TAB 24

LINEAR ALGEBRA

The equations of motion for the Two Body Problem can be represented as the modal matrix

then

in two dimensions. Since is a conservative field, then

which actually does not work because is conservative only in the plane of the orbit. To see this create a hodograph with two separate circles for the bi-elliptic Earth to Mars trajectory. If were a conservative force then the intersection of the two circles would coincide, but it does not. To see why, consider the Cayley Hamilton theory, which says that a matrix satisfies its own characteristic equation. There is a contradiction that the f and g equations solve the Two Body Problem, but not the f and g matrix. That is, the fact that a matrix satisfies it's own differential equation (the C H Theory) means the matrix is an envelope curve.

Note that the transformation to the symmetric plane is linear. This implies other systems can be approximated by a matrix (e.g. in orbital estimation theory) which is of a linear operation of independent vectors ~ that being the case, this system can be studied using linear programming techniques.

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© 2004 WH Clark