Phase space trajectory question

In summary, the phase space picture for a simple pendulum shows that adjacent trajectories never diverge, making the evolution of the pendulum predictable. This is due to the fact that a small variation in initial conditions does not produce a significantly different trajectory. However, if two phase space paths for the pendulum were to cross, the total energy for those paths would be the same but would no longer be constant if the paths diverged. This is not consistent with the behavior of an ideal pendulum.
  • #1
Grand
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0
In my lecture they give the phase space picture for a simple pendulum
http://mathematicalgarden.files.wordpress.com/2009/03/pendulum-portrait3.png?w=500&h=195

and then say that adjacent trajectories never diverge and therefore evolution is predictable. I wanted to ask, is the statement that adjacent trajectories never diverge just a consequesnce of the fact that motion of the simple pendulum is predictable, i.e. a small variation in initial conditions will not produce totally different trajectory. Or am I just being stupid and the answer is diferent?
 
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  • #2
Suppose two phase space paths for the pendulum crossed at a point in phase space. At that point the total energy for the two paths is the same, E = T + V. But if the paths diverge then the total energy can no longer be constant, but the total energy for an ideal pendulum is constant.

Does that fly for you?
 

Related to Phase space trajectory question

1. What is a phase space trajectory?

A phase space trajectory refers to the path traced by a system in phase space, which is a mathematical space where all possible states of a physical system are represented. It is often used in physics and engineering to study the behavior of complex systems over time.

2. How is a phase space trajectory different from a regular trajectory?

A regular trajectory describes the motion of a single particle in physical space, while a phase space trajectory describes the evolution of a system in phase space. This means that a phase space trajectory takes into account all possible states of the system, rather than just the position and velocity of a single particle.

3. What information can be obtained from a phase space trajectory?

A phase space trajectory can provide valuable information about the dynamics and stability of a system. It can also reveal patterns and relationships between different variables of the system, which can help in understanding its behavior and making predictions.

4. How is a phase space trajectory plotted?

A phase space trajectory is typically plotted using a phase space diagram, which is a graph with the variables of the system plotted on the axes. The trajectory is then traced by plotting the state of the system at different points in time.

5. Can a phase space trajectory be used to predict the future behavior of a system?

Yes, a phase space trajectory can be used to make predictions about the future behavior of a system. By analyzing the trajectory, patterns and trends can be identified, which can help in predicting the future state of the system. However, the accuracy of these predictions depends on the complexity and stability of the system.

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