Question

Set up Lagrange's equations in cylindrical coordinates for a particle of mass \(m\) in a potential field \(V(r, \theta, z) .\) Hint: \(v=d s / d t ;\) write \(d s\) in cylindrical coordinates.

Short answer

The Lagrange equations in cylindrical coordinates are \( m \ddot{r} - mr \dot{\theta}^2 + \frac{\partial V}{\partial r} = 0 \), \[ \frac{d}{dt} \left( m r^2 \dot{\theta} \right) = \frac{\partial V}{\partial \theta} \), and \( m \ddot{z} = -\frac{\partial V}{\partial z} \).

Step-by-step solution

- Understand the variables in cylindrical coordinates

In cylindrical coordinates, a point in space is represented by \( (r, \theta, z) \) where \( r \) is the radial distance, \( \theta \) is the azimuthal angle, and \( z \) is the height.

- Express kinetic energy in cylindrical coordinates

The kinetic energy \( T \) of a particle can be written in terms of its velocities in cylindrical coordinates: \[ T = \frac{1}{2}m ( \dot{r}^2 + r^2 \dot{\theta}^2 + \dot{z}^2 ). \]

- Write the Lagrangian

The Lagrangian \( L \) is given by the difference between the kinetic energy and the potential energy: \[ L = T - V = \frac{1}{2}m ( \dot{r}^2 + r^2 \dot{\theta}^2 + \dot{z}^2 ) - V(r, \theta, z). \]

- Set up Lagrange's equations

Lagrange's equations are given by \[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, \] where \( q_i \) represents each coordinate \( r, \theta, \) and \( z \).

- Apply Lagrange's equations to each coordinate

For \( r \): \[ \frac{d}{dt} \left( m \dot{r} \right) - m r \dot{\theta}^2 - \frac{\partial V}{\partial r} = 0, \] leading to the equation \[ m \ddot{r} - mr \dot{\theta}^2 = -\frac{\partial V}{\partial r}. \] For \( \theta \): \[ \frac{d}{dt} \left( m r^2 \dot{\theta} \right) - \frac{\partial V}{\partial \theta} = 0, \] leading to the equation \[ \frac{d}{dt} \left( m r^2 \dot{\theta} \right) = \frac{\partial V}{\partial \theta}. \] For \( z \): \[ \frac{d}{dt} \left( m \dot{z} \right) - \frac{\partial V}{\partial z} = 0, \] leading to the equation \[ m \ddot{z} = -\frac{\partial V}{\partial z}. \]

Key concepts

Kinetic Energy

Kinetic energy is the energy that a particle possesses due to its motion. In cylindrical coordinates, the kinetic energy of a particle with mass \(m\) moving in a field is expressed in terms of its velocities in the \(r\), \(\theta\), and \(z\) directions. The formula for kinetic energy is:

\[ T = \frac{1}{2}m ( \dot{r}^2 + r^2 \dot{\theta}^2 + \dot{z}^2 ). \]

Here:

• \( \dot{r} \): Rate of change of radial distance with respect to time
• \( \dot{\theta} \): Rate of change of azimuthal angle with respect to time
• \( \dot{z} \): Rate of change of height with respect to time

By summing the squares of these velocities, weighted by the mass, and dividing by 2, we get the kinetic energy. This form ensures all directions contribute appropriately to the particle's energy.

Potential Energy

Potential energy is the energy that a particle possesses due to its position in a field. In many problems, the potential energy \(V\) is a function of the coordinates. For example, in a gravitational field near the Earth's surface, it might be a function of \(z\), while in an electric field, it could depend on \(r\) and \(\theta\) as well.

The potential energy for a particle in cylindrical coordinates is typically denoted as \( V(r, \theta, z) \), indicating that it can vary with the radial distance \(r\), the azimuthal angle \(\theta\), and the height \(z\). This function encapsulates the influence of external forces acting on the particle. Understanding potential energy is crucial, as it directly affects the Lagrangian mechanics framework.

Lagrangian Mechanics

Lagrangian mechanics is a reformulation of classical mechanics that provides powerful tools for solving mechanical problems. The cornerstone of this approach is the Lagrangian function \(L\), which is defined as the difference between the kinetic energy \(T\) and the potential energy \(V\):

\[ L = T - V. \]

For a particle in cylindrical coordinates, this becomes:

\[ L = \frac{1}{2}m ( \dot{r}^2 + r^2 \dot{\theta}^2 + \dot{z}^2 ) - V(r, \theta, z). \]

Lagrangian mechanics simplifies the equations of motion by transforming them into a set of generalized coordinates, making it easier to deal with complex systems. By using the Euler-Lagrange equation:

\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, \]

we can derive the equations of motion for each coordinate.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height element \(z\). Points in space are described by three components:

• \(r\): Radial distance from the origin to the projection of the point in the \(xy\)-plane
• \(\theta\): Azimuthal angle, the angle between the positive \(x\)-axis and the line from the origin to the projection in the \(xy\)-plane
• \(z\): Height above the \(xy\)-plane

This system is particularly useful for problems with cylindrical or rotational symmetry. By converting Cartesian coordinates to cylindrical ones, many physical problems become simpler to solve.

Lagrange's Equations

Lagrange's equations are a set of second-order differential equations used to describe the dynamics of a system. These equations are derived from the Lagrangian function and are vital in Lagrangian mechanics. For a particle in cylindrical coordinates, these equations are applied to each coordinate (\(r\), \(\theta\), and \(z\)) as follows:

• For \(r\): \[\frac{d}{dt} \left( m \dot{r} \right) - m r \dot{\theta}^2 - \frac{\partial V}{\partial r} = 0.\]
• For \(\theta\): \[ \frac{d}{dt} \left( m r^2 \dot{\theta} \right) - \frac{\partial V}{\partial \theta} = 0.\]
• For \(z\): \[ m \ddot{z} = -\frac{\partial V}{\partial z}. \]

These equations describe how the system evolves with time, taking into account both kinetic and potential energies. By solving these equations, one can predict the motion of the particle in the field.