# Patching two Lambert's problem solutions together

I am following this paper on the implementation of a genetic algorithm to find a good MGA trajectory.

The way they approach the problem is by solving the Lambert problem for each planetary transfer. Then, they patch the solutions using the patched-conic approximation (section 2.A: "For the multiple gravity-assist model, two Lambert solutions are essentially “patched” together using the standard patched conic method. This results in a powered hyperbolic orbit for each gravity assist in which a ∆V is allowed only at the perigee passage for each gravity assist. The ∆V at each gravity assist is part of the final cost function, but is usually driven to a near zero value.").

As I understand it, for a trajectory Earth-Jupiter with a Mars swing-by, they essentially solve the Earth-Mars and Mars-Jupiter transfers, then patch the two solutions. Which results in some ΔV need at perigee passage (at Mars).

However, given that the Lambert solutions already provide the arrival and the departure velocities required at flyby, isn't the ΔV required the difference between the two velocities?

I think I am not understanding properly the role of the patched-conic at joining two Lambert problem solutions. Can someone clarify it?

You must be very careful & specific as to what arrival and the departure velocities are given by the Lambert solver. Most typical are heliocentric velocities, so you can determine the heliocentric conic section (heliocentric orbit). You then "patch" this conic to the planeto-centric hyperbolic conic section by converting to planeto-centric $$v_{\infty}$$ values, then to periapsis planeto-centric velocities. At that point you can subtract them to find a $$\Delta V$$, though typically this is driven to near zero like the paper says.