The Euler-Lagrange method in Pontryagin’s formulation

The classical variational problem with nonholonomic constraints is solvable by the Euler-Lagrange method in Pontryagin’s formulation; however, in this case Lagrange multipliers are merely measurable functions. In this paper, we put forward a modified Euler-Lagrange method, in which the original problem involves a Lagrangian dependent only on the independent components of the velocity vector. Under this approach, the Lagrange multipliers make up an absolutely continuous vector function. Our method is applied to the problem of horizontal geodesics for a nonholonomic distribution on a manifold. These equations are established as having two types of connections: connection on the distribution and connection on the manifold; this was not accounted for by other researchers.     

Jacobi fields for a nonholonomic distribution

The variational problem with nonholonomic constraints was considered in detail by Bliss. A distribution is a special case of constraints. Horizontal geodesics on a manifold with flat metric and constant tensor of nonholonomity are considered. It is proved that, in the classical adjoint problem, conjugate points appear, which does not involve any loss of optimality. The second variation of the length (or energy) functional of admissible (horizontal) geodesics for a distribution on a smooth manifold is expressed in terms of the distribution curvature tensor.     

The arrival time brachistochrones in general relativity

We consider a general relativistic version of the classical brachistochrone problem, whose solutions are causal curves, parameterized by a constant multiple of their proper time and with 4-acceleration perpendicular to a given observer field, that extremize the arrival time measured by an observer at the final endpoint. This kind of brachistochrones presents characteristics different from the travel time brachistochrones, that were studied in [8, 9, 10]. In this article we formulate the variational problem in a general context; moreover, in the case of a stationary metric, we prove two variational principles and we determine the second order differential equation satisfied by the arrival time brachistochrone. Using these variational principles and techniques from Critical Point Theory we establish some results concerning the existence and the multiplicity of travel time brachistochrones with a given energy between an event and an observer.     

The index growth and multiplicity of closed geodesics

In the recent paper [31] of Long and Duan (2009), we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in [31] of Long and Duan (2009) on rational, and in [12] of Duan and Long (2007), [39] of Rademacher (2010), and [40] of Rademacher (2008) on completely non-degenerate closed geodesics on spheres and CP² to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.     

Entire parabolic trajectories as minimal phase transitions

For the class of anisotropic Kepler problems in $\mathbb{R }^d\setminus \{0\}$ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full characterization of such a codimension-one manifold of potentials and we show how to parameterize it with respect to the degree of homogeneity.     

A Support Theorem for the Geodesic Ray Transform on Functions

Let (M,g) be a simple Riemannian manifold. Under the assumption that the metric g is real-analytic, it is shown that if the geodesic ray transform of a function f∈L ²(M) vanishes on an appropriate open set of geodesics, then f=0 on the set of points lying on these geodesics. The approach is based on analytic microlocal analysis.     

Nonholonomous geodesics as solutions to Euler-Lagrange integral equations and the differential of the exponential mapping

Equations of horizontal geodesics on a Riemannian (or pseudo-Riemannian) manifold with nonholonomous distribution are obtained using the Euler-Lagrange method in Pontryagin’s formulation. It is shown that if the distribution and the metric tensor of the distribution are C ^k -smooth, k ≥ 1, then any regular solution to the variational problem is C ^{k + 1}-smooth. The differential of the exponential mapping is obtained for nonholonomous distribution with the condition of cyclicity with respect to “vertical” coordinates. This differential is nonsingular provided that the distribution is strongly bracket generating.     

Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.     

Korpelevich’s method for variational inequality problems on Hadamard manifolds

The concept of pseudomonotone vector field on Hadamard manifold is introduced. A variