Analytic mechanics is a mathematical science, but it is of high importance
for engineers as it provides analytic solutions to fundamental problems of
engineering mechanics. At the same time it establishes generally applicable
procedures. Mathematical physics texts, such as  and , laid the founda-
tion for these analytic approaches addressing physical problems.
In the following sections we ﬁnd analytic solutions for classical mechanical
problems of elasticity utilizing Hamilton’s principle. The most ﬁtting applica-
tion is the excitation of an elastic system by displacing it from its equilibrium
position. In this case the system will vibrate with a frequency characteristic
to its geometry and material, while constantly exchanging kinetic and poten-
The case of non-conservative systems, where energy loss may occur due to
dissipation of the energy, will not be discussed. Hamilton’s principle may be
extended to non-conservative systems, but the added diﬃculties do not en-
hance the discussion of the variational aspects, which is our main focus.
11.1 Elastic string vibrations
We now consider the vibrations of an elastic string. Let us assume that the
equilibrium position of the string is along the x axis, and the endpoints are
located at x =0andx = L. We will stretch the string (since it is elastic) by
displacing it from its equilibrium with some
value, resulting in a certain force F exertedonbothendpointstoholditin
place. We assume there is no damping and the string will vibrate indeﬁnitely
if displaced, i.e., the system is conservative.
A particle of the string located at the coordinate value x at the time t has
a yet unknown displacement value of y(x, t). The boundary conditions are: