Enrico Pagani

Inhomogeneous waves in viscoelastic media

Abstract Inhomogeneous, plane, monochromatic waves travelling in viscoelastic media are considered. Through a description in terms of complex potentials, detailed expressions of phase speed and attenuation are derived by having recourse to thermodynamic restrictions and to the properties of the complex propagation vector under inversion of the frequency. The complex amplitude vector for transverse and longitudinal waves is also discussed. Next, reflection and refraction at the common boundary of different types of media are investigated. As an application, the problem of a longitudinal wave propagating within an inviscid fluid and entering a viscoelastic solid is analyzed numerically. In particular it is shown that an analysis of the dependence of the reflected wave on the frequency leads to the determination of the relaxation time for an exponential-type relaxation function.

Is the Riemann tensor derivable from a tensor potential

In recent years, following an earlier result of C. Lanczos concerning the representation of the Weyl tensor in arbitrary space-times, it has been conjectured that the Riemann tensor itself admits a linear representation in terms of the covariant derivatives of a suitable “potential” tensor of rank 3. This conjecture is shown to be false, at least for a class of spacetime geometries including several physically significant ones.

Legendre transformation and analytical mechanics: A geometric approach

A revisitation of the Legendre transformation in the context of affine principal bundles is presented. The argument, merged with the gauge-theoretical considerations developed by Massa et al., provides a unified representation of Lagrangian and Hamiltonian mechanics, extending to arbitrary nonautonomous systems the symplectic approach of Tulczyjew.

On the problem of stability for higher-order derivative Lagrangian systems

The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an example, that the indefiniteness of the energy does not forbid the stability.

Surface waves on a solid half‐space

Inhomogeneous waves in viscoelastic solids are considered with the aim of a thorough characterization of surface waves in a solid half‐space. To ascertain the existence of surface waves and to investigate their properties, a general scheme is established that is appropriate for numerical developments. Viscoelastic and elastic solids are examined in detail and previous results on admissible surface waves are generalized.

Geometric constrained variational calculus I: Piecewise smooth extremals

A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagins equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagins algorithm in hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.

Geometric constrained variational calculus. II: The second variation (Part I)

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

Geometric constrained variational calculus. III: The second variation (Part II)

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.