At first glance it would appear that, when the dynamical equations of a system are known, we are able to accurately predict its state at any future moment in time. A closer examination, however, reveals the counter-example of molecular motion in gases. Although all the equations of motion of individual molecules and the laws of their collisions are known in this case, it is useless to solve these equations in an attempt to predict the precise positions and velocities of molecules at some future moment. The deterministic prediction fails in this case because of the extreme sensitivity of such a system to small variations in its initial conditions. The slightest perturbation in the coordinates and velocities of the molecules is sufficient to completely change their motion.
KeywordsPhase Space Hamiltonian System Hamiltonian Function Canonical Transformation Phase Trajectory
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