Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

[Phys-L] equal area + angular momentum + conservation +- constancy

Before we get started, remember that we are always better off
getting rid of cross products and replacing them with wedge
products. In particular, angular momentum should be thought
of as the bivector

L = r ∧ p [1]

and it can be visualized as an oriented /area/. For a
Keplerian orbit, the L bivector lies in the plane of the
orbit. The connection to Kepler's equal-area law is
direct and profound.

On 11/30/2012 04:29 PM, Bernard Cleyet quoted Pari Spolter:

It is shown that equal areas are swept in equal intervals of time
only near ....

In a multi-planet system:

-- On a planet-by-planet basis, angular momentum is /conserved/ but
it is not constant (since it can be transfered to other objects in
the system).

-- The angular momentum of the system as a whole is conserved and
also constant (since it is a closed system).

To the extent that Kepler's 400-year-old wording in terms of equal areas
expresses planet-by-planet constancy rather than conservation, there is
not the slightest reason to expect that it would hold exactly.

On the other hand, if you /sum/ the areas, you get the total angular
momentum, which is constant as well as conserved. For more about the
distinction between conservation and constancy, see

Approximately every student on earth has misconceptions about this.
It's a simple distinction, but nobody was born knowing it.

Angular momentum is a vector perpendicular to the plane formed by v
and r and is conserved, indicating that there is no torque in the
direction vertical to the plane of the orbits.

That's not exactly right either.

For one thing, for real planets, the orbits don't all lie in one plane.

Secondly, the law of conservation of angular momentum expresses a whole
lot more than the absence of some narrow class of stray torques.

Also, remember: We are much better off thinking of torques and angular
momenta as bivectors. For one thing, cross products are not well defined
except in three dimensions, whereas wedge products work nicely in any
dimension from two on up. Furthermore, cross products involve a right-
hand rule, and it seems silly to involve any such thing in a system where
the fundamental physics is left/right symmetric.

Help stamp out cross products.