Anyone who has done a course in physics at secondary schools met the so-called "simple pendulum", one of the most commonly used examples in teaching.
Known since antiquity, it has inspired the fundamental insights of Galileo Galilei that later brought the great man in the formulation of the properties of isocronism, and opened the way for concepts of motion and momentum that are the foundation of classic mechanics.
Apart from these historical notes, since our first meeting with the pendulum what we notice most is its "simplicity": the motion of the pendulum can be described by means of a simple equation. Depending on certain initial conditions - angle, mass, length - you can predict all the future behaviour of the pendulum without possibility of errors. The behaviour is periodic in time, provided you can neglect some forces that have no direct relation with the mechanism itself, but are inevitable in a real pendulum: friction and resistance.
However, you can incorporate these components into the motion of the pendulum, which from ideal and linear becomes real (at least a little' more, because the whole mass is considered as condensed in a single point and concentrated in the lower summit of the shaft) and non-linear. The equation is significantly modified, but at first glance one could not say that the effects may disrupt our first understanding of the pendulum: it should remain a simple and absolutely predictable dynamic system.
I invite you to make direct experience of such pendulum playing with the simulator available on the website MyPhysicLab.com.
Depending on the choice of system parameters the behaviour of the pendulum becomes very similar to a totally random behavior, although it remains absolutely deterministic. This is a chaotic dynamics, where you can not predict the kinematic conditions of the system (e.g. speed, position, acceleration) even if initial conditions are well-known.
This is apparently in disagreement with the deterministic nature of the pendulum.
If one looks at the trajectories of state variables of the system they never repeat: this can be demonstrated mathematically. There is no periodicity for the oscillations of the pendulum. The trajectories will never overlap, infinitely. The state variables are no longer constrained to repeat the same routes, but free to experience an infinite variety of possible trajectories within the state space. The pendulum, from simple, has become something much more complex.
Known since antiquity, it has inspired the fundamental insights of Galileo Galilei that later brought the great man in the formulation of the properties of isocronism, and opened the way for concepts of motion and momentum that are the foundation of classic mechanics.
Apart from these historical notes, since our first meeting with the pendulum what we notice most is its "simplicity": the motion of the pendulum can be described by means of a simple equation. Depending on certain initial conditions - angle, mass, length - you can predict all the future behaviour of the pendulum without possibility of errors. The behaviour is periodic in time, provided you can neglect some forces that have no direct relation with the mechanism itself, but are inevitable in a real pendulum: friction and resistance.
However, you can incorporate these components into the motion of the pendulum, which from ideal and linear becomes real (at least a little' more, because the whole mass is considered as condensed in a single point and concentrated in the lower summit of the shaft) and non-linear. The equation is significantly modified, but at first glance one could not say that the effects may disrupt our first understanding of the pendulum: it should remain a simple and absolutely predictable dynamic system.
I invite you to make direct experience of such pendulum playing with the simulator available on the website MyPhysicLab.com.
Depending on the choice of system parameters the behaviour of the pendulum becomes very similar to a totally random behavior, although it remains absolutely deterministic. This is a chaotic dynamics, where you can not predict the kinematic conditions of the system (e.g. speed, position, acceleration) even if initial conditions are well-known.
This is apparently in disagreement with the deterministic nature of the pendulum.
If one looks at the trajectories of state variables of the system they never repeat: this can be demonstrated mathematically. There is no periodicity for the oscillations of the pendulum. The trajectories will never overlap, infinitely. The state variables are no longer constrained to repeat the same routes, but free to experience an infinite variety of possible trajectories within the state space. The pendulum, from simple, has become something much more complex.
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