Question

Consider two particles of equal masses, \(m_{1}=m_{2},\) attached to each other by a light straight spring (force constant \(k\), natural length \(L\) ) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\), and rewrite it in terms of the \(\mathrm{CM}\) and relative positions, \(\mathbf{R}\) and \(\mathbf{r},\) using polar coordinates \((r, \phi)\) for \(\mathbf{r} .\) (b) Write down and solve the Lagrange equations for the CM coordinates \(X, Y\). (c) Write down the Lagrange equations for \(r\) and A. Solve these for the two special cases that \(r\) remains constant and that \(\phi\) remains constant. Describe (i) the contant the corresponding motions. In particular, show that the frequency of oscillations in the second case is \(\omega=\sqrt{2 k / m_{1}}.\)

Short answer

For constant \( r \), circular motion; for constant \( \phi \), oscillatory motion with \( \omega = \sqrt{\frac{2k}{m}} \).

Step-by-step solution

Write the Lagrangian for System

The Lagrangian, \( L \), is defined as the difference between the kinetic energy (T) and the potential energy (V) of the system: \[ L = T - V \]For two particles of mass \( m \), the kinetic energy is given by:\[ T = \frac{1}{2}m \dot{\mathbf{r}}_1^2 + \frac{1}{2}m \dot{\mathbf{r}}_2^2 \]The potential energy due to the spring is:\[ V = \frac{1}{2}k(r - L)^2 \]Here, \( r = |\mathbf{r}_2 - \mathbf{r}_1| \) is the extension of the spring from its natural length \( L \). Thus the Lagrangian can be written as:\[ L = \frac{1}{2}m \dot{\mathbf{r}}_1^2 + \frac{1}{2}m \dot{\mathbf{r}}_2^2 - \frac{1}{2}k(r - L)^2 \]

Express Lagrangian using CM and Relative Coordinates

Introduce the center of mass (CM) coordinate \( \mathbf{R} \) and relative coordinate \( \mathbf{r} \):\[ \mathbf{R} = \frac{\mathbf{r}_1 + \mathbf{r}_2}{2}, \quad \mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1 \]Express the velocities as:\[ \dot{\mathbf{r}}_1 = \dot{\mathbf{R}} - \frac{1}{2}\dot{\mathbf{r}}, \quad \dot{\mathbf{r}}_2 = \dot{\mathbf{R}} + \frac{1}{2}\dot{\mathbf{r}} \]Substitute these into the Lagrangian:\[ L = m \dot{\mathbf{R}}^2 + \frac{1}{2}m \dot{\mathbf{r}}^2 - \frac{1}{2}k(r - L)^2 \]

Write Lagrange Equations for CM Coordinates

The Lagrangian is now separated into CM and relative motion. The motion of the CM governed by coordinates \( X \) and \( Y \):The Lagrange equations for \( X \) and \( Y \) from \( L = m (\dot{X}^2 + \dot{Y}^2) + \text{(terms with \( \mathbf{r} \))} \) are:\[ \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{X}} \right) = \frac{\partial L}{\partial X}, \quad \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{Y}} \right) = \frac{\partial L}{\partial Y} \]Since there are no forces acting in the \( X \) and \( Y \) directions, these reduce to:\[ m \ddot{X} = 0, \quad m \ddot{Y} = 0 \]The solutions are straight-line motion: \( X(t) = X_0 + V_X t \) and \( Y(t) = Y_0 + V_Y t \).

Write Lagrange Equations for Relative Motion in Polar Coordinates

Use \( \mathbf{r} = (r, \phi) \) as polar coordinates, with \( \dot{r} \) and \( r \dot{\phi} \) being the radial and angular components of velocity respectively. The kinetic energy becomes:\[ \frac{1}{2}m(\dot{r}^2 + (r \dot{\phi})^2) \]The Lagrangian for \( r \) and \( \phi \) is then:\[ L = m \dot{\mathbf{R}}^2 + \frac{1}{2}m(\dot{r}^2 + r^2 \dot{\phi}^2) - \frac{1}{2}k(r - L)^2 \]

