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When you put a cold drink on the kitchen counter
the counter surface temperature will decrease.
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But, if the cold drink is removed, the counter
will eventually return to room temperature.
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If instead we place a cup of tea on the counter,
the counter temperature rises; but if we remove
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the cup of tea, the counter top eventually
returns to room temperature. We say that the
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counter at room temperature is a stable equilibrium.
In this video, we discuss the world from the
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perspective of equilibrium and stability,
and in particular linear stability.
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This video is part of the Linearity video
series. Many complex systems are modeled or
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approximated as linear because of the mathematical
advantages.
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All the world is an initial value problem,
and the matter merely state variables. However,
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and less poetically, there are alternative
interpretations of physical, and indeed social,
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systems that can prove very enlightening.
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The purpose of this video is to introduce
you to the framework of equilibrium and stability
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analysis. We hope this motivates you to study
the topic in greater depth.
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To appreciate the material you should be familiar
with elementary mechanics, ordinary differential
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equations, and eigenproblems.
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Let's look at the iced drink and hot teacup
example from the perspective of equilibrium
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and stability.
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In this example, the governing partial differential
equation is the heat equation shown here.
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At equilibrium, the solution of the governing
equations is time-independent, that is, the
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partial time derivative is zero.
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This tells us that the del square temperature
term must also be zero, which is only possible,
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given the boundary conditions, if the entire
counter is at room temperature.
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Stability refers to the behavior of the system
when perturbed from a particular equilibrium
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near the uniform counter temperature; a stable
system returns to the equilibrium state; an
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unstable system departs from the equilibrium
state.
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We say that this equilibrium is stable because
if we perturb the temperature by increasing
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or decreasing the temperature slightly, it
will return to room temperature, the equilibrium
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state, after enough time.
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Here we intuitively understand that the equilibrium
is stable from our own experiences. But for
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other situations, we need mathematical methods
for determining whether or not equilibria
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are stable.
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One way to determine if the equilibrium of
this partial differential equation is stable
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is to apply an energy argument.
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Manipulation of this heat equation permits
us to derive a relationship that describes
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how the "mean square departure" of the counter
temperature from room temperature evolves
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over time, as shown here.
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Note u(x) is the deviation of the temperature
in the counter from room temperature, Ω is
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the counter region, and Γ is the counter
surface; d1 and d2 are positive constants—determined
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by the thermal properties of the counter.
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Because the right--hand side of this equation
is negative, it drives the temperature fluctuations—the
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integral of u(x) squared over the counter
region—to zero.
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This energy argument is the mathematical prediction
of the behavior we observe physically.
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Linear stability theory refers to the case
in which we limit our attention to initially
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small perturbations. This allows us to model
the evolution with linear equation! Linearizing
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the governing equations has many mathematical
advantages.
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Let's consider the following framework for
linear stability analysis.
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choose a physical system of interest;
develop a (typically nonlinear) mathematical
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model;
identify equilibria;
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linearize the governing equations about these
equilibria;
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convert the initial value problem to an eigenproblem;
inspect the eigenvalues and associated eigenmodes
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(or eigenvectors)
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Let's see how to use this framework as we
proceed through the example of the real, physical
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pendulum seen here.
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You see a large bob, the rod, and a flexural
hinge, which is designed to reduce friction
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losses.
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Let's develop the mathematical model.
We show here the simple pendulum consisting
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of a bob connected to a massless rod; we denote
the (angular) position of the bob by theta(t),
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and the angular velocity of the bob by ω(t);
g is the acceleration due to gravity; and
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L is the effective rod length for our simple
pendulum.
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The effective length L is chosen such that
the simple pendulum replicates the dynamics
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of the real, physical pendulum; L is calculated
from the center of mass, the moment of inertia,
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and the mass of the compound pendulum.
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The dynamics may be expressed as a coupled
system of ordinary differential equations
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that describe how the angular displacement
and angular velocity evolve over time.
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These equations are nonlinear due to the presence
of sin(θ) and the drag function fdrag(ω).
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The drag function is quite complicated.
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For large ω—it is equal to c|omega|omega,
where c is a negative constant.
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But for very small angular velocities, near
points on the trajectory where the pendulum
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isn't moving, or at least not moving fast,
drag is given by to b omega, where b is a
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negative constant.
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Next, let's explore the validity of this model.
Here you see a comparison between a numerical
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simulation, and an experiment, courtesy of
Drs. Yano and Penn, respectively.
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The numerical simulation is created by calibrating
the drag function to the experimental data.
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The agreement is quite good for both small
and large initial displacement angles.
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However, because we are fitting the dissipation
to the data, this comparison does not truly
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validate the mathematical model.
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To validate the mathematical model, we must
focus on a system property that is largely
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independent of the here, small, dissipation,
such as the natural frequency, or period,
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of the pendulum motion.
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And a comparison of the natural frequency
of the physical pendulum to that predicted
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by the numerical model shows that the natural
frequencies agree quite well, for any initial
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displacement angle, even large.
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We now look for equilibrium states, solutions
that are independent of time. To find these
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equilibria we set the left--hand side of the
equations to zero and solve for θ and ω.
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We can readily conclude that there are two
equilibria:
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(θ, ω) = (0, 0), which we denote the "bottom"equilibrium;
and
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(θ, ω) = (π, 0), which we denote the "top"equilibrium.
