PORTUGALIAE MATHEMATICA Vol. 61 Fasc. 3 – 2004 Nova S´ erie CRUISING IN A CENTRAL FORCE FIELD Stanislav Furta and Gaetano Zampieri Recommended by Carlos Rocha Abstract: We study a particle in a central force field which has a cruise motion, namely which is constrained to keep a constant kinetic energy. It is an integrable dynam- ics. We describe the global geometry of the problem by introducing special variables and a new time. This permits us to prove some general facts such as the existence and the orbital stability of circular motions. As an application a Bertrand-like problem is solved. Moreover, some noteworthy potential functions are dealt with as the Newton gravity of a single celestial body. 1 – Introduction A natural Lagrangian function and the nonholonomic constraint to keep the kinetic energy constant, define a natural thermostatted system, see [Z2] and [DM]. These system have nonlinear constraints on the velocities (see [Z1] Section 4, [GZ], and the references therein). Our main aim is to study the thermostatted motion of a (test) particle in a central force field. We ask the kinetic energy, so the speed, of the particle to be constant, something which seems related to “cruise controls”, at least loosely. In this case of a single particle, we prefer to speak of cruise motion then of thermostatted motion. Received : March 26, 2003; Revised : May 23, 2003. AMS Subject Classification : 37J60, 70F25, 70H06. Keywords and Phrases : nonholonomic dynamics; central force fields. * Supported by the GNFM of the “Istituto Nazionale di Alta Matematica”.