Lagrangian Description of the Variational Equations

A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all the virtual displacements joining any two integral curves. Our main result establishes that both the Euler-Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the first prolongation of the original LagrangianAfter discussing certain features of the formulation, we introduce the so-called inherited constants of the motion and relate them to the Noether constants of the extended system

The initial fate of an anisotropic JBD universe

The dynamical effects on the scale factors due to the scalar $\phi$-field at the early stages of a supposedly anisotropic Universe expansion in the scalar-tensor cosmology of Jordan-Brans and Dicke is studied. This universe shows an {\sl isotropic} evolution and, depending on the value of the theorie's coupling parameter $\omega$, it can begin from a singularity if $\omega>0$ and after expanding shrink to another one; or, if $\omega <0$ and $-3/2< \omega\leq -4/3$, it can evolve from a flat spatially-infinite state to a non extended singularity; or, if $-4/3 < \omega < 0$, evolve from an extended singularity to a non singular state and, at last, proceed towards a singularity

Jacobi equations using a variational principle

A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jacobi equations as well as in analysing the stability of the systems and can be related to the vertical extension of the Lagrangian formalism. To exemplify two of such aspects, we uncover a constant of motion in the Jacobi equations of autonomous systems and we recover the well-known sufficient conditions of stability of two dimensional orbits in classical mechanics.Comment: 7 pages, no figure

Picture of the low-dimensional structure in chaotic dripping faucets

Chaotic dynamics of the dripping faucet was investigated both experimentally and theoretically. We measured continuous change in drop position and velocity using a high-speed camera. Continuous trajectories of a low-dimensional chaotic attractor were reconstructed from these data, which was not previously obtained but predicted in our fluid dynamic simulation. From the simulation, we further obtained an approximate potential function with only two variables, the drop mass and its position of the center of mass. The potential landscape helps one to understand intuitively how the dripping dynamics can exhibit low-dimensional chaos.Comment: 8 pages, 3 figure