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

Thornton & Marion, Classical Dynamics, Fifth 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 orthogornal.
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 system 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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