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We develop a class of C1-continuous time integration methods that areapplicable to conservative problems in elastodynamics. These methods are basedon Hamilton's law of varying action. From the action of the continuous systemwe derive a spatially and temporally weak form of the governing equilibriumequations. This expression is first discretized in space, considering standardfinite elements. The resulting system is then discretized in time,approximating the displacement by piecewise cubic Hermite shape functions.Within the time domain we thus achieve C1-continuity for the displacement fieldand C0-continuity for the velocity field. From the discrete virtual action wefinally construct a class of one-step schemes. These methods are examined bothanalytically and numerically. Here, we study both linear and nonlinear systemsas well as inherently continuous and discrete structures. In the numericalexamples we focus on one-dimensional applications. The provided theory,however, is general and valid also for problems in 2D or 3D. We show that themost favorable candidate -- denoted as p2-scheme -- converges with order four.Thus, especially if high accuracy of the numerical solution is required, thisscheme can be more efficient than methods of lower order. It further exhibits,for linear simple problems, properties similar to variational integrators, suchas symplecticity. While it remains to be investigated whether symplecticityholds for arbitrary systems, all our numerical results show an excellentlong-term energy behavior.

C1-continuous space-time discretization based on Hamilton's law of varying action