in this paper we study isotropic lagrangian submanifolds , in complex space forms . it is shown that they are either totally geodesic or minimal in the complex projective space , if . when , they are either totally geodesic or minimal in . we also give a classification of semi-parallel lagrangian h-umbilical submanifolds.

This paper suggests an inverse optimal proportional–integral–derivative (PID) control design method to track trajectories in Lagrangian systems. The inverse optimal PID controller exists if and only if the Lagrangian system is extended disturbance input-to-state stable. First, we find the Lyapunov function and the control law that satisfy the extended disturbance input-to-state stability by usi...

We describe a class of mechanical systems for which the “method of controlled Lagrangians” provides a family of control laws that stabilize an unstable (relative) equilibrium. The controlled Lagrangian approach involves making modifications to the Lagrangian for the uncontrolled system such that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the close...

We describe a class of mechanical systems for which the “method of controlled Lagrangians” provides a family of control laws that stabilize an unstable (relative) equilibrium. The controlled Lagrangian approach involves making modifications to the Lagrangian for the uncontrolled system such that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the close...

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of...

Recently, calculations which consider the implications of anomalous trilinear gauge-boson couplings, both at tree-level and in loop-induced processes, have been criticized on the grounds that the lagrangians employed are not SU L (2) × U Y (1)gauge invariant. We prove that, in fact, the general Lorentz-invariant and U (1) em invariant but not SU L (2) × U Y (1) invariant action is equivalent to...

We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa–Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa– Holm system in Lagrangian variable...

We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler–Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in...

We consider minimizers for the natural time-dependent Lagrangian system in R with Lagrangian L(x, v, t) = |v|/β − U(x, t), where β > 1. For minimizers on a time interval of length T with one endpoint fixed, we prove that the absolute values of velocities are bounded by K log T , provided that the potential U(x, t) and its gradient are uniformly bounded. We also show that the above estimate is a...

In this note, I briefly revisit the possibility of a physical 12-dimensional F-theory, in view of a recent joint investigation with H. Sati [1, 2, 3] on expansion from type II string theory to F-theory. In particular, I shall discuss Lagrangians and signatures, in particular a new possible phenomenological scenario of compactificaton from dimension (9, 3) to (3, 1).