# Ch. 3. Sect. 3.3: Harmonic Oscillations in Two Dimensions. Sect. 3.4: Phase Diagrams. - PowerPoint PPT Presentation

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## Ch. 3. Sect. 3.3: Harmonic Oscillations in Two Dimensions. Sect. 3.4: Phase Diagrams.

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### Title: Lecture 12 Subject: Ch. 3. Sect. 3.3: Harmonic Oscillations in Two Dimensions. Sect. 3.4: Phase Diagrams. Author: Charles W. Myles Last modified by – PowerPoint PPT presentation

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Title: Ch. 3. Sect. 3.3: Harmonic Oscillations in Two Dimensions. Sect. 3.4: Phase Diagrams.

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(No Transcript)
2
SHO in 2d Sect. 3.3
• Look at particle motion in 2d (xy plane), under a
force, linear in the displacement (Hookes
Law)
• For simplicity, assume the force constants the
are same in the x y directions.
• Can do the problem in plane polar coordinates or
in rectangular coordinates
• F -kr ? Fx -kx -kr cos?
• Fy -ky -kr sin?
• Lets work in rectangular coordinates
• Newtons 2nd Law equations Components of
• F -kr mr ma

3
• x y components of Newtons 2nd Law
• -kx mx -ky my
• Or x (?0)2 x 0 y (?0)2 y 0
• with (?0)2 ? k/m
• Solutions x(t) A cos(?0t - a)
• y(t) B cos(?0t - ß)
• The amplitudes A, B, the phases a, ß are
determined (similar to 1d) by initial conditions.
• The motion is simple harmonic in each of the 2 d.
Both oscillations at the same frequency but (in
general) different amplitudes.

4
• The equation for the path in the xy plane is
obtained by eliminating t in x(t) A cos(?0t -
a) and
• y(t) B cos(?0t - ß). Defining d ? a - ß,
algebra gives B2x2 -2ABxy cosd A2y2
A2B2sin2d
• The general path is complicated! Some special
cases
• d ?p/2 ? An ellipse (x/A)2 (y/B)2 1
• If A B, this is a circle.
• d 0 ? A straight line y (B/A) x
• d ?p ? A straight line y -(B/A) x
• Except for special cases, the general path is an
ellipse!

5
Paths in the xy Plane for A B Various d
6
• More general motion in 2d (xy plane), under a
Hookes Law force
• Assume different force constants for x y
• Fx -kx x Fy -ky y
• Newtons 2nd Law equations
• -kx x mx and -ky y my
• Or x (?x)2x 0 and y (?y)2 y 0
• with the definitions (?x)2 ? (kx/m), (?y)2 ?
(ky/m)
• Solutions x(t) A cos(?xt - a) y(t) B
cos(?yt - ß)

7
• x(t) A cos(?xt - a) y(t) B cos(?yt - ß)
• Path in the xy plane is no longer an ellipse, but
a Lissajous curve.
• If the motion repeats itself in regular time
intervals, this curve is closed.
• This will happen only if (?x/?y) is a rational
fraction (if the frequencies are ?
commensurable)
• If (?x/?y) ? a rational fraction, the curve will
be open. In this case, the mass will never pass
twice through the same point with the same
velocity.
• ? An infinitesimal change in initial conditions
can result in qualitatively different motion (a
possible sign of chaos, discussed in Ch. 4!)

8
• A typical Lissajous figure A B, ?y (¾)?x, a
ß

9
• If the 2 frequencies arent commensurable (if
their ratio deviates from a rational fraction by
even an infinitesimal amount)
• The path in the xy plane will not be closed it
will eventually fill a rectangle of size
• 2A ? 2B
• For the path to be closed, (?x/?y) must be a
rational fraction to infinite precision!
• The shape of the Lissajous curve strongly depends
on the phase difference d ? a - ß

10
• Some typical Lissajous figures
• A B ?y 2?x d 0, (p/3), (p/2)

11
Phase Diagrams Sect. 3.4. Back to 1d!
• 1d Oscillator Because Newtons 2nd Law eqtn of
motion is a 2nd order diff. eqtn, the state of
motion of the oscillator is completely specified
if two quantities are given at an initial time
t0 x(t0), v(t0) x(t0).
• ? Its convenient useful to consider x(t) x(t)
v(t) to be coordinates in an abstract 2d phase
space.
• For a 3d particle, the phase space would be 6
dimensional!
• At any time t, the 1d oscillator motion is
completely specified by specifying a point in
this 2d phase space (the x - v or x - x plane).

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• At time t, the oscillator motion is specified by
specifying a point P P(x,x) in this phase
space.
• As time progresses, P P(x,x) will move in this
plane trace out a phase path or phase
trajectory.
• Different initial conditions ? Different phase
paths
• The totality of all possible phase paths of the
particle ? Phase Portrait or Phase Diagram of the
particle. Studying such diagrams gives insight
into the physics of the particle motion.
• This concept isnt limited to oscillators, but is
clearly valid for any particle.
• A very useful concept in statistical mech (Phys.
4302)

13
• Look in detail at the phase diagram for the 1d
simple harmonic oscillator
• x(t) A sin(?0t - d) v(t) x(t) ?0A
cos(?0t - d)
• Eliminating t from these 2 equations gives
• x2/A2 x2/(A2?02) 1
• This is a family of ellipses in the x - x plane!
• The phase diagram for the1d oscillator a family
of ellipses, each of which is a separate phase
path, for different initial conditions.

14
• The phase diagram for the 1d oscillator is a
family of ellipses, as in the Figure.
• Note Oscillator total energy E (½)kA2.
• Also, ?02 (k/m) ? The ellipse equation can be
written
• x2/(2E/k) x2/(2E/m) 1

15
• Writing the elliptical phase path as
• x2/(2E/k) x2/(2E/m) 1
• ? PHYSICS Each phase path (ellipse) corresponds
to a definite total energy E of the oscillator
(different initial conditions!).
• No 2 phase paths of the oscillator can cross!
• If they could cross, this would mean that for
given initial conditions x(t0), v(t0), the motion
could proceed on different phase paths. This is
impossible, since the solutions to a linear 2nd
order differential equation are unique!

16
• In constructing a phase diagram Choose x as the
x-axis x v as the y-axis.
• The motion of a typical point P(x,x) is always
clockwise! Because, for a harmonic oscillator,
for x gt 0, x v decreases if x lt 0, x v
increases!
• Earlier we obtained x(t) A sin(?0t - d)
• v(t) x(t) ?0A cos(?0t - d) by integrating
Ns 2nd Law Eq. (a 2nd order differential
equation)
• (d2x/dt2) (?0)2 x 0
• But, we can get the phase path in a simpler way.

17
• Can use (dx/dt) x v (1)
• (dx/dt) (dv/dt) - (?0)2x (2)
• Divide (2) by (1) (dx/dx) - (?0)2(x/x)
• A 1st order differential eqtn for x(x) v(x)
• xdx - (?0)2xdx
• Solution x2/A2 x2/(A2?02) 1 as before!
• For the SHO, we can easily use either the 2nd
order differential eqtn or the 1st order one just
described. For motion in more complicated
situations, its sometimes easier to find the
solution to the path directly from a 1st order
equation.