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*From*: John Denker <jsd@av8n.com>*Date*: Sat, 01 Dec 2012 14:38:17 -0700

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

http://www.av8n.com/physics/conservative-flow.htm

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.

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