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I am studying this article on orbital mechanics and can't understand the meaning of some terms related to mathematics (perhaps because my native language doesn't have such terms). I created the same thread on math.stackexchange.com and there I was recommended to duplicate this question here.

1.

The user constructs a targeting scheme [in the program] consisting of independent targeting variables (components of the impulses) and dependent variables (targeting goals). Generally a scheme is constructed such that the number of independent variables equals the number of dependent variables to provide the differential corrector [DC] with a 'square' problem to solve.

'square' problem – here does the author mean a system of equations where the number of unknown variables is equal to the number of equations in the system?

2.

Swingby [the programm] propagates trajectories numerically with full operations-level force modeling invoked.

I don't have any ideas.

3.

In his development of the third-order solution to the equations of motion for LPOs, Richardson shows...

I don't understand the phrase "third-order solution". The author is talking about "solution of a third-order equations"?

4.

In summary, given the four parameters of the desired Lissajous, the third-order theory yields the state coordinates of the desired target insertion point.

Is the author here talking about a theory based on equations (probably differential) of the third order?

5.

The angle phi is measured in the RLP [a reference frame] xy-plane clockwise from the -x-axis, and the angle psi is the phase with respect to the z-axis.

I don't have any ideas.

6.

However the departure burn and the Z-burn are differentially corrected simultaneously, making for a 3-by-3 differential correction problem.

3-by-3 problem – could it be a system of three equations with three unknown variables?

7.

... position calculated by the third order approximation.

This is a phraseology (like the phrase "zeroth-order approximation" for a very inaccurate solution) or reference to third order polynomium approximation?

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Some of this appears to be more colloquial than strict math. Here are my guesses

'square problem' - just means N inputs and N outputs requires a square matrix for transformation

'operations-level forcemodeling' -- crank up all the computers and software on hand

'third-order equations /theory' - most likely taking the analytic equations and using the first three terms in a Taylor or Fourier series expansion.

'phase with respect to Z-axis' -- since you mention Lissajous figures earlier, this probably means the periodic motion along the z-axis described as $z = sin(\omega t + \psi)$

'three by three' - you are correct- another square matrix of size 3,3 .

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