La importancia de fabricarse un “CsO”

rank, which have very special relevance in physics and other areas. Let us start from an important example. The multidimensional harmonic oscillator has total energy which can be presented as

E = K + U , where K = 1

2m_ij ˙q_i˙q_j and U = 1

2k_ijq_iq_j, i = 1, 2, ..., N . (3.37) Here the dot indicates time derivative, m_ij is a positively defined mass matrix and k_ij is a positively defined interaction bilinear form. The expression “positively defined” means that, e.g., k_ijq_iq_j ≥ 0 and the zero value is possible only if all q_i are zero. From the viewpoint of physics it is necessary that both m_ij and k_ij should be positively defined. In case of m_ij this guarantees that the system

can not have negative kinetic energy and in case of k_ij positive sign means that the system performs small oscillations near a stable equilibrium point.

The Lagrange function of the system is defined as L = K − U and the Lagrange equations m_ijq¨_j = k_ijq_j

represent a set of interacting second-order differential equations. Needless to say it is very much de-sirable to find another variables Q_a(t), in which these equations separate from each other, becoming a set of individual oscillator equations

Q¨_a+ ω₀^(a)Q_a= 0 , a = 1, 2, ..., N . . (3.38) The variables Q_a(t) are called normal modes of the system. The problem is how to find the proper change of variables and pass from q_i(t) to Q_a(t).

The first observation is that the time dependence does not play significant role here and that the problem of our interest is purely algebraic. The solution does exist and it can be achieved even if only one of the two tensors m_ij and k_ij is positively defined in the sense explained above. In what follows we will assume that m_ij is positively defined and will not assume any restrictions of this sort for k_ij. Our consideration will be mathematically general and is applicable not only to the oscillator problem, but also to many others, e.g. to the moment of inertia of rigid body (and to other cases). The last problem is traditionally characterized as finding a principal axes of inertia, so the general procedure of diagonalization for a symmetric positively defined second-rank tensor is often identified as reducing the symmetric tensor to the principal axes.

The procedure which is more complete than the one presented here can be found in many textbooks on Linear Algebra. We will present a slightly reduced version, which has an advantage of being also must simpler. Our strategy will be always to consider a 2D version first and then generalize it to the general D-dimensional case.

As a first step we will deal with the 2D case, when q_i = q₁, q₂. Let us consider the positively defined term m_ijq_iq_j and perform the rotation

q₁ = x⁰₁cos α − x⁰₂sin α

q₂ = x⁰₁sin α + x⁰₂cos α . (3.39) The angle α can be always chosen such that the bilinear expression

˜

m_ij ˙x_i ˙x_j = k_ij ˙q_i ˙q_j (3.40)

become diagonal, ˜k₁₂= 0. Indeed,

˜k₁₂ = (k₁₁+ k₂₂) cos α sin α + k₁₂(cos²α − sin²α) , hence the condition we need has the form

tan 2α = − 2 k₁₂

(k₁₁+ k₂₂). (3.41)

As a result of rotation (3.39) with the angle (3.41) we arrive at the new matrix ˜m_ij which is diagonal and, moreover, all its diagonal elements ˜m₁₁ and ˜m₂₂ are positive. Next we have to

perform the rescalings of the variables x₁ = p

k˜₁₁y₁ and x₂ = p

˜k₂₂y₂. As a result the new expression for the k-term is y₁² + y₂². The most important feauture of this formula is that it is invariant under rotations. Let us call the new matrix which emerges in the k-sector after the two changes of variables ˜k_ij. If we now perform one more rotation which will diagonalize the ˜k_ij matrix, the form of the m_ij-matrix does not change. Hence we can see that one can diagonalize the two 2D matrices m_ij and k_ij at the same time. Moreover, it is easy to note that we did not use the positive definiteness of the m_ij and not of the k_ij.

The last part of our consideration concerns the case of an arbitrary dimension D. Let us first perform the operations described above in the sector of the variables q₁ and q₂. As a result we arrive at the new m and k matrices with zero (12)-elements. The most important observation is that when we perform rotations and rescalings in the (13)-sector, the (12)-elements of the m and k matrices remain zero.

Chapter 4

Curvilinear coordinates and local coordinate transformations

4.1

Curvilinear coordinates and change of basis

The purpose of this chapter is to consider an arbitrary change of coordinates

x^0α= x^0α(x^µ) , (4.1)

where x^0α(x) are not necessary linear functions as before. In is supposed that x^0α(x) are smooth functions (this means that all partial derivatives ∂x^0α/∂x^µ are continuous functions) maybe except some isolated points.

It is easy to see that all the definitions we have introduced before can be easily generalized for this, more general, case. Say, the scalar field may be defined as

ϕ⁰¡ x⁰¢

= ϕ (x) . (4.2)

Also, vectors (contra- and covariant) are defined as a quantities which transform according to the rules

a^{0 i}¡ x⁰¢

= ∂x⁰ⁱ

∂x^j a^j(x) (4.3)

for the contravariant and

b⁰_l¡ x⁰¢

= ∂x^k

∂x^0lb_k(x) (4.4)

for the covariant cases. The same scheme works for tensors, for example we define the (1, 1)-type tensor as an object, whose components are transforming according to

Aⁱ_j⁰0

¡x⁰¢

= ∂xⁱ⁰

∂x^k

∂x^l

∂x^0j A^k_l (x) etc. (4.5)

The metric tensor g_ij is defined as

g_ij(x) = ∂X^a

∂xⁱ

∂X^b

∂x^j g_ab, (4.6)

where g_ab = δ_ab is a metric in Cartesian coordinates. Also, the inverse metric is g^ij = ∂xⁱ

∂X^a

∂x^j

∂X^b δ^ab,

where X^a (as before) are Cartesian coordinates corresponding to the orthonormal basis ˆn_a. Of course, the basic vectors ˆn_a are the same in all points of the space.

It is so easy to generalize notion of tensors and algebraic operations with them, because all these operations are defined in the same point of the space. Thus the only difference between general coordinate transformation x^0α = x^0α(x) and the special one x^0α = ∧^α_β⁰x^β + B^α0 with ∧^α_β⁰ = const and B^α0 = const, is that, in the general case, the transition coefficients ∂xⁱ/∂x^0j are not necessary constants.

One of the important consequences is that the metric tensor g_ij also depends on the point.

Another important fact is that antisymmetric tensor ε^ijk also depends on the coordinates ε^ijk= ∂xⁱ Using E₁₂₃ = 1 and according to the Theorem 3.10 we have

ε₁₂₃ =√

g and ε¹²³= 1

√g, where g = g(x) = det kg_ij(x)k . (4.7) Before turning to the applications and particular cases, let us consider the definition of the basis vectors corresponding to the curvilinear coordinates. It is intuitively clear that these vector must be different in different points. Let us suppose that the vectors of the point-dependent basis e_i(x) are such that the vector a = aⁱe_i(x) is coordinate-independent geometric object. As far as the rule for the vector transformation is (4.3), the transformation for the basic vectors must have the form

e⁰_i = ∂x^l

∂x⁰ⁱe_l. (4.8)

In particular, we obtain the following relation between the point-dependent and permanent or-thonormal basis

e_i = ∂X^a

∂xⁱ nˆ_a. (4.9)

Exercise 1. Derive the relation (4.8) starting from (4.3).

In document UNIVERSIDAD DE CÓRDOBA (página 122-139)