Phase space trajectory question

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SUMMARY

The discussion centers on the predictability of phase space trajectories for a simple pendulum. It establishes that adjacent trajectories in phase space do not diverge, affirming the predictable nature of the pendulum's motion. The conversation highlights that this predictability stems from the conservation of total energy, where both kinetic and potential energies remain constant. The inquiry into whether the non-divergence of trajectories is merely a consequence of predictability is addressed, confirming that the two concepts are intrinsically linked.

PREREQUISITES
  • Understanding of phase space concepts
  • Knowledge of simple pendulum mechanics
  • Familiarity with energy conservation principles
  • Basic grasp of dynamical systems theory
NEXT STEPS
  • Research the mathematical representation of phase space for dynamical systems
  • Study the implications of energy conservation in oscillatory motion
  • Explore the concept of Lyapunov stability in phase space
  • Learn about the differences between conservative and non-conservative systems
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Students of physics, particularly those studying classical mechanics, as well as educators and researchers interested in dynamical systems and energy conservation principles.

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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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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?
 

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