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
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
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  (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.