This paper extends the world-volume action for Dp-branes to the case of N coincident Dp-branes, where the world-volume theory involves a U(N) gauge theory. The guiding principle is that the action should be consistent with T-duality. The resulting action includes nonderivative interactions for nonabelian scalar fields and shows that Dp-branes naturally couple to all RR potentials, including those with form degrees larger than $ p+1 $. The dynamics of Dp-branes in nontrivial background fields are considered, and it is shown that external fields can polarize the branes. A simple example demonstrates that a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, interpreted as a spherical D2-D0 bound state. The paper also discusses the consistency of the action with T-duality, showing that it requires additional terms involving commutators of nonabelian scalars. The Born-Infeld and Chern-Simons actions are extended to nonabelian cases, and the resulting actions are shown to be consistent with T-duality. The paper also compares the results with those from matrix theory and discusses the implications for the dynamics of Dp-branes in nontrivial backgrounds.This paper extends the world-volume action for Dp-branes to the case of N coincident Dp-branes, where the world-volume theory involves a U(N) gauge theory. The guiding principle is that the action should be consistent with T-duality. The resulting action includes nonderivative interactions for nonabelian scalar fields and shows that Dp-branes naturally couple to all RR potentials, including those with form degrees larger than $ p+1 $. The dynamics of Dp-branes in nontrivial background fields are considered, and it is shown that external fields can polarize the branes. A simple example demonstrates that a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, interpreted as a spherical D2-D0 bound state. The paper also discusses the consistency of the action with T-duality, showing that it requires additional terms involving commutators of nonabelian scalars. The Born-Infeld and Chern-Simons actions are extended to nonabelian cases, and the resulting actions are shown to be consistent with T-duality. The paper also compares the results with those from matrix theory and discusses the implications for the dynamics of Dp-branes in nontrivial backgrounds.