  The weak form of the SDOF and MDOF equations of motion are obtained. The
original initial conditions problem is transformed into a boundary value
problem. The boundary value problem is then solved and transformed back to the
initial conditions one. Subsequently, a general method for obtaining numerical
methods using an arbitrary number of linearly independent approximating
functions is outlined.
  This is part one of a series of three papers, in the second of which a
numerical method is obtained, using Bernstein polynomials of arbitrarily high
order. The numerical evidence for the convergence of the method will be
presented in the third part paper.
