State of the general dynamic beam equation, rail in this case, can be expressed by the

Euler-Lagrange equation:

The device is a fully actuated mechanical system described by the following

Euler-Lagrange equation [32]:

This identification seems natural by realizing that both effective fields give the same

Euler-Lagrange equation in the BPS limit.

Here, [W.sub.[PHI]](S) will be called Willmore energy with a potential [PHI] and in Section 3 the first variation formula and

Euler-Lagrange equation associated with [W.sub.[PHI]] are computed.

The purpose is to maximize the objective functional (1) on conditions of (2) and (3) by finding such function of c that delivers the wanted maximum of the profit formation; and for the general view integral of (4) there are the necessary conditions for the extremum existence in the view of the well-known

Euler-Lagrange equation [25]:

Euler-Lagrange equation with first order constraint [[PHI].sub.j] ([q.sub.i], t) = 0; j = 1,...,m,where [[lambda].sub.i] is indefinite multiplier; [q.sub.i] is generalized coordinate of mechanism; [q.sub.i] is the first derivative of the generalized coordinates and the angular velocity of the links mechanism.

Using the calculus of the variations, the

Euler-Lagrange equation for this energy functional is obtained as follows [19] (the details of solution are shown in Appendix A):

These calculations are based on

Euler-Lagrange Equation (3.1), and forces applied for control of force/moments acting on crane.

Suppose that the equation [[DELTA].sub.n][y] = 0 is the

Euler-Lagrange equation of some variational problem corresponding to a Lagrangian L.

Certainly, each of these observing views a particle in P to be an independent particle, which enables us to establish the dynamic equation (1.2) by

Euler-Lagrange equation (2.3) for [P.sub.i], 1 [less than or equal to] i [less than or equal to] m, respectively, and then we can apply the system of differential equations

The

Euler-Lagrange equation of the bienergy is given by [T.sub.2]([phi]) = 0 .

Since the Lagrangian contains no cross-terms in the kinetic energy combining [??] and [??], the

Euler-Lagrange equation partitions into two parts.