Problem Statement
Consider an undamped linear oscillator governed by the equation:
\( m x'' + c x' + k x = 0 \)
where:
- \(m\) is the mass
- \(c\) is the damping coefficient
- \(k\) is the spring constant
Assume the system is **undamped** (\(c = 0\)) and let \(m = 1\), \(k = 1\).
Introduce a new variable \(y = x'\) to rewrite the second-order equation as a system of first-order equations:
\[
\begin{aligned}
x' &= y, \\
y' &= -\frac{k}{m} x - \frac{c}{m} y
\end{aligned}
\]
Determine the trajectory of the system in the phase plane and show why it forms a circle.
Phase-Plane Trajectory of an Undamped Oscillator
Step 1: Write down the system
We consider the oscillator with
\( m = 1, \quad k = 1, \quad c = 0 \).
The phase-plane system is:
\( x' = y, \quad y' = -x \)
Step 2: Define a candidate conserved quantity
We define the total energy:
\( E(x,y) = \frac{1}{2} (x^2 + y^2) \)
Step 3: Differentiate \(E\) with respect to time
Using the chain rule:
\(\frac{dE}{dt} = \frac{\partial E}{\partial x} x' + \frac{\partial E}{\partial y} y'\)
Compute the partial derivatives:
\(\frac{\partial E}{\partial x} = x, \quad \frac{\partial E}{\partial y} = y\)
Substitute:
\(\frac{dE}{dt} = x x' + y y'\)
Step 4: Substitute the system equations
From the system:
\( x' = y, \quad y' = -x \)
Substitute into \(dE/dt\):
\(\frac{dE}{dt} = x(y) + y(-x)\)
Step 5: Simplify
\(\frac{dE}{dt} = xy - xy = 0\)
Step 6: Conclude conservation
Since \(\frac{dE}{dt} = 0\), we have
\(E = \text{constant} \quad \Rightarrow \quad \frac{1}{2}(x^2 + y^2) = C\)
Multiply both sides by 2:
\(x^2 + y^2 = 2C\)
This is the equation of a circle centered at the origin.
Step 7: Physical interpretation
- The radius of the circle is \(\sqrt{2C}\).
- Motion is periodic and stays on the circle forever.
- Velocity (\(y\)) is horizontal in phase plane, acceleration (\(y'\)) is vertical.
- Energy is conserved: kinetic + potential = constant.
Step 8: Takeaway
\(x^2 + y^2 = \text{constant} \quad \Rightarrow \quad \text{phase-plane trajectory is a circle}\)
For an undamped oscillator, the motion is perfectly periodic, and the phase-plane trajectory is a circle centered at the origin.