Conservative Fieldgreenspun.com : LUSENET : quaternions : One Thread
what is conservative field?
-- Waqas Ch. (firstname.lastname@example.org), June 12, 2003
A concervative force field has four equivalent definitions.
1. The curl is zero. With quaternions, the curl is isolated from other derivatives by using a commutator. If box === (d/dt, Del), and [A, B] === (A B - B A)/2, then for a force F will be conservative if:
[Box, F] = 0.
2. There is a scalar potential function for a force F. This works because Del X (Del phi) = 0. Take the derivative of a M/R potential, and Newton's law results.
3. The line integral of any closed loop is zero.
4. The line integral along different paths is the same.
If one holds, the rest do.
-- Doug Sweetser (email@example.com), June 23, 2003.