This work has been motivated by application of DBI Lagrangians, and their dynamics, in cosmology and the theory of inflation. The focus is on the study of the dynamics of a scalar tachyonic field with a non-standard Lagrangian of a DBI type. For some relevant analyses, it is suitable to use lower-dimensional models, including a zero-dimensional classical mechanical analogue. Original calculations for several specific and important (tachyon) potentials are presented. These potentials are also exactly solvable in the framework of Friedmann cosmology, and they have physical motivation in inflationary cosmology. For the purpose of quantization of the (tachyonic) dynamical models, the so-called locally-equivalent Lagrangians of the standard type are considered. As one example, the classical and quantum formalism for an inverted harmonic oscillator with time-dependent frequency is given in more details.