Lagrangian Description of the Variational Equations
نویسندگان
چکیده
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. Typeset using REVTEX Corresponding author
منابع مشابه
A Novel Variational Method for Deriving Lagrangian and Hamiltonian Models of Inductor-Capacitor Circuits
We study the dynamical equations of nonlinear inductor-capacitor circuits. We present a novel Lagrangian description of the dynamics and provide a variational interpretation, which is based on the maximum principle of optimal control theory. This gives rise to an alternative method for deriving the dynamic equations. We show how this generalized Lagrangian description is related to generalized ...
متن کاملA Canonical Transformation Relating the Lagrangian and Eulerian Description of Ideal Hydrodynamics
In fluid mechanics, two ways exist to specify the fields. The one most often used is the Eulerian description, in which the fields are considered as functions of the position in space, and time. The Lagrangian description, on the other hand, is based on the observation that many quantities specifying the fluid refer more fundamental ly to small identifiable pieces of matter, the "fluid particle...
متن کاملLagrangian multiforms and multidimensional consistency
We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for such systems in terms of Lagrangian multiforms. We discuss the connection of this formalism with the notion of multidimensional consistency, and the role of ...
متن کاملLagrangian Navier-stokes Diffusions on Manifolds: Variational Principle and Stability
We prove a variational principle for stochastic flows on manifolds. It extends V. Arnold’s description of Lagrangian Euler flows, which are geodesics for the L2 metric on the manifold, to the stochastic case. Here we obtain stochastic Lagrangian flows with mean velocity (drift) satisfying the NavierStokes equations. We study the stability properties of such trajectories as well as the evolution...
متن کاملAn Introduction to the Generalized Lagrangian-mean Description of Wave, Mean-flow Interaction'
The generalized Lagrangian-mean description is motivated a n d illustrated by means of some simple examples of interactions between waves and mean flows, confining attention for the most part to waves of infinitesimal amplitude. The direct manner in which the theoretical description leads to the wave-action concept and related results, and also to the various 'noninteraction' theorems, more acc...
متن کامل