Question

Consider a mass \(m\) moving in two dimensions with potential energy \(U(x, y)=\frac{1}{2} k r^{2},\) where \(r^{2}=x^{2}+y^{2} .\) Write down the Lagrangian, using coordinates \(x\) and \(y,\) and find the two Lagrange equations of motion. Describe their solutions. [This is the potential energy of an ion in an "ion trap," which can be used to study the properties of individual atomic ions.]

Short answer

The Lagrangian is \( \mathcal{L} = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - \frac{1}{2} k (x^2 + y^2) \). Equations of motion are \( m\ddot{x} = -kx \) and \( m\ddot{y} = -ky \), which are simple harmonic oscillators.

Step-by-step solution

Understanding the Potential Energy

The potential energy given is \( U(x, y) = \frac{1}{2} k r^2 \) where \( r^2 = x^2 + y^2 \). This represents the energy associated with the position of the mass in two-dimensional space.

Writing the Kinetic Energy

In two dimensions, the kinetic energy \( T \) for a mass \( m \) is given by \( T = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) \), where \( \dot{x} \) and \( \dot{y} \) are the velocities in the \( x \) and \( y \) directions.

Formulating the Lagrangian

The Lagrangian \( \mathcal{L} \) is the difference between the kinetic and potential energy: \( \mathcal{L} = T - U \). Substituting the expressions from previous steps, \( \mathcal{L} = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - \frac{1}{2} k (x^2 + y^2) \).

Finding the Lagrange Equations of Motion

Using the Euler-Lagrange equation \( \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0 \), apply it for both coordinates. For \( x \), this results in \( m \ddot{x} = -kx \), and for \( y \), it results in \( m \ddot{y} = -ky \).

Solving the Equations of Motion

The equations \( m \ddot{x} = -kx \) and \( m \ddot{y} = -ky \) describe harmonic oscillators in both \( x \) and \( y \) directions. The general solutions are \( x(t) = A \cos(\omega t + \phi_x) \) and \( y(t) = B \cos(\omega t + \phi_y) \) where \( \omega = \sqrt{\frac{k}{m}} \), and \( A, B, \phi_x, \phi_y \) are constants determined by initial conditions.

Key concepts

Two-Dimensional Motion

When a mass moves in two dimensions, it means it has freedom to travel both in the "x" direction and the "y" direction. Imagine tracking the path of an object on a flat surface: it can go left, right, forward, or backward. This is the essence of two-dimensional motion.

In our exercise, we deal with a particle that moves within this plane. Its position at any time can be represented by coordinates \(x\) and \(y\). Understanding how these coordinates change over time can help us describe the particle's overall motion.

The potential energy \(U(x, y)\) is a function of these coordinates, meaning it gives us a measure of the energy based on the object's position. The kinetic energy, however, depends on the changes in these coordinates — specifically, the velocities which are \(\dot{x}\) and \(\dot{y}\). By combining both kinetic and potential energies, we create a comprehensive picture of the system using Lagrangian mechanics.

Harmonic Oscillator

A harmonic oscillator is a system that experiences a restoring force proportional to the displacement from an equilibrium position. In simple terms, if you pull it out of its rest position, it will try to go back, like a bouncy spring.

In our problem, the equations of motion \( m \ddot{x} = -kx \) and \( m \ddot{y} = -ky \) clearly show harmonic behavior. These equations mean the force trying to bring the particle back to center increases with the distance it moves from the center. It’s like the stronger you stretch a spring, the harder it pulls back.

These equations describe the behavior in both the \(x\) and \(y\) directions, independently. Each direction can oscillate with its own frequency, but, as they have the same constants, they share a frequency \(\omega = \sqrt{\frac{k}{m}}\). It tells us how fast these oscillations occur and is determined by the mass \(m\) and spring constant \(k\). This results in periodic motion represented as a cosine function.

Euler-Lagrange Equation

The Euler-Lagrange equation is a fundamental equation in Lagrangian mechanics. It allows us to derive the equations of motion for a system from a given Lagrangian.

The Lagrangian \(\mathcal{L}\) is given by the difference between kinetic energy \(T\) and potential energy \(U\). For our system, \(\mathcal{L} = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - \frac{1}{2} k (x^2 + y^2)\). This captures the dynamics of our problem in a single formula.

The Euler-Lagrange equation itself is \(\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0\). You apply this to each coordinate: \(q\) becomes \(x\) and \(y\). For \(x\), this leads to \(m \ddot{x} = -kx\), and similarly for \(y\).

What's fascinating about this is how it connects the change in motion (acceleration and force) to the properties of the physical system. It gives a beautiful way to look at the physics of complex systems.