Mathematical Methods of Physics – Fall 2010 – Dr. E.J. Zita
Week 4 – Thursday 21 Oct 2010
1) Start DiffEq Ch.2
2) Mechanics Ch.3 Oscillations
2.1 Modeling via systems – Predator/Prey revisted (p.152)
Equilibrium solution is describes system of homogeneous equations.
If those equations are linear, you could use linear algebra (matrices) to solve them. (Predator
prey system isn’t linear, but coupled – eating and being eaten terms.)
p.157 – Find nontrivial equilibria for Modified predator-prey system. (Do 2.2#12)
Discuss behaviors of populations in phase plots p.158
SHO: mass+spring p.159
Rewrite second-order equation in one unknown (y) as 2 first-order equations in y and v:
often easier to solve.
Even if you can’t solve the equations analytically, you can phase-plot them for insight.
Do problems 2.1 #1-6 p.164 for (ii) Predator-Prey. Do #20 (phase plane for SHO) p.167
Vector Fields follow lines on phase plots (p.171); vector length ~ speed
Equilibrium points (where derivatives vanish) p.177.
Equilibrium curves are asymptotes. P.180
Solution curves cannot cross equilibrium curves by the uniqueness theorem. (Sec.2.4)
2.2 Exercises p.182: Find equl. Points, and sketch some curves with HPGSystemSolvers
p.187: Analytic methods for special systems (thank heavens)
•
•
•
Guess and check! (always check anyway, whatever your method. It’s free.)
Decoupled systems – trivial – solve each equation independently
Partially decoupled – solve the independent one first, then sub it into the remaining
equation (p.191)
p.193: Damped Harmonic Oscillator can be written as a partially decoupled system (y and v)
Euler’s method for systems p.198
(Careful – Y is a vector containing the two equations x(t) and y(t), and F = dY/dt is a vector
containing their two derivatives. Don’t confuse vector capital Y with equation small y)
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As last week, Y1 = Y0 + Δt (dY/dt) = Y0 + Δt F:
Find the new Y1 from the old value Y0 by
adding the slope (rate of change) times the timestep Δt.
Euler’s method is more accurate with smaller timesteps Δt. … continue next week (6)
PIQs for DiffEq:
Sometimes a second order equation can be rewritten as a first order system of equations to
make numeric & qualitative tests easier. (D.E. 161) (AU)
1. The use of a solution curve helps conceptualize a differential equation. The visual of the
phase portrait and equilibrium point are examples used in the RF phase plane on pg. 156 in
Diff. Eq.
2. It is interesting that a harmonic oscillator is expressible as either a 2nd order differential
equation or coupled first order equations. I suppose this means that all coupled equations are
really just higher order equations.
3. The "guess" method we used the last few weeks seems is more powerful than first glance
would suggest since it is essentially how you are able to solve the dampend spring oscillation
problem. (diff eq, p. 194)
1. A direction field presents a better/tidier visualization than a vector field. The only
disadvantage is the loss of information about the speed of the solution due to the smaller size
of the arrows. (Diff. Eq. pg. 173)
2. Only for very specific forms of coupled systems can we find analytical solutions. Isn't that
kinda depressing? (diff eq, p.187) (O7%)
We were surprised to learn that even though the x(t) graphs of the Lorenz systems with
varying initial conditions are unpredictably unique, the three dimensional solution curves of
the Lorenz systems are all similar. (D.E. 210) (AU)
Euler's method for systems gives us numerical approximations for periodic systems that are
not necessarily described by cos, sin and so on. (IPL)
Edward Lorenz didn't just discover that simple differential equations can behave chaotically,
but laid the foundation for chaos theory in his study of atmospheric science. (IPL)
There is a fun example problem dealing with an oscillating skyscraper on page 206 in the
diff-eq book. The example uses Euler's method and deals with a dampened oscillation. The
problem also shows how different phase plane regions can behave differently and through off
your calculations. (UW)
After reading about the P-delta effect, we would like to move our class outside. (AU)
(despite cold or rain?)
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Mechanics Ch2: Oscillations
p.87: Simple Harmonic Oscillator: F = ma → mx = −kx
We derived Tuesday that the displacement x of a mass m on a spring k has solution x = A sin
ωt, with the angular frequency ω = k .
m
1
Recall that ω = 2π f and f = .
T
p.92: Simple pendulum is similar – derive frequency of oscillation for small displacements.
p.85: Powerful general technique: For any potential with a minimum at x0, you can find the
effective spring constant for oscillations about x0 by expanding the potential in a MacLaurin
series about x0 (or a Taylor serries about x0 =0):
U(x) = a0 + a1 x + a2 x2 + a3 x3 + … and then noting that
−dU
F ( x) =
= −2a2 x = −kx therefore 2a2 = k.
dx
p.88: Constants of motion and Initial conditions (IC): do #3.2 p.139
p.89: Discuss SHO as the projection of a rotating vector. Do #3.4 for homework?
p.93: Energy in SHO: Kinetic K and potential U energies slosh back and forth.
Assuming total mechanical energy is conserved (3.3.3): Etot = Kmax = Umax
p.96: 3.4 Damped Harmonic Motion: (Mechanical energy is no longer conserved!)
New differential equation: F = ma → mx + cx + kx = 0 (where damping c ∞ speed) is
c
k
and ω02 = .
x + 2γ x + ω02 x = 0 where γ ≡
often rewritten as 2m
m
p.98: New technique yields solutions of the form
x(t ) = A1e − (γ − q )t + A2 e − (γ + q ) t where q = γ 2 − ω02 . Three kinds of solutions:
I: γ 2 > ω02 : Overdamping (slower decay)
II: γ 2 = ω02 : Critical damping (faster decay)
III: γ 2 < ω02 : Underdamping (oscillation – the interesting case)
p. 102: Quality factor is high if oscillator is close to resonance / weakly damped / losing
little energy (also see p.118-121)
Q=
ω
2π
2πτ
=
= ωdτ = d where τ = time constant for energy loss and Td = oscillation period.
Td
2π
2γ
τ
ωd
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3.5 Phase space – an important and versatile concept. (p.106)
Example: plot speed vs x. SHO has closed trajectories.
Unstable equilibria have open-ended trajectories.
Underdamped Harmonic Oscillator (p.108)
Oscillating, but decaying: Multiply a sinusoidal function times a negative exp function.
See #3.9 p.139. Do 3.11 or 3.12?
3.6: Forced Harmonic Motion: Resonance (p.113)
Including damping and forcing, the full DiffEq is now: mx + cx + kx = F0 cos ωt
* two frequencies:
- ω0 = natural frequency of unforced system
- ω = driving frequency
* system will, in general,
- act crazy when driving first starts (“transients”), then
- settle down into a “steady state”
* at the driving freq, when it is low
* at the natural freq, when the driving freq is high (p.114)
* if ω= ω0, then RESONANCE: system can be driven to high amplitudes
Work through details of Example 3.6, pp.113-115? (Neglecting damping)
p.116: Damped, Driven harmonic Oscillator (here, eiωt notation really comes in handy)
These oscillators can be made chaotic. PHET DEMO
They are the most general form, and can be tricky to solve.
Electrical-Mechanical Analogs p.123 (see Table 3.6.1 p.124)
Inductance L ~ Mass: inertia (ZL = iωL)
Capacitance C ~ 1/spring constant: restoring force (ZC = 1/iωC
Resistance R: Rate of energy dissipation (P=I2R) ~ damping constant v2
Energy and Equilibrium: Equil: F=0.
KE: Integrate dv/dx
(Do # 3.24 p.141?)
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PIQs for Mechanics:
Points: & Insights
If we can classify the oscillator as a simple harmonic oscillator, then we can easily apply a
known set of equations to find out things like period, frequency, and various constants. (IPL)
A constant force shifts the equilibrium position of a harmonic oscillator. On page 91 of A.M.
the authors point out that the equation describing the motion, mẍ+kx=0, is the same with a
constant force (gravity for example) as without, we just measure the displacement of x from a
different point. A periodic force however, can cause a change in amplitude when the
frequency of the force is close to the frequency of the oscillator. On page 114 of A.M. the
authors write that the differential equation of motion with a periodic force is given by
mẍ+kx=F0 cosωt. (AU)
The situation described in example 3.5.1. (A.M. 107) is analogous to a ball balanced on the
tip of an upside down cone. Technically if the ball is exactly balanced on the tip it will stay
there indefinitely, but if the ball moves off of the tip of the cone even slightly, it will
experience a force (gravity) that moves it away from the equilibrium point. (Automatic
Unicorns)
Questions:
1. What is the principle of super positions? (A.M.)
2. What does the damping factor lambda represent? What is damping? (A.M. pg. 97)
3. What does the lower case s with two dots represent? ms(dot dot)=-mgsin§ (A.M. pg. 92,
ex 3.2.2) (O7%)
How do you tell whether something is over damped or under damped? (IPL)
What is the difference between non linear drag and kaotic motion? (RR)
In the general solution for harmonic motion (A.M. 90) we understand how the authors do the
following:
x(t)=Asin(ω0 t+ϕ0 )=Asinϕ0 cosω0 t+Acosϕ0 sinω0 t,
But which identities do the authors use to arrive at the next equation?: (AU)
x(t)=Ccosω 0 t+Dsinω 0 t
with tanϕ 0 =C/D, A²=C²+D²
When Analyzing damped harmonic motion we encounter the differential equation
a+2γv+ω02x=0. The authors outline a method on page 98 of A.M. that allows us to solve this
equation by using D = d/dx as an operator, and then factoring the left hand side of the
equation. The authors write that "[b]ecause the order of operation is arbitrary, the general
solution is a sum of solutions obtained by setting the result of each first-order operation on x
equal to zero" and go on to solve the equation. How does the order of operations being
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arbitrary imply that the general solution is a sum of solutions? Have we learned any similar
techniques in D.E.? (AU)
On page 103 of A.M. the authors write that the quality factor Q is "2π times the inverse of the
ratios given in the expression above." How do they justify this? (AU)
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