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The formation of a tree leaf

Qinglan Xia (2007)

ESAIM: Control, Optimisation and Calculus of Variations

In this article, we build a mathematical model to understand the formation of a tree leaf. Our model is based on the idea that a leaf tends to maximize internal efficiency by developing an efficient transport system for transporting water and nutrients. The meaning of “the efficient transport system” may vary as the type of the tree leave varies. In this article, we will demonstrate that tree leaves have different shapes and venation patterns mainly because they have adopted different efficient...

The H–1-norm of tubular neighbourhoods of curves

Yves van Gennip, Mark A. Peletier (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We study the H–1-norm of the function 1 on tubular neighbourhoods of curves in ℝ 2 . We take the limit of small thicknessε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε3), the ends (ε4), and the curvature (ε5). The second result is a Γ-convergence result, in which the central curve may vary along the sequence ε → 0. We prove that a rescaled version of the...

The H–1-norm of tubular neighbourhoods of curves

Yves van Gennip, Mark A. Peletier (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We study the H–1-norm of the function 1 on tubular neighbourhoods of curves in ℝ 2 . We take the limit of small thickness ε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε3), the ends (ε4), and the curvature (ε5). The second result is a Γ-convergence result, in which the central curve may vary along the sequence ε → 0. We prove that a rescaled version of...

The Hamilton-Cartan formalism in the calculus of variations

Hubert Goldschmidt, Shlomo Sternberg (1973)

Annales de l'institut Fourier

We give an exposition of the calculus of variations in several variables. The introduction of a linear differential form studied by Cartan makes possible an invariant treatment of the Hamiltonian formalism. Noether’s theorem, the Hamilton-Jacobi equation and the second variation are discussed and a Poisson bracket is defined.

The interface crack with Coulomb friction between two bonded dissimilar elastic media

Hiromichi Itou, Victor A. Kovtunenko, Atusi Tani (2011)

Applications of Mathematics

We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.

The inverse problem in the calculus of variations: new developments

Thoan Do, Geoff Prince (2021)

Communications in Mathematics

We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for n = 2 . We then examine a new class of solutions in arbitrary dimension n and give some non-trivial examples in dimension 3.

The Lazy Travelling Salesman Problem in ℝ 2

Paz Polak, Gershon Wolansky (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We study a parameter (σ) dependent relaxation of the Travelling Salesman Problem on  ℝ 2 . The relaxed problem is reduced to the Travelling Salesman Problem as σ → 0. For increasing σ it is also an ordered clustering algorithm for a set of points in ℝ 2 . A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided σ is large enough. ...

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