a b s t r a c tWe investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable t _ xÀ; _ ðxxÞ-, and € ðxxÞ-dependence besides with the usual t-, x-, and _x-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.

Dual actions with respect to U(1) gauge fields for the Born–Infeld and Dp-brane theories are reexamined. Taking into account an additional condition, i.e. a corollary to the field equation of the auxiliary metric, one obtains an alternative dual action that does not involve the infinite series in the auxiliary metric given by [M. Abou Zeid and C.M. Hull, Phys. Lett. B 428 (1998) 277], but just picks out the first term from the series formally. New effective interactions of the theories are revealed. That is, the new dual action gives rise to an effective interaction in terms of one interaction term rather than infinitely many terms of different (higher) orders of interactions physically. However, the price paid for eliminating the infinite series is that the new action is not quadratic but highly nonlinear in the Hodge dual of a (p - 1)-form field strength. This non-linearity is inevitable under the requirement that the two dual actions are equivalent.