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# n-bodies collision

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My problem does not correspond to well known n-bodies problem, but those are some n bodies. I have n circles represented with radius, velocity and position each. They are moving in different directions, through same area. I want to find minimal (optimal) change of velocities and directions, so that there is no collision during a given time interval. Has anyone seen something like this solved? What technique is fast enough (I have some ideas, but not fast)? Can non-linear programming be used. I wanted to minimize f(da,dv)=sum_by_i(da_i^2+dv_i^2) where da=(da_1,...,da_n) is change of angles, and similar for dv, under conditions dist(X_i(a_i+da_i,v_i+dv_i,t),X_j(a_j+da_j,v_j+dv_j,t))>=r_i+r_j for all t in given interval where a_i are current angles of directions, and v_i are current velocities. Problem is in free parameter t in dist(...), because this is not in specification of the NLP problem. If i took it as an variable i would get solution -da_min,dv_min,t_0, but I want X_i and X_j on given distance all the time not just t_0- X_i and X_j are positions of centers depending on curent angle, velocity, and time (and ofcourse starting position). If i have not been quite clear, but you are interested to help I could try to simplify. Thanks, Mladen

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