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PHY354H1
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Erich Poppitz
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Lecture

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Physics

PHY354H1

Erich Poppitz

Winter

Description

Lecture 21: Energy Phase Diagrams, and En-
ergy Conservation in 3D
▯ relevant text sections: Morin 5.3
The view from phase space
▯ For a conservative one dimensional system, the energy
1
E = mv + V (x) (1)
2
is conserved following the motion. Notice that E is written as a function
of v and x. We can imagine plotting isosurfaces of E as functions of v
and x in the format of a contour plot. In this plot, the trajectories of
particles will go along surfaces of constant energy. Let’s consider three
examples
1. A spring with V = 1kx has energy
2
E = 1mv + kx1 2 (2)
2 2
and depcribes an ellipse for each value of E > 0. The points
x = ▯ 2E=(km) are the turning points where v = 0. We might
have expected this behaviour, since we already determined that
sinusoidal motion corresponds to ellipses in phase space. To plot
this, we use the routine energyho.py, which uses the function
meshgrid to create a grid of values of x and v:
1 The top panel shows the potential energy V (x), and the bot-
tom panel shows the total energy E(x;v). The oscillation occurs
around the energy minimum, proceeding along energy contours in
a clockwise direction.
2. For projectile motion we have
1 2
E = mv + mgz;
2
where v = z_. This is the equation for a family of parabolas lying
on their side. The code is found in energy projectile.py and
the plot is
2 Here, there is no energy minimum or oscillatory motion, so the
trajectories just correspond to the rising and falling motion of
vertically projected particles. There is a single turning point for
each trajectory, corresponding to the top of the trajectory where
z = E=mg.
3. Finally, consider the simple pendulum (WITHOUT THE SMALL
ANGLE APPROXIMATION) with energy
1 2_2
E = 2ml ▯ + mgl(1 ▯ cos▯) (3)
In this case, the potential energy is periodic in ▯. Here, we plot
_
the energy in ▯ ▯▯ phase space: the code is energy pendulum.py
and the plot is
3 We plot E=(mgl) in the bottom panel. For E < 2mgl, the energy
of the oscillator is not su▯cient for the pendulum to roll, i.e. to get
‘over the top’. The pendulum motion is vibrational and bounded
by the turning points ▯ such that cos▯ = 1▯E=(mgl) and therefore
_
▯ = 0. For E > 2mgl we have rolling motion. A pendulum started
from ▯ = 0 with E > 2mgl will \escape" its local potential well
and undergo rolling motion. When ▯ = 0, E = 1mv . This
2 p
implies that v needs to be greater than the escape velocity 2 gl
for rolling motion to occur. What would the escape velocity be
for a pendulum started at ▯ = ▯=4?
The boundary between the two types of motion is E = 2mgl or
E=(mgl) = 2: this is known as the separatrix. Notice how the
\▯" crossings of the separatrix occurs at the local maxima of the
potential energy | these are the unstable positions ▯ = ▯▯ of the
4 pendulum.
Energy Conservation in 3D
▯ Last time when we considered 1D motion, we looked at forces that
only depended on position: F = F(x) and showed that these forces are
conservative and hence can be written in terms of a potential energy:
dV (x)
F(x) = ▯
dx
▯ We are now going to consider what happens in 3D. Similar to the 1D
case, the only type of force that has any chance of being conservative
is one that depends on position only (i.e. doesn’t depend explicitly on
velocity or time). So we will consider forces F = F(~r) where r is the
position vector:
r = xx^ + y^+ zz^
▯ It turns out that in 3D, its not enough that the force only depend on
position to ensure the force is conservative. Another condition must
be met which we will determine next time. For now we will assume we
are dealing with a conservative force.
Deriving the Energy Equation for 3d Motion
▯ If I consider Newton’s law in 3d: F = m~ v and keep in mind that it is
just shorthand for 3 scalar equations (one for the x direction, one for
the y direction, one for the z direction), then I can write the 3 equations
as:
dvx(x;y;z)
F xx;y;z) = m dt (4)

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