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[Phys-L] LRL vector == another Kepler symmetry + Check Your Work

Hi --

As everybody knows, the Kepler problem conserves angular momentum and
conserves energy. The same is true of any other problem with a central
force that is the gradient of some potential.

What is rather less well known is that the Kepler problem has another
symmetry, a hidden four-dimensional symmetry. This leads to conservation
of the Laplace–Runge–Lenz vector.

In case you're wondering whether this is useful, it is. I recently got
yet another object lesson in the importance of the "Check Your Work"
principle. In my Kepler spreadsheet, I decided to calculate the velocity.
I decided to uphold the Check Your Work principle by calculating the LRL
vector ... and lo and behold, it wasn't constant. To make a long story
short, I had dyslexified one of the equations, writing cos instead of
sin and sin instead of cos. Even after I knew there was a bug, I had to
stare at the equation for a while before I figured out what was wrong.

Note that since the LRL vector is a vector, it is two or three times
more sensitive than the energy as a check on the calculations. In
particular, for a zero-eccentricity orbit, the energy check would not
have caught the aforementioned error.

In addition to the LRL check and the energy check, you can also do an
approximate check by approximating the velocity via finite differences:
Δr/Δt. That is not a super-accurate check on the numerical methods,
if you're trying to calculate the velocity to machine precision, but
it's plenty good enough for catching typos.

The main lesson is that it's good practice to study the problem long
enough to figure out a set of incisive checks, and actually run the

As a related point: Set a good example for the students. Pride
goeth before a fall. If you pretend to be so smart you can get
the right answer without doing the checks, (a) you're going to
make a mistake sooner or later, and (b) "even" if you don't get
caught, and maybe /especially/ if you don't get caught, you are
modeling bad behavior in front of the students.

Also related: Rather than just preaching about the principle of
the thing, it helps to show students concrete examples of what
it looks like in practice. Even if they don't understand the LRL
vector check, they should understand the kinetic energy / binding
energy check.

A lesson for students is to /leave the checks in the spreadsheet/
so that they can re-check the work later, and so that others can
re-check it. Resist the temptation to make the spreadsheet "smaller"
or "more elegant". Hide the columns if necessary, but don't delete

Such checks cannot solve all the world's problems: As Dykstra
was fond of pointing out, testing can demonstrate the presence
of bugs, but it can never prove the absence of bugs.

OTOH testing is a whole lot better than no testing. Typesetting a
web page and then plugging the equations into a spreadsheet is a
serious check. It increases the reliability by at least an order
of magnitude.


I upgraded my web document to include a section that collects most of
the relevant equations in one place, using a consistent set of symbols:

All of the equations are findable in the references, but you would
need to look through three or four references to find everything you
need, and you would need to wade through a lot of stuff you don't
need ... and they don't all use the same notation.


Wildly tangential remark: For those who are interested in the
history of such things: I never cease to be amazed by the date on
Kepler's equation: 1609. A transcendental equation in 1609! By
way of contrast, Galileo was working at about the same time, and
was not exactly the village idiot, but there is AFAICT no evidence
that Galileo ever wrote an equals sign in his life. Most people
who were masters of the mathematics of the day relied on geometrical
proofs rather than algebra and equations.

The "=" sign was invented in 1557 and was not fully established
until circa 1700.

The telescope was invented in 1608/1609 and played no role in Tycho
Brahe's observations or Kepler's analyses. These were amazingly
tough and determined guys.