Thornton & Marion (5th Edition), Chapter 01, Problem 06 Solution

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Thornton & Marion, Classical Dynamics, 5th Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 06. Another proof of the orthogonality condition

The problem asks you to

  • prove that Equation (1.15) can be obtained by using the fact that the transformation matrix preserves the length of the line segment.

This problem assumes

  • that the coordinate systems are both orthogonal.

We should know about

  1. Transformation (rotation) matrix
  2. Orthogonality condition

Solution

We assume a point P is represented in the x_i coordinate system by P(x_1, x_2, x_3), and it can be also represented in the x'_i coordinate system by P(x'_1, x'_2, x'_3).

These coordinate systems have the same origin. The length of the line segment from the origin and the point P is

l^2 = x^2_1 + x^2_2 + x^2_3 = \sum_{i=1}^{3} x^2_i
l'^2 = x'^2_1 + x'^2_2 + x'^2_3 = \sum^3_{i=1} x'^2_i
 

Since the transformation matrix preserves the length of the line segment,

l^2 = l'^2
thus,

\sum^3_{i=1} x^2_i = \sum^3_{i=1} x'^2_i
 

In the equation of transformation,

x'_i = \sum_j \lambda_{ij} x_j
using the fact that the index j is a dummy variable, so we can write the above equation like this:

\sum^3_{i=1} x'^2_i = \sum^3_{i=1} ( \sum^3_{j=1} \lambda_{ij} x_j) ( \sum^3_{k=1} \lambda_{ik} x_k) 
 

Therefore,

\sum^3_{i} x^2_i = \sum^3_{i=1} ( \sum^3_{j=1} \lambda_{ij} x_j) ( \sum^3_{k=1} \lambda_{ik} x_k) = \sum_{j,k}^3 x_j x_k (\sum_i^3 \lambda_{ij} \lambda_{ik})

This equation is satisfied only if

\sum_i \lambda_{ij}\lambda_{ik} = \delta_{jk}
 

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