The finite element technique is applied to functional which govern dynamical problems, where time is an independent variable. The present paper demonstrates improved accuracy in mass-spring-damper systems and exemplifies "rendevous problems" of a "travelling" particle in a medium. The motion is governed by Hamilton's principle. The time interval is fixed. Functionals are constructed from Hamilton's extended principle and appropriate conditions stating the various constraints. Initial value problems may be incorporated by writing a functional in accordance with Gurtin's method. Shape functions are polynomials in time and can be extended to other spatial variables when present. As a result of a variation to the functional, a system of algebraic (not necessarily linear) equations is formed. This system is solved simultaneously to yield its motion within the boundaries of the given time interval.