Steady motions of a continuous medium, resonances and Lagrangian turbulence

A method which enables one to establish a non-regularity property of the motion of fluid particles (known as chaotic advection or Lagrangian turbulence) for typical steady flows is developed. The method is based on expanding solutions of the equations of motion of a continuous medium in powers of a small parameter and using the conditions for the destruction of invariant resonant tori when perturbations are added. It is shown that the velocity field, defined as the solution of the Burgers equations, generates a generally non-regular dynamical system. For an ideal barotropic fluid in an irrotational force field, the method proposed yields a well-known necessary condition for chaotization: the velocity field is collinear with its curl. Special attention is given to investigating the chaotization of typical steady flows of a heat-conducting perfect gas.     

The theory of problems in the calculus of variations whose Lagrangian function involves second order derivatives: A new approach

Summary Variational principles whose Lagrangian functions involve higher order derivatives have, in the past, been applied to certain aspects of the theory of elementary particles. The corresponding Lagrangian functions must satisfy certain conditions if consistency with the classical electromagnetic interaction terms is sought, and it is found that these conditions are closely related to the requirement that the action integral be invariant under a parameter transformation. If, however, the latter condition is accepted, the usual expression for the Hamiltonian function vanishes identically, resulting in a complete break-down of the canonical equations. Thus an alternative approach to the theory of parameter-invariant problems in the calculus of variations whose Lagrangians depend on second order derivatives is developed. A general Finsler metric is introduced in a natural manner, which provides a geometrical background to the theory as well as useful analytical techniques. It is possible to define an alternative Hamiltonian function corresponding to which a canonical formalism is developed. The method of equivalent integrals is generalised, giving rise to a new and rigorous derivation of theEuler-Lagrange equations, which in turn leads to a generalisation of the so-called excess-function and the analogue of the well-known condition of Weierstrass in the calculus of variations. To Enrico Bompiani on his scientific Jubilee.     

Integrable hydrodynamic submodels with a linear velocity field

Invariant and partially invariant solutions to the equations of gas dynamics with a linear velocity field are defined by a matrix satisfying a homogeneous integrable Riccati equation. The classification is carried out of solutions by the acceleration vector in the Lagrangian coordinates. Some example is given of an invariant solution for which the selected volume “collapses” to an interval.     

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated.     

The dynamics of systems with large gyroscopic forces and the realization of constraints

Lagrangian systems with a large multiplier N on the gyroscopic terms are considered. Simplified equations of motion of general form with holonomic constraints are obtained in the first approximation with respect to the small parameter ɛ = 1/N. The structure of the solutions of the precessional equations is examined.     

Mathews' Identity for Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Some aspects of the invariant theory of a prehomogeneous vector space of Heisenberg parabolic type are studied. In particular, it is shown that a classical identity given by George Ballard Mathews for the space of binary cubic forms has a natural explanation in terms of the Bruhat decomposition associated with the parabolic subgroup and consequently admits a generalization to all prehomogeneous vector spaces of this type. The results are expected to play a role in the definition of an analogue of the Kelvin transform for certain conformally invariant systems of differential equations that have previously been associated with these spaces.     

Non-Abelian tensor gauge fields

A recently proposed extension of Yang-Mills theory contains non-Abelian tensor gauge fields. The Lagrangian has quadratic kinetic terms, as well as cubic and quartic terms describing nonlinear interaction of tensor gauge fields with the dimensionless coupling constant. We analyze the particle content of non-Abelian tensor gauge fields. In four-dimensional space-time the rank-2 gauge field describes propagating modes of helicity 2 and 0. We introduce interaction of the non-Abelian tensor gauge field with fermions and demonstrate that the free equation of motion for the spinor-vector field correctly describes the propagation of massless modes of helicity 3/2. We have found a new metric-independent gauge invariant density which is a four-dimensional analog of the Chern-Simons density. The Lagrangian augmented by this Chern-Simons-like invariant describes the massive Yang-Mills boson, providing a gauge invariant mass gap for a four-dimensional gauge field theory.     

On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler–Poincaré theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler–Poincaré theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin–Noether theorem given in Sect. 4 of Holm et al. (Adv. Math. 137:1–81, 1998).     

