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Nexus
Ook
A square cat
A Nexus mindmap
[E. Putterman and O. Raz,
The square cat
, Am. J. Phys.
76
(2008)]
0
Follows
0
Can the cat change it's orientation without
changing it's total angular momentum?
0
1
2
3
1
0
1
2
3
0
0
Location
of masses
1
We know this will be conserved
since no external torques act on
the cat while it falls
3
0
1
2
2
Total angular momentum
2
0
1
0
1
2
Motion
3
A scalar field associates a scalar
with each point in space
0
e.g. temperature
1
Scalar field
0
A vector field associates a vector
with each point in space
0
0
e.g. rotation
1
Vector field
1
Not uniquely defined
0
0
Notation
1
0
"grad"
0
0
"div"
1
0
"curl"
2
0
"Laplacian"
3
Common types
2
Derivatives
2
Fields
4
0
0
Integrating along the two
paths is the same as
integrating along the boundary
0
1
3
i.e. expand each component in a taylor
series expansion of a function of
two
variables. Ignore terms quadratic in a small
quantity.
0
The surface is approximately flat for the
element, if the element is small enough
2
1
0
Calculate the integral for
each segment. First each
line element selects a
component due to the
dot product, then use the
linear expansion for the
component.
0
1
Almost everything
cancels because
we went in a loop
0
The result
looks like the
z
-component
of the
curl
1
2
If we define the area as having a direction we can get
rid of the z-component and write it as a dot-product
0
3
2
Circulation of vector field
0
After combining all the little surface elements
1
This is called Stoke's theorem
0
1
Stoke's theorem
5
0
1
0
0
0
1
0
1
Use Stoke's theorem
1
Now we have the
mathematical
machinery to tackle
the problem of the cat!
4
0
Cat can rotate!!
2
0
1
0
0
1
2
0
Check
3
Solution
6
A square cat