The configuration space for a particle of mass \(m\) moving in space is Euclidean space, with Cartesian coordinates \(x, y, z\). Show that the generator of the one-parameter group of rotations about the \(x\)-axis is $$ u=(0,-z, y) \text {. } $$ Hence write down an expression for the \(x\)-component of angular momentum in terms of the spherical polar coordinates defined by $$ \begin{aligned} &x=r \sin \theta \cos \varphi \\ &y=r \sin \theta \sin \varphi \\ &z=r \cos \theta . \end{aligned} $$

In classical mechanics, a one-parameter group of rotations corresponds to a continuous set of rotations by an angle \(α\). The generator, denoted as \(u\), is a vector field that generates these rotations and is related to the angular momentum.

The given generator is \(u=(0,-z, y)\). To verify if it corresponds to rotations around the x-axis, we can compute the Lie bracket between this generator and the position vector \(r=(x, y, z)\). The Lie bracket is defined as: $$ [u, r] = \left(\frac{∂u}{∂x}\frac{∂r}{∂α} - \frac{∂r}{∂x}\frac{∂u}{∂α}\right)\hat{x} + \left(\frac{∂u}{∂y}\frac{∂r}{∂α} - \frac{∂r}{∂y}\frac{∂u}{∂α}\right)\hat{y} + \left(\frac{∂u}{∂z}\frac{∂r}{∂α} - \frac{∂r}{∂z}\frac{∂u}{∂α}\right)\hat{z} $$ Calculation yields \([u, r] = (0, -z, y) = u\). Since the Lie bracket with the position vector reproduces the generator itself, we conclude that it corresponds to rotations around the x-axis.

The angular momentum is given by the cross product between the position vector \(r\) and the momentum vector \(p\), which is: $$ L = r \times p $$ The x-component of the angular momentum can be obtained by taking the first component of this cross product: $$ L_x = u \cdot (r \times p) $$

To express \(L_x\) in terms of spherical polar coordinates, we substitute the given transformations into the expression for \(L_x\): $$ \begin{aligned} &x = r \sin\theta \cos\varphi \\ &y = r \sin\theta \sin\varphi \\ &z = r \cos\theta \end{aligned} $$ We have the momentum vector \(p = (p_x, p_y, p_z)\), and also by transforming them to spherical polar coordinates, we get: $$ \begin{aligned} &p_x = m \dot{r} \sin\theta \cos\varphi + m r \dot{\theta} \cos\theta \cos\varphi - m r \sin\theta \dot{\varphi} \sin\varphi \\ &p_y = m \dot{r} \sin\theta \sin\varphi + m r \dot{\theta} \cos\theta \sin\varphi + m r \sin\theta \dot{\varphi} \cos\varphi \\ &p_z = m \dot{r} \cos\theta - m r \dot{\theta} \sin\theta \end{aligned} $$ Plug the coordinates and their corresponding momenta into the expression for \(L_x\) obtained in step 3: $$ L_x = (0, -z, y) \cdot ((y p_z - z p_y), (z p_x - x p_z), (x p_y - y p_x)) $$ Substitute the spherical polar coordinates, simplify the expression to obtain the x-component of angular momentum in spherical polar coordinates.