/itex] About any other point, angular momentum = angular momentum about centre of mass plus angular momentum of centre of mass: [itex]\mathbf{L}_P\ =\ I_{c.o.m.}\,\mathbf{\omega}\,+\, \mathbf{r}_{c.o.m.}\times m\mathbf{v}_{c.o.m.}[/itex The inertia tensor is associated with any mass and is part of the angular momentum (L) eqn L=Iw.. I inertia tensor (3x3 matrix) and w angular v... When setting the tensor we are only concerned about what the mass will do when rotated freely? onatirec (Mechanical) 9 Oct 20 15:0 The directions of the three mutually orthogonal unit vectors define the three so-called principal axes of rotation of the rigid body under investigation. These axes are special because when the body rotates about one of them (i.e., when is parallel to one of them) the angular momentum vector becomes parallel to the angular velocity vector