In high-dynamic low earth orbit (LEO) satellite communication (SATCOM) systems, frequent channel state information (CSI) acquisition consumes a large number of pilots, which is intolerable in resource-limited SATCOM systems. To tackle this problem, we propose to track the state-dependent parameters including Doppler shift and channel angles, by exploiting the physical and approximate on-orbit mobility characteristics for LEO satellite and ground users (GUs), respectively. As a prerequisite for tracking, we formulate the state evolution models for kinematic (state) parameters of both satellite and GUs, along with the measurement models that describe the relationship between the state-dependent parameters and states. Then the rough estimation of state-dependent parameters is initially conducted, which is used as the measurement results in the subsequent state tracking. Concurrently, the measurement error covariance is predicted based on the formulated Cram$\acute{\text{e}}$r-Rao lower bound (CRLB). Finally, with the extended Kalman filter (EKF)-based state tracking as the bridge, the Doppler shift and channel angles can be further updated and the CSI can also be acquired. Simulation results show that compared to the rough estimation methods, the proposed joint parameter and channel tracking (JPCT) algorithm performs much better in the estimation of state-dependent parameters. Moreover, as to the CSI acquisition, the proposed algorithm can utilize a shorter pilot sequence than benchmark methods under a given estimation accuracy.