On wave solutions of gauge-invariant generalization of field theories with asymmetric fundamental tensor in a generalized peres space-time

Summary The gauge invariant generalization of field theories with asymmetric fundamental tensor developed by Buchdahl has been considered and its plane wave-like solutions in the sense of Takeno are investigated in generalized Peres space-time, recently considered by the author. It has been shown that under certain conditions these solutions become identical with those of strong field equations of Einstein in the same space-time. It has been also shown that this space-time satisfying the field equations of Buchdahl admits a parallel null vector field and is gravitationally null which further, transforms to other well known forms of space-time under a new time coordinate Z=z-t. Entrata in Redazione il 2 afosto 1976. Work is supported by State Council of Science and Technology (U.P.), India.     

Lie group analysis of the axisymmetric flow

The non-linear equations of motion describing the incompressible axisymmetric flow in a flexible and extensible circular cylindrical tube is considered. By employing Lie theory, the full one-parameter infinitesimal transformation group leaving the equations of motion invariant is derived along with its associated Lie algebra. Subgroups of the full group are then used to obtain a reduction by one in the number of independent variable in the system. These reductions are continued until a system of ordinary differential equations is reached. A series type approximate solution of these ordinary differential equations is obtained which leads to a series type approximate solutions in R²{0} to momentum equations.     

A non-conventional discontinuous Lagrangian for viscous flow

Although many attempts for finding a variational formulation of Navier-Stokes equations have been made, a Lagrangian for viscous flow has not been established yet. An auspicious suggestion was made by Scholle [1] by extending Seliger and Whitham's Lagrangian [2] with additional terms ending up with some partial success: on the one hand, the phenomenon ‘viscosity’ occurs in a qualitatively correct manner, on the other hand the equations of motion resulting from the variation of Hamilton's principle differ from Navier-Stokes equations and therefore, their solutions reveal noticeable quantitative differences to those of Navier-Stokes equations. In this paper the Lagrangian [1] is modified by applying an innovative idea by Anthony [3], motivated by the reformulation of the Lagrangian in terms of complex fields, which can also be understood as the inversion of Madelungs idea [4] of reformulating the complex Schrödinger's equation into a hydrodynamic form. The prize one has to pay is that the resulting Lagrangian is discontinuous and therefore the mathematical treatment of the related variational problem challenging. Furthermore, an additional parameter, ω₀, has to be introduced. However, it is demonstrated that Navier-Stokes equations are recovered by the limit ω₀ → ∞, whereas the case of finite ω₀ can be interpreted as a generalization towards non-equilibrium thermodynamics [3]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

KAM theory in configuration space

A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser. This research has been supported by the Nuffields Foundation under grant SCI/180/173/G and by the Stiftung Volkswagenwerk.     

On complicated models of continuous media in the general theory of relativity

In the context of the general theory of relativity, the system of Euler's equations is obtained from the variational equation under the assumption that the Lagrangian of the material depends on supplementary (as compared with classical theories) thermodynamic parameters, and when possible irreversible processes are taken into account. It is shown that, for a thermodynamically closed system, the equations of momenta for a continuous medium are a consequence of the field equations. The form of the energymomentum tensor of the material is considered when the arguments include the Lagrangian of the derivatives of the supplementary thermodynamic parameters.     

Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Variational and Hamiltonian models in the dynamics of magnetizable conducting fluid

The variational model and the Hamiltonian canonical equation of motion are updated using the Lagrangian invariant for three-dimensional unsteady adiabatic flows of magnetizable, ideally conducting, compressible inviscid fluid. The results are applied to derive Hamiltonian noncanonical equations of motion in physically defined variables. Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 44–46, 1994.     

A connection-theoretic approach to reduction of second-order dynamical systems with symmetry

We deal with reduction of Lagrangian systems that are invariant under the action of the symmetry group. Unlike the bulk of the literature we do not rely on methods coming from the calculus of variations. Our method is based on the geometrical analysis of regular Lagrangian systems, where solutions of the Euler-Lagrange equations are interpreted as integral curves of the associated second-order differential equation field. In particular, we explain so-called Lagrange-Poincaré reduction [1] and Routh reduction [3] from the viewpoint of that vector field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dual form of Noether's theorem with applications to continuum mechanics

Considering simultaneously the equations of motion of the physical system and of the non-physical adjoint system, we introduce a general form of Noether's theorem by constructing a “dual Lagrangian” functional with a corresponding invariant of motion which preserves its value along the trajectories of combined physical and unphysical systems. The statement of invariance of this functional reduces to the classical statement of Noether's theorem if the system is self-adjoint; some possible generalizations are indicated. Applications to continuum mechanics are discussed within the framework of Noble's dual variational formulation.