Lecture 26 Friday - March 31, 2006 Written or last updated: March 31, 2006

P442 – Analytical Mechanics - II Scalars, Vectors, and All That c Alex R. Dzierba

Some Notation

We start by laying out notation some of which many of you may already be familiar with and others may not. We are also going to make the assumption for now that we are dealing with three dimensional space.

Implied summation by repeated index The components of a vector a are aj where j = 1, 3. When an index is repeated in expressions that usually implies a summation. Here are some examples:

aibi ⇒ a1b1 + a2b2 + a3b3 ∂a ∂a ∂a ∂a i ⇒ 1 + 2 + 3 ∂xi ∂x1 ∂x2 ∂x3

Assuming that cij and dij are elements of a :

cijdjk ⇒ ci1d1k + ci2d2k + ci3d3k

and the result would yield the ik-th element of the product matrix. Here is product of a matrix times a matrix times a vector and the result is a vector:

3 3 X X cijdjkak ⇒ cijdjkak j=1 k=1

Kronecker Delta This is defined as

 1 if i = j δ = (1) ij 0 otherwise

As will be noted below, the δij are components of a second-order .

1 Levi-Civita Tensor   0 if any two labels are equal ijk = 1 if the labels are an even of 1,2,3 (2)  −1 if the labels are an odd permutation of 1,2,3

The Levi-Civita tensor has 3 × 3 × 3 = 27 components of which 21 equal zero, three equal +1 and three equal −1.

The determinant of a matrix A can be written as:

a11 a12 a13

|A| = a21 a22 a23 = ijka1ia2ja3k (3)

a31 a32 a33 where a repeated index implies a summation.

Vectors and Tensors

Suppose that the 3 × 3 matrix Lij is the transformation matrix that specifies how the components (ai) of a vector are changed when the Cartesian axes are rotated. Then:

0 ai = Lijaj

A vector is a first-order tensor. If a and b are any two vectors, the nine numbers aibj (products of their components) form a tensor and in this case it will be a second-order tensor whose elements we will call Aij. Under a rotation the new elements of our tensor are:

0 Aij = LikLjmAkm

The δij are components of a second-order tensor. The ijk are components of a third-order tensor.

Examples of Tensors in

The is a tensor. Recall how it was defined:

Z Ijk = ρ(x)(x · xδjk − xjxk)dV V

In the above δjk and xjxk are components of a tensor.

We saw the the L and angular velocity ω are related through the moment of inertia tensor:

2 Li = Iikωk

In electrodynamics we are used to Ohm’s law which states that I = V/R where I is current, V is voltage and R is resistance. This is often recast at J = σE where J is current per area, σ is conductivity (the inverse of resistivity) and E is the electric field. But what we wrote down here only applies to materials that are isotropic. But for anisotropic crystals the conductivity is different along different crystal axes and the material is more correctly described by:

Ji = σikEk

where σjk is a second-order tensor. There is a similar equation relating the magnetic moment per volume M to the magnetic field B through the magnetic susceptibility χ which is also a tensor:

Mi = χikBk

In the above examples we have relations between vectors involving second-order tensors. We have just started talking about the theory of elasticity that relates two second-order tensors through a fourth-order tensor. One of the second order tensors is the strain tensor which describes the local deformation of an elastic body at any interior point as:

  1 ∂ui ∂uj Tij = + 2 ∂uj ∂ui where the displacement vector u describes the strain of a small volume element located by the vector x. We also have a second-order tensor Sij called the tensor where the i-th component of stress (pressure or shear force) acting normal to or along the plane whose normal is specified by the index j.

The stress and strain tensor are related by:

Tij = γijkmSkm

The fourth-order tensor γijkm completely describes the elastic properties of the material. Although this beast requires 34 = 81 numbers in principle, symmetry arguments reduce this number to 21. This number is even smaller for crystals with higher symmetry and an isotropic material only requires two numbers to describe its elastic properties.

Cross Product

The components of the cross product of two vectors a and b can be written as:

(a × b)i = ijkajbk = ajbk − akbj (4)

3 we see that we have here a mathematical object written as products of components of two vectors – so what we have here is a second-order tensor. So let’s write this as:

cjk = ajbk − akbj (5)

Forget for the moment that we are talking about a vector defined as the cross product of two vectors but rather look at this as a tensor. This particular tensor is clearly anti-symmetric since cij = −cji. That means the diagonal elements are zero since cii = −cii ⇒ cii = 0. We start with 3 × 3 = 9 elements of which three are zero so that leaves us with six but only three are needed by the antisymmetry property. So we only need three numbers.

The three numbers, it turns out, match the dimensionality of our vectors and furthermore they transform, under rotations, just like a vector.

Transforming a Cross Product Vector In doing this example, let me use the x, y, z for indices rather than 1, 2, 3. So we have:

cxy = axby − aybx

cyz = aybz − azby

czx = azbx − axbz

Suppose we do a rotation of coordinates rotating through angle θ about the z-axis. Then look at how our vectors transform starting with a:

ax0 = ax cos θ + ay sin θ

ay0 = ay cos θ − ax sin θ

az0 = az

There will be similar equations for b in the primed frame. Here is what you find when you evaluate:

cx0y0 = ax0 by0 − ay0 bx0 = cxy and you also find:

cy0z0 = cyz cos θ + czx sin θ

cz0x0 = czx cos θ − cyz sin θ

So cyz and czx and cxy behave like the x, y, z components, respectively, of a vector. So this tensor transforms like a vector. But this fortuitous accident only happens in three . The four- analogy

4 would be an anti-symmetric tensor with 4 × 4 = 16 elements which requires six numbers (the four diagonal elements are zero and leaving 12 but of these we only need six). That means this tensor in four dimension cannot transform like a four-dimensional vector.

Polar and Axial Vectors

So our in three dimensional world we see that vectors, like displacement and momentum, transform like vectors. Cross products, like torque and angular momentum transform like vectors as well. But there is a way that cross products are different.

If we invert coordinates, so that x → −x and y → −y and z → −z then vectors like displacement x and velocity v and momentum p change sign. This is also true for the electric field E. Coordinate inversions are sometimes called parity operations or space reflections. For these vectors a → −a. Such vectors, that change sign under a coordinate inversion, are called polar vectors. Cross product vectors, on the other hand, are called axial vectors or pseudovectors – they don’t change sign under a parity operation.

This is all because cross-products require a handedness in their definition. This is true of torque τ , angular momentum L, angular velocity ω and the magnetic field B, for example. For polar vectors all you need is a length and direction. Axial vectors also require a handedness. The cross product of two vectors is given by the right-hand rule. Handedness remains unchanged under space reflections.

Also note that you cannot get to a space reflection through rotations.

Scalars and Pseudoscalars

A scalar (like mass or energy or the of two polar vectors) does not change sign under a space inversion. But it is possible to define a scalar that does change sign and such scalars are called pseudoscalars. An example is the product of three polar vectors: a · (b × c).

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