Question

1. For 2-dimensional rotations by angle ϕ₁ and ϕ₂ prove that the rotation matrices have the property, R(ϕ₁+ ϕ₂ ) = R(ϕ₁)R(ϕ₂). In other words, the combination of two rotations is a matrix of the same form. This is it is also a rotation.

2. Prove that for Galilean transformations with velocity u in the x-direction

$\frac{\partial^2 }{\partial t'^2}=u^2\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial t^2}+2u\frac{\partial^2 }{\partial x\partial t}$

3. Using Maxwell's equations in differential form prove that the electric and magnetic fields in free space follow the wave equation.

Answer 1

1. Let the rotations be $\theta_1$ and $\theta_2$. We can write the rotation matrices as,

$R(\theta_1) = \begin{bmatrix} cos \theta_1 & sin \theta_1 \\ -sin \theta_1 & cos \theta_1 \end{bmatrix}$

$R(\theta_2) = \begin{bmatrix} cos \theta_2 & sin \theta_2 \\ -sin \theta_2 & cos \theta_2 \end{bmatrix}$

The combined Rotation is given by,

$R(\theta_1 + \theta_2) = R(\theta_1)R(\theta_2)$

$\Rightarrow R(\theta_1)R(\theta_2) = \begin{bmatrix} cos \theta_1 & sin \theta_1 \\ -sin \theta_1 & cos \theta_1 \end{bmatrix} \begin{bmatrix} cos \theta_2 & sin \theta_2 \\ -sin \theta_2 & cos \theta_2 \end{bmatrix}$

$\Rightarrow R(\theta_1)R(\theta_2) = \begin{bmatrix} cos \theta_1 cos\theta_2 - sin\theta_1 sin \theta_2 & cos\theta_1 sin \theta_2 + sin \theta_1 cos\theta_2 \\ -sin \theta_1cos\theta_2 - cos \theta_1 sin \theta_2 & -sin\theta_1 sin \theta_2 + cos \theta_1 cos \theta_2 \end{bmatrix}$

Applying Trigonometric identities we get,

$\Rightarrow R(\theta_1)R(\theta_2) = \begin{bmatrix} cos(\theta_1 + \theta_2 ) & sin(\theta_1 + \theta_2) \\ -sin(\theta_1 + \theta_2) & cos(\theta_1 + \theta_2) \end{bmatrix}$

$\Rightarrow R(\theta_1 + \theta_2) = \begin{bmatrix} cos(\theta_1 + \theta_2 ) & sin(\theta_1 + \theta_2) \\ -sin(\theta_1 + \theta_2) & cos(\theta_1 + \theta_2) \end{bmatrix}$

Hence, we can say the final rotation matrix for two rotations is the multiplication between two rotation matrices representing individual rotations.