Narrow Results

In this paper, we consider infinitesimal bending of the second-order of curves and knots. The total curvature of the knot during the second-order infinitesimal bending is discussed and expressions for the first and the second variation of the total curvature are given. Some examples aimed to illustrate infinitesimal bending of knots are shown using figures. Colors are used to illustrate curvature values at different points of bent knots and the total curvature is numerically calculated.

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyze a case where the domain is not compact. More precisely, we shall prove that a large family of proper biharmonic maps $\phi :ℝ\to {𝕊}^2$ is strictly stable with respect to compactly supported variations. In general, the computations involved in this type of problems are very long. For this reason, we shall also define and apply to specific examples a suitable notion of index and nullity with respect to equivariant variations.

We survey results on infinitesimal deformations ("Jacobi fields") of harmonic maps, concentrating on (i) when they are integrable, i.e., arise from genuine deformations, and what this tells us, (ii) their relation with harmonic morphisms — maps which preserve Laplace's equation.

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler–Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence we obtain a simple non-computational proof of the relation between the null subspace of the second variation of the action and the presence of JVFs (and conjugate points) along an extremal. We work in the framework of skew-symmetric algebroids.

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.

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.

We characterize the second variation of an higher order Lagrangian by a Jacobi morphism and by currents strictly related to the geometric structure of the variational problem. We discuss the relation between the Jacobi morphism and the Hessian at an arbitrary order. Furthermore, we prove that a pair of Jacobi fields always generates a (weakly) conserved current. An explicit example is provided for a Yang–Mills theory on a Minkowskian background.

The paper proves second order sufficient conditions for the strong optimality of singular extremals of the first kind. The conditions are given both in Hamiltonian formulation and by means of the coercivity of a suitable coordinate-free second variation. The conditions are close to the necessary ones in the usual sense, namely we require strict inequalities where the necessary conditions have mild inequalities.