Independent parameters of the rotation tensor ##R_{ij}##

[brotherbobby]

Homework Statement

Show how the $\text{orthogonality condition}$ of the rotation matrix $R_{ij} R_{ik} = \delta _{jk}$ diminishes up to $\mathbf{three}$ the number of independent parameters of the matrix.

Relevant Equations

An arbitrary vector $\vec x$ upon rotated by the matrix $\mathbb R$ results in a new vector $\vec x'$ given by $\vec x' = \mathbb{R} \vec x$, or in component notation : $x'_i = \rm R_{ij} x_j$, where $\rm R_{ij}$ are the components of the rotation matrix.

Due to orthogonality, these components satisfy the requirement : $R_{ij} R_{ik} = \delta _{jk}$.

I am afraid I had no credible attempt at solving the problem.

My poor attempt was writing the matrix $\mathbb R$ as a $3 \times 3$ square matrix with elements $a_{ij}$ and use the matrix form of the orthogonality relation $\mathbb R^T \mathbb R = \mathbb I$, where $\mathbb I$ is the identity matrix with diagonal elements 1 and off diagonal elements 0. Those a's are the parameters of the rotation matrix. I obtained 9 equations for the a's, some identical, but found that almost all the a's would cancel to leave only 1 independent component!

Any help would be welcome.

 

[strangerep]

This type of problem can be solved simply by counting the number of independent parameters that you start with (which I hope you've already realized is $3\times 3 = 9$), and subtracting the number of independent constraint equations, which in this case is ... ?

Hint: is $\delta_{jk}$ symmetric, anti-symmetric, or neither?

 

[brotherbobby]

Very sorry for the late reply. Thank you for the hint above, which really leads easily to the solution.

We know that the set of relations (constraints) that variables have, lead to a decline in how many of them are independent. The more such relations (expressed as equations), the lower the number of independent variables involved.

In our problem above, these variables are the components of the rotation tensor $R_{ij}$. But they satisfy some relations, compactly written as $R_{ij} R_{ik} = \delta_{jk}$. How many relations are these? Nine, on first glance, but they are not all independent. The kronecker tensor being symmetric, we have the number of independent equations as its number of independent components : $N = \tfrac{3(3+1)}{4} = 6$.

These six relations serve to reduce the number of independent parameters of the rotation matrix by $9 - 6 = \boxed{3}$.