An identity is derived which yields a correspondence between symmetries and conservation laws for self-adjoint differential equations. This identity does not rely on use of a Lagrangian as needed to obtain conservation laws by Noether’s theorem. Moreover, unlike Noether’s theorem, which can only generate conservation laws from local symmetries, the derived identity generates conservation laws f...

The aim of this paper is to present a new approach construct constants motion associated with scaling symmetries dynamical systems. Scaling maps could be the equations but not its Lagrangian action. We have constructed Noether inspired theorem in vertical extended space that can used obtain for these symmetries. obtained as particular case our construction. To illustrate how procedure works, we...

In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian same time correlation tensors are equal. This statement allows us to transform the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between the local homogeneity of Frisch and global homogeneity. It is sho...

In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian same time correlation tensors are equal. This statement allows us to transform the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between the local homogeneity of Frisch and global homogeneity. It is sho...

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multidimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statisti...

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistica...

Recent advances in geometric mechanics, motivated in large part by applications in control theory, have introduced new tools for understanding and utilizing the structure present in mechanical systems. In particular, the use of geometric methods for analyzing Lagrangian systems with both symmetries and non-integrable (or nonholonomic) constraints has led to a uniied formulation of the dynamics ...