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This problem is very similar to what is known in the physics literature (including in real atom-optics experiments) as the "wedge billiard."

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Formulas for linear stability of billiards with potentials (such as gravity) are given in Holger Dullin's paper here:

Holger R Dullin, "Linear stability in billiards with potential." Nonlinearity, Volume 11, Number 1, 11, 151. 1998.

Abstract: A general formula for the linearized Poincaré map of a billiard with a potential is derived. ...
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This problem is very similar to what is known in the physics literature (including in real atom-optics experiments) as the "wedge billiard."

Edit

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Linear

Formulas for linear stability of billiards with potentials (such as gravity) may be analysed according to are given in Holger Dullin's paper here
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This problem is very similar to what is known in the physics literature (including in real atom-optics experiments) as the "wedge billiard."

Edit: Some more comments -

Linear stability of billiards with potentials (such as gravity) may be analysed according to Holger Dullin's paper here
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