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Hi,

The following problem is from Halliday and Resnick's Physics, Vol. I, 1967 edition, page 379, Problem 35. It has defeated me for the last month.

"A particle undergoes motion in the x-y plane. It's equation of motion is given by

[tex]x = A_x cos(w_x t)[/tex]

[tex]y = A_y cos(w_y t)[/tex]

(i)If the ratio of the two angular frequencies is a rational number, then prove that the motion is periodic.

(ii)If the ratio of the two angular frequencies is irrational, then prove that the particle will pass through every point in the rectangle given by [itex]x = \pm A_x[/itex] and [itex]y = \pm A_y[/itex], but will never pass through a given point with the same velocity twice."

(i) is elementary, it's (ii) that's giving me the creeps. What is effectively being asked is a proof that the motion will exhibit deterministic chaos, taking a non-repeating but constrained trajectory through the 4D phase space, something like strange attractor behaviour.

They don't teach non-linear dynamics in our high-school syllabus, so I'm not aware of any standard theorems or tools that can be used to tackle such problems. As a first analysis, I've identified four components to the proposition:

a) The motion is constrained inside the rectangle. This is obvious.

b) The motion is non-periodic. This is easily proved by reductio ad absurdum.

c)Given any value of (x,y) lying in the specified rectangle, There exists some value of t at which the particle will be at (x,y).

d)The particle does not return to the same point in the phase space at two distinct instants of time.

Since the equation of motion is deterministic, presumably (b) implies (d). So that leaves me with (c) to prove. How do I do it? Thanks for your help.

I've also attached my attempt at a proof of (b) below:

"Let us assume that the motion is periodic, with period T. As the only periods of the real cosine function are integral multiples of [itex]2\pi[/itex], we have

[tex]T = \frac {2n\pi}{w_x} = \frac {2m\pi}{w_y}

\implies \frac {w_x}{w_y} = \frac {m}{n}[/tex]

This implies that the ratio of the angular frequencies is equal to the ratio of two integers, a rational number. But this contradicts the given condition that the said ratio is irrational. Therefore our assumption can not be right. Therefore the motion is non-periodic."

Molu

The following problem is from Halliday and Resnick's Physics, Vol. I, 1967 edition, page 379, Problem 35. It has defeated me for the last month.

"A particle undergoes motion in the x-y plane. It's equation of motion is given by

[tex]x = A_x cos(w_x t)[/tex]

[tex]y = A_y cos(w_y t)[/tex]

(i)If the ratio of the two angular frequencies is a rational number, then prove that the motion is periodic.

(ii)If the ratio of the two angular frequencies is irrational, then prove that the particle will pass through every point in the rectangle given by [itex]x = \pm A_x[/itex] and [itex]y = \pm A_y[/itex], but will never pass through a given point with the same velocity twice."

(i) is elementary, it's (ii) that's giving me the creeps. What is effectively being asked is a proof that the motion will exhibit deterministic chaos, taking a non-repeating but constrained trajectory through the 4D phase space, something like strange attractor behaviour.

They don't teach non-linear dynamics in our high-school syllabus, so I'm not aware of any standard theorems or tools that can be used to tackle such problems. As a first analysis, I've identified four components to the proposition:

a) The motion is constrained inside the rectangle. This is obvious.

b) The motion is non-periodic. This is easily proved by reductio ad absurdum.

c)Given any value of (x,y) lying in the specified rectangle, There exists some value of t at which the particle will be at (x,y).

d)The particle does not return to the same point in the phase space at two distinct instants of time.

Since the equation of motion is deterministic, presumably (b) implies (d). So that leaves me with (c) to prove. How do I do it? Thanks for your help.

I've also attached my attempt at a proof of (b) below:

"Let us assume that the motion is periodic, with period T. As the only periods of the real cosine function are integral multiples of [itex]2\pi[/itex], we have

[tex]T = \frac {2n\pi}{w_x} = \frac {2m\pi}{w_y}

\implies \frac {w_x}{w_y} = \frac {m}{n}[/tex]

This implies that the ratio of the angular frequencies is equal to the ratio of two integers, a rational number. But this contradicts the given condition that the said ratio is irrational. Therefore our assumption can not be right. Therefore the motion is non-periodic."

Molu

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