I show that the general implicit-function problem (or parametrized fixedpoint problem) in one complex variable has an explicit series solution given by a trivial generalization of the Lagrange inversion formula. I give versions of this formula for both analytic functions and formal power series.

This paper deals with twisting motion of a falling cat robot by two torque inputs around her waist. The cat robot consists of two rigid columns and has internally two actuators at the joint to generate torque inputs around the normal coordinates. This system is a nonholonomic system whose angular momentum is conserved. W e obtain the Lagrange’s equation of motion with nonholonomic constraint an...

can be found in closed form by means of certain transformations on generating functions and the extraction of coefficients (see the end of section 3) suggested by Ira Gessel, and they comment: “He [Gessel] attributes this elegant technique, the ‘method of coefficients,’ to G. P. Egorychev.” Actually, in his book Integral Representation and Computation of Combinatorial Sums, Egorychev [4] (see [...

The infinity Laplace equation ∆∞u = 0 arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the L∞ variational problem of minimizing the functional ess-supU |Du|. The more general functional ess-supUF (x, u, Du) leads similarly to the so-called Aronsson equation AF [u] = 0. In this paper we show that these PDE operators and various interesting generalization...

We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange problem of the calculus of variations on time scales).

In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers, we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity, and dynamic equation of the robot manipulator. The state equation (...