Problem 4 - Classical Mechanics

Question

A particle of mass \(m\) moves in three dimensions under the action of a central, conservative force with potential energy \(V(r)\). Find the Hamiltonian function in terms of spherical polar co-ordinates, and show that \(\varphi\), but \(\operatorname{not} \theta\), is ignorable. Express the quantity \(J^{2}=\) \(m^{2} r^{4}\left(\dot{\theta}^{2}+\sin ^{2} \theta \dot{\varphi}^{2}\right)\) in terms of the generalized momenta, and show that it is a second constant of the motion.

Step-by-Step Solution

Answer

In summary, we found the kinetic energy of a particle moving in three dimensions in spherical polar coordinates and constructed the Hamiltonian function. We then determined generalized momenta and found that while the coordinate \(\varphi\) is ignorable, \(\theta\) is not. We expressed \(J^2\) in terms of generalized momenta and verified that it is a constant of the motion.

Step 1: Find the kinetic energy in spherical polar coordinates

First, we need to express the kinetic energy of the particle in spherical polar coordinates. To do this, we can recall the transformation between Cartesian and spherical polar coordinates: \(x = r\sin\theta\cos\varphi\) \(y = r\sin\theta\sin\varphi\) \(z = r\cos\theta\) Now, we differentiate these equations with respect to time to find the velocity components in Cartesian coordinates: \(\dot{x} = \dot{r}\sin\theta\cos\varphi + r\cos\theta\cos\varphi\dot{\theta} - r\sin\theta\sin\varphi\dot{\varphi}\) \(\dot{y} = \dot{r}\sin\theta\sin\varphi + r\cos\theta\sin\varphi\dot{\theta} + r\sin\theta\cos\varphi\dot{\varphi}\) \(\dot{z} = \dot{r}\cos\theta - r\sin\theta\dot{\theta}\) Now we can find the kinetic energy T in terms of spherical polar coordinates by using the velocity components: \(T = \dfrac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)\) After some algebraic manipulations, we get: \(T=\dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right)\)

Step 2: Construct the Hamiltonian function

Now that we have found the kinetic energy in spherical polar coordinates, let us write down the Hamiltonian function, which is the sum of the kinetic and potential energies: \(H = T + V = \dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right) + V(r)\)

Step 3: Find the generalized momenta

To determine whether \(\varphi\) and \(\theta\) are ignorable coordinates, we need to find the generalized momenta associated with these coordinates. The generalized momentum associated with a generalized coordinate \(q\) is defined as: \(p_q = \dfrac{\partial L}{\partial\dot{q}}\) where \(L\) is the Lagrangian, which can be written as \(L = T - V\). First, let us find the generalized momentum associated with \(\varphi\): \(p_{\varphi} = \dfrac{\partial}{\partial\dot{\varphi}}\left(\dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right) - V(r)\right) = mr^2\sin^2\theta\dot{\varphi}\) Next, let's find the generalized momentum associated with \(\theta\): \(p_{\theta} = \dfrac{\partial}{\partial\dot{\theta}}\left(\dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right) - V(r)\right) = mr^2\dot{\theta}\)

Step 4: Determine ignorable coordinates

A coordinate is ignorable if its generalized momentum does not depend on the coordinate itself. From our results above, we see that: \(p_{\varphi} = mr^2\sin^2\theta\dot{\varphi}\) does not depend on \(\varphi\). Therefore, \(\varphi\) is an ignorable coordinate. On the other hand, \(p_{\theta} = mr^2\dot{\theta}\) does depend on \(\theta\) through the term \(\sin^2\theta\). Thus, \(\theta\) is not an ignorable coordinate.

Step 5: Express \(J^2\) in terms of generalized momenta

We are asked to express the given quantity \(J^2\) in terms of generalized momenta: \(J^2 = m^2r^4\left(\dot{\theta}^2 + \sin^2\theta\dot{\varphi}^2\right)\) Using the expressions for \(p_{\varphi}\) and \(p_{\theta}\) we found earlier, we can write \(J^2\) as: \(J^2 = \dfrac{1}{m^2r^4}\left(p_{\theta}^2 + \dfrac{p_{\varphi}^2}{\sin^2\theta}\right)\)

Step 6: Show \(J^2\) is a constant of the motion

A quantity is a constant of the motion if its time derivative is zero. To show that \(J^2\) is a constant of the motion, we need to differentiate it with respect to time and show that the derivative is zero. Differentiating \(J^2\) with respect to time gives: \(\dfrac{dJ^2}{dt} = \dfrac{d}{dt}\left(\dfrac{1}{m^2r^4}\left(p_{\theta}^2 + \dfrac{p_{\varphi}^2}{\sin^2\theta}\right)\right)\) Using the chain rule and noting that the time derivatives of \(p_{\theta}\) and \(p_{\varphi}\) are zero (since they are constants of motion), we find that the time derivative of \(J^2\) is indeed zero: \(\dfrac{dJ^2}{dt} = 0\) Therefore, \(J^2\) is a constant of the motion.

Key Concepts

Spherical Coordinates

Understanding spherical coordinates is essential in describing a particle's motion in three dimensions. This system uses three variables: radial distance \( r \), polar angle \( \theta \), and azimuthal angle \( \varphi \). These coordinates relate to Cartesian coordinates \( x, y, \) and \( z \) as follows:

• \( x = r\sin\theta\cos\varphi \)
• \( y = r\sin\theta\sin\varphi \)
• \( z = r\cos\theta \)

In motion analysis, the velocities in spherical coordinates are obtained by differentiating these relations with respect to time. This leads to expressions for \( \dot{x}, \dot{y}, \) and \( \dot{z} \) components of velocity, which are then used to calculate the kinetic energy.
Expressing functions like kinetic energy in spherical coordinates simplifies handling problems involving symmetry, such as central forces, where the system inherently favors these coordinates.

Central Force

A central force depends only on the distance from a fixed point, not the direction. Such forces are spherically symmetric and are often described by a potential energy function \( V(r) \), solely dependent on radial distance \( r \).
This symmetry simplifies the problem, as the force acts along the radial direction only. Consequently, the problem's rotational components (\( \theta \) and \( \varphi \)) introduce conserved quantities due to the absence of forces in those directions.
Central force problems are crucial for understanding systems like gravitational and electrostatic interactions. In these scenarios, employing spherical coordinates allows us to effectively analyze and solve the motion equations with minimal complexity.

Generalized Momenta

In Lagrangian and Hamiltonian mechanics, generalized momenta play a crucial role. They are linked to the generalized coordinates and represent conserved quantities under specific symmetries.

• The generalized momentum \( p_{\varphi} = mr^2\sin^2\theta\dot{\varphi} \) remains constant due to symmetry in the \( \varphi \) direction, rendering \( \varphi \) an ignorable coordinate.
• For the \( \theta \) angle, \( p_{\theta} = mr^2\dot{\theta} \), which does depend on \( \theta \), making it non-ignorable.

These momenta simplify calculations and provide insights into conserved quantities like angular momentum, pivotal in analyzing rotational dynamics in systems influenced by central forces.

Constant of Motion

A constant of motion is a quantity that remains unchanged as the system evolves. In our problem, \( J^2 \) is identified as a constant of motion, given by:

• \( J^2 = \dfrac{1}{m^2r^4}(p_{\theta}^2 + \dfrac{p_{\varphi}^2}{\sin^2\theta}) \)

To prove its constancy, we show the time derivative of \( J^2 \) is zero.
This reflects the underlying symmetries of the problem, linking the conserved quantities to the central nature of the force. Understanding these constants is critical for analyzing the stability and evolution of the system over time.