Damped Oscillator: Var of Area in Phase Space Over Time

In summary, the question asks for a demonstration that the area in phase space of a cluster of orbits for the damped simple harmonic oscillator, described by the equations \dot{x} = (1/m) y and \dot{y} = -kx - (r/m) y, varies in time as A(t) = A(0) e^{(-r/m)t}. The correct equation has been updated in the conversation. The solution may involve integration, but the question remains unclear.
  • #1
Sojourner01
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0

Homework Statement


Show that the area in phase space of a cluster of orbits for the damped simple harmonic oscillator given in the lecture varies in time as:
[tex]A(t) = A(0) e^{(-r/m)t}[/tex]

Homework Equations


[tex]\dot{x} = (1/m) y[/tex]
[tex]\dot{y} = -kx - (r/m) y[/tex]

The Attempt at a Solution


I don't understand the question. I don't see how an orbit - which is a line - can have an area. I'm guessing that the result is found by some integration of the given equations, but since I don't see how the statement of the question makes any sense, I can't follow the logic.

Edit: bah. Copied down the wrong equation. The correct one is now quoted. Still, knowing the right one isn't enlightening me on what to do.
 
Last edited:
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  • #2
Nobody?

Is the question nonsense, or have I not explained properly?
 

Related to Damped Oscillator: Var of Area in Phase Space Over Time

1. What is a damped oscillator?

A damped oscillator is a physical system that exhibits oscillatory motion while gradually losing energy due to external forces such as friction or air resistance. It is characterized by a decrease in amplitude or a decrease in the frequency of oscillation over time.

2. How is the damping of an oscillator measured?

The damping of an oscillator is typically measured by the decay rate, which is the rate at which the amplitude of the oscillations decreases over time. It can also be measured by the quality factor, which is the ratio of the energy stored in the oscillator to the energy lost per cycle.

3. What is the role of phase space in a damped oscillator?

Phase space in a damped oscillator refers to the space that represents all possible states of the system, including its position and velocity. It is a useful tool for analyzing the behavior of the oscillator over time, as it allows us to visualize how the system's variables change over time.

4. How does the area in phase space change over time in a damped oscillator?

In a damped oscillator, the area in phase space decreases over time due to the dissipation of energy. This can be seen as the trajectory of the system in phase space gradually spirals towards the equilibrium point, where the amplitude of oscillations becomes zero.

5. What is the significance of the variance of area in phase space over time in a damped oscillator?

The variance of area in phase space over time in a damped oscillator is a measure of the rate at which the system loses energy. It is a useful parameter for studying the behavior of the oscillator and can provide insights into the damping mechanism of the system.

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