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PPT – Ch. 3. Sect. 3.3: Harmonic Oscillations in Two Dimensions. Sect. 3.4: Phase Diagrams. PowerPoint presentation | free to download - id: 6d1435-MzYzZ

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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

- 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.

- 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!

Paths in the xy Plane for A B Various d

- 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)

- 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!)

- A typical Lissajous figure A B, ?y (¾)?x, a

ß

- 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

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

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).

- 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)

- 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.

- 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

- 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!

- 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.

- 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.