Solve for Radial and Angular Motion

Consider two cases:**Case 1**: \( r = \text{constant}\)- The system behaves like a rotating rigid body. The centrifugal force is balanced by the spring force, leading to no oscillation. In this case, \( \phi(t) = \text{constant} \).**Case 2**: \( \phi = \text{constant}\)- Only radial motion occurs, resulting in oscillations. The Lagrangian simplifies to include only terms in \( r \), and solving the Lagrange equation, we find the oscillation frequency:\[ \omega = \sqrt{\frac{2k}{m}} \]

Interpret Results

The solution implies the nature of motion based on different constraints in the system. For constant \( r \), circular motion occurs with no changes to \( \phi \). When \( \phi \) remains constant, the body behaves as if it's performing simple harmonic oscillation with angular frequency \( \omega \) derived above.

Key concepts

Lagrange Equations

Lagrange equations are a cornerstone of classical mechanics. They provide a powerful framework to describe a system's dynamics based on its kinetic and potential energy. The equations emerge from the principle of least action, which states that the trajectory of a system between two states is one that minimizes the action. This action is the integral of the Lagrangian over time.

• The Lagrangian, typically denoted as \( L \), is the difference between kinetic energy \( T \) and potential energy \( V \).
• Mathematically, \( L = T - V \).
• The Lagrange equations are given by: \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) = \frac{\partial L}{\partial q} \), where \( q \) represents the generalized coordinates.

In the provided exercise, these equations are used to analyze the motion of two particles connected by a spring. By expressing the Lagrangian in terms of both the center of mass and relative motion, the complexity of the problem reduces significantly, allowing a clearer insight into different types of motion possible in the system.

Polar Coordinates

Polar coordinates offer a convenient way to describe motion, especially when dealing with problems involving radial distances and angles. In the context of the exercise, we represent the relative position of the particles using polar coordinates, characterized by the radial length \( r \) and angular position \( \phi \).

• Radial coordinate \( r \) measures the distance from origin to the point.
• Angular coordinate \( \phi \) defines the direction relative to a reference direction, usually the positive x-axis.
• The velocity components in polar coordinates include a radial component \( \dot{r} \) and a tangential component \( r \dot{\phi} \).

These help translate the problem into a form suitable for deriving equations of motion for different scenarios. In this exercise, it's particularly handy to understand oscillatory motion and cases where either \( r \) or \( \phi \) is kept constant.

Center of Mass

The center of mass (CM) is a central concept in mechanics, representing the average position of all mass in a system. This point moves as if all the external forces are applied at it. In systems with symmetrical interactions, like our spring-coupled particles, the CM simplifies the dynamics.

• For two masses \( m_1 \) and \( m_2 \), the CM \( \mathbf{R} \) is given by \( \mathbf{R} = \frac{m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2}{m_1 + m_2} \).
• With equal masses, the formula simplifies to \( \mathbf{R} = \frac{\mathbf{r}_1 + \mathbf{r}_2}{2} \).
• The CM allows decoupling of translational and relative motion, reducing complexity by focusing on motion's primary axes.

Using CM coordinates, we isolate the inertial drift of the entire system, represented by straightforward equations \( m \ddot{X} = 0, \) \( m \ddot{Y} = 0 \), indicating no net force impact on CM due to symmetry and internal forces only.

Harmonic Oscillator

A harmonic oscillator is a system that experiences restoring forces proportional to the displacement from its equilibrium, leading to oscillations around this equilibrium. In the exercise, when considering \( \phi \) as constant, the system behaves like a harmonic oscillator.

• The spring exerts a force described by Hooke's Law, \( F = -k(r - L) \), where \( k \) is the spring constant, and \( L \) is the rest length.
• The restoring force enables the system to oscillate with simple harmonic motion characteristics.
• The natural frequency of these oscillations is \( \omega = \sqrt{\frac{2k}{m}} \), derived from solving the Lagrange equations with \( r \) as the dynamic variable and \( \phi \) constant.

Understanding this behavior connects to broader concepts in physics, as harmonic oscillators model countless systems, from basic springs to quantum systems, capturing regular, predictable rhythmic motion.