64 research outputs found
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 -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 , it can begin from a singularity if and
after expanding shrink to another one; or, if and , it can evolve from a flat spatially-infinite state to a non extended
singularity; or, if , 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
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