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The mathematical model actually has infinitely
many equilibria corresponding to theta values
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that are integral multiples of pi.
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But for our stability analysis, two will suffice.
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Are these equilibrium solutions stable or
unstable for the physical pendulum? Pause
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the video here and discuss.
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They are stable if a small nudge will result
in commensurately small bob motion; They are
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unstable, if a small nudge will result in
incommensurately large bob motion. So we can
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predict that the bottom equilibrium is stable;
the top equilibrium, unstable. If our mathematical
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model is good, it should predict the same
behavior as the physical system.
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But how do we mathematically analyze stability
of our model? Let's work through the linear
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stability analysis framework for the bottom
equilibrium, θ = 0 and ω = 0. First, we
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linearize the equations about the equilibrium.
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The linearized equations are only valid near
the equilibrium, theta = 0 and omega =0, i.e.
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for small displacements, theta prime, with
small angular velocities, omega prime.
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The first equation of our system is already
linear.
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So we only need to worry about linearizing
the sin(0+θ') term and the dissipation term
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of the second equation for small theta prime
and omega prime.
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What is the linear approximation of sin(theta')
about theta=0. Hint: use a Taylor series.
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Pause the video and write down your answer.
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If we assume that theta prime is small, we
can approximate sin(θ') by theta' — the
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deviation of θ from 0— by ignoring the
higher order terms in the Taylor series expansion
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of sin theta' about 0.
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Because we are linearizing near omega prime
sufficiently close to zero, the dissipation
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term is asymptotic to b omega prime, as mentioned
previously. We then substitute these expressions
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into our dynamical equations to obtain the,
linear equations indicated.
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We must supply initial conditions, and the
initial angle and angular velocity must be
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small in order for this linearized system
of equations to be applicable. We now write
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the linear equations in matrix form, in order
to prepare for the next step — formulation
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as an eigenproblem.
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To do this, we assume temporal behavior of
the form eλt. This yields an eigenvalue problem
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for λ.
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Note that our matrix is 2×2 and hence there
will be two eigenvalues and two associated
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eigenvectors, or eigenmodes.
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Once we obtain the eigenvalues and eigenvectors
we may reconstruct the solution to the linearized
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equations, as shown here; the constants c1
and c2 are determined by the initial conditions.
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However, to determine stability, a simple
inspection of the eigenvalues suffices:
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What happens to e to the lambda t when each
of the eigenvalues has negative real part?
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Pause the video.
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We observe that eλt decays in this case,
and the system is stable.
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What happens if any of the eigenvalues has
positive real part? Pause the video.
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If an eigenvalue has positive real part — in
which case eλt corresponds to exponential
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growth away from the equilibrium
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— the system is unstable; note that even
one eigenvalue with a positive real part is
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sufficient to deem the system unstable, since
sooner or later, no matter how small initially,
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the growing exponential term will dominate.
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Lastly, if the real part of the eigenvalue
with largest real part is zero — in which
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case eλt is of constant magnitude — the
system is marginally stable and requires further
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analysis. We now proceed for our particular
system.
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For this problem it is easy to find the eigenvalues
analytically.
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We observe that both λ1 and λ2 have a small
negative real part - due to our negative damping
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coefficient, b.
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Thus, the system is stable as we predicted
based on the physical pendulum.
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For our experimental system, b L over g is
much less than 1.
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And thus the damping does indeed have little
effect on the period of motion. If we recall
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the connection between the complex exponential
and sine and cosine, we may conclude that
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the linearized system response is a very slowly
decaying oscillation. This linear approximation
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to our governing equations can predict not
just the stability of the system, but also
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because the system is stable, the evolution
of the system—assuming of course small initial
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conditions
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Indeed, comparing our original non-linear
numerical solution to the numerical solution
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obtained from the linear model, we find that
the agreement between the two is very good
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for the low-amplitude case.
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As advertised, we obtain a solution to the
linear model with which we can predict the
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motion resulting from small perturbations
away from equilibrium. But as you see here,
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the accuracy of the linear theory is indeed
limited to small perturbations
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— for even moderately larger angles, the
linear model no longer adequately represents
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the behavior of the pendulum.
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But this is to be expected since the linear
approximation of these governing equations
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is only valid when theta prime and omega prime
are very close to zero.
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To Review
We have thus linearized the governing equations
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for the pendulum near the equilibrium theta
= 0, omega = 0.
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By solving an eigenvalue problem, we showed
that the equilibrium was stable at this point.
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The linear equations even predict the behavior
of the pendulum near this stable equilibrium.
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This video has focused primarily on the modal
approach to stability analysis, which is the
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simplest and arguably the most relevant approach
in many applications. However, there are other
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important approaches too, such as the "energy"approach,
briefly mentioned at the outset of the video.
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We shall leave to you to analyze the case
of the "top"equilibrium, which we predicted
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to be unstable.
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You'll find that there's an unstable mode
AND a stable mode. The unstable mode will
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ultimately dominate, but the stable mode is
still surprising. Our intuition suggests than
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any perturbation should grow. The stable mode
requires a precise specification of both initial
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angular displacement AND initial angular velocity.
This explains the apparent contradiction between
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the mathematics and our expectations.