  An application of variational principle to bifurcation of periodic solution
in Lagrangian mechanics is shown. A few higher derivatives of the action
integral at a periodic solution reveals the behaviour of the action in function
space near the solution. Then the variational principle gives a method to find
bifurcations from the solution. The second derivative (Hessian) of the action
has an important role. At a bifurcation point, an eigenvalue of Hessian tends
to zero. Inversely, if an eigenvalue tends to zero, the zero point is a
bifurcation point. The third and higher derivatives of the action determine the
properties of the bifurcation and bifurcated solution.
