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 $(r,\varphi )$ 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 $\varphi$ remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is $\omega =\sqrt{2k/m_1}$.

Short answer

The Lagrangian is simplified using CM and relative coordinates. Constant $r$ leads to stable rotation; constant $\varphi$ leads to oscillation with $\omega =\sqrt{2k/m}$.

Step-by-step solution

Write the Lagrangian in terms of ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$

The kinetic energy $T$ of the system is given by the sum of the kinetic energy of each particle. Thus, $T=\frac{1}{2}m{\dot{\mathbf{r}}}_1^2+\frac{1}{2}m{\dot{\mathbf{r}}}_2^2$. The potential energy $V$ is due to the spring and is given by $V=\frac{1}{2}k(|{\mathbf{r}}_2-{\mathbf{r}}_1|-L{)}^2$. Hence, the Lagrangian $\mathcal{L}$ can be written as $\mathcal{L}=T-V=\frac{1}{2}m({\dot{\mathbf{r}}}_1^2+{\dot{\mathbf{r}}}_2^2)-\frac{1}{2}k(|{\mathbf{r}}_2-{\mathbf{r}}_1|-L{)}^2$.

Change to CM and Relative Coordinates

Let's introduce the center of mass position $\mathbf{R}=\frac{1}{2}({\mathbf{r}}_1+{\mathbf{r}}_2)$ and the relative position $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$. The velocities are $\dot{\mathbf{R}}=\frac{1}{2}({\dot{\mathbf{r}}}_1+{\dot{\mathbf{r}}}_2)$ and $\dot{\mathbf{r}}={\dot{\mathbf{r}}}_2-{\dot{\mathbf{r}}}_1$. In polar coordinates $\mathbf{r}$, let $\mathbf{r}=(r,\varphi )$. The Lagrangian can now be written as $\mathcal{L}=m{\dot{\mathbf{R}}}^2+\frac{1}{4}m{\dot{\mathbf{r}}}^2-\frac{1}{2}k(r-L{)}^2$.

Lagrange Equations for CM Coordinates

For the center of mass coordinates, we use ${\mathcal{L}}_{\mathrm{CM}}=m{\dot{\mathbf{R}}}^2$. The equations of motion are derived using $\frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{X}})-\frac{\partial \mathcal{L}}{\partial X}=0$ and $\frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{Y}})-\frac{\partial \mathcal{L}}{\partial Y}=0$. These simplify to $m\ddot{X}=0$ and $m\ddot{Y}=0$. The solutions are $X(t)=X_0+V_{X}t$, $Y(t)=Y_0+V_{Y}t$.

Lagrange Equations for Relative Coordinates $r$ and $\varphi$

For the relative coordinates, we have ${\mathcal{L}}_r=\frac{1}{4}m{\dot{r}}^2+\frac{1}{4}m{r}^2{\dot{\varphi }}^2-\frac{1}{2}k(r-L{)}^2$. The Lagrange equations give $\frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{r}})-\frac{\partial \mathcal{L}}{\partial r}=0$ and $\frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{\varphi }})-\frac{\partial \mathcal{L}}{\partial \varphi }=0$, which lead to: $m\ddot{r}=mr{\dot{\varphi }}^2-k(r-L)$ and $\frac{d}{dt}(m{r}^2\dot{\varphi })=0$.

Solve the Equations for Special Cases

For $r$ constant, $\ddot{r}=0$ implies $mr{\dot{\varphi }}^2=k(r-L)$. For $\varphi$ constant, $\dot{\varphi }=0$ simplifies $m\ddot{r}=-k(r-L)$ to simple harmonic motion: $\ddot{r}=-\frac{k}{m}(r-L)$ with frequency $\omega =\sqrt{\frac{k}{m/2}}=\sqrt{\frac{2k}{m}}$.

Describe the Motions

For the motion with constant $r$, the system rotates with the angular velocity satisfying $mr{\dot{\varphi }}^2=k(r-L)$. For the motion with constant $\varphi$, the system oscillates harmonically around $r=L$ with frequency $\omega =\sqrt{\frac{2k}{m}}$. This describes a radial oscillation or "breathing mode".

Key concepts

Center of Mass Coordinates

In physics, the concept of "center of mass" coordinates is quite straightforward yet profoundly beneficial in simplifying complex problems. Imagine a system of several particles. Instead of tracking each particle's movement, we can focus on a single point representing the entire system's mass, called the center of mass (CM). This simplifies the analysis significantly.

For two particles with positions ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, the center of mass coordinate $\mathbf{R}$ is defined using the formula:

• $\mathbf{R}=\frac{1}{2}({\mathbf{r}}_1+{\mathbf{r}}_2)$

This equation assumes equal mass particles, which makes it especially simple because the weights in the average are equal. The velocities of these particles can collectively result in a center of mass velocity $\dot{\mathbf{R}}$, defined as:

• $\dot{\mathbf{R}}=\frac{1}{2}({\dot{\mathbf{r}}}_1+{\dot{\mathbf{r}}}_2)$

Using CM coordinates is vital because it allows us to decouple the motion related to the system's overall journey through space from internal motions, such as oscillations or rotations.

Harmonic Motion

Harmonic motion is a type of periodic oscillation that is key to understanding phenomena like springs and pendulums. It occurs when an object is subject to a restoring force proportional to its displacement from an equilibrium position.

In the context of our two-particle system connected by a spring, harmonic motion arises when considering the system's internal dynamics. For our system, if the relative position $r$ remains constant during motion, it indicates that there’s no change in distance between the particles due to the spring's restoring force. Conversely, when the angle $\varphi$ stays constant but $r$ changes, this describes oscillations about the natural length $L$ of the spring.

Harmonic motion is described by a simple harmonic oscillator modeled by the equation:

• $\ddot{r}=-\frac{k}{m}(r-L)$

Here, $k$ is the spring constant, $m$ is the mass, and $L$ is the natural length of the spring. The frequency of these oscillations, crucial for understanding how quickly the system oscillates, is given by:

• $\omega =\sqrt{\frac{2k}{m}}$

This frequency is critical as it dictates the speed of the radial or breathing mode oscillations of the particles around their equilibrium separation.

Relative Coordinates

Relative coordinates simplify the study of internal dynamics between two or more particles without worrying about the system's movement as a whole. They define the movement of one particle concerning another, making it a powerful tool in analyzing complex mechanical systems.

For our spring-connected particle system, relative coordinates help to focus on the internal spring motion. Define the relative position $\mathbf{r}$ as the difference in positions of the two particles:

• $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$

In polar form, this can be expressed as $(r,\varphi )$, where $r$ is the radial distance between the particles, and $\varphi$ is the angle of the line connecting them with respect to a fixed axis.

The equations of motion in terms of these coordinates provide insights into how the system behaves under different conditions. For instance, if $r$ changes while $\varphi$ remains fixed, this results in simple harmonic motion, revealing intrinsic oscillations. On the other hand, if $\varphi$ changes and $r$ remains constant, this describes rotational motion, revealing how energy is distributed between kinetic and potential forms.

Using relative coordinates, especially in a Lagrangian framework, abstracts complex interactions into manageable analyses, assisting in deriving equations that describe a system's complete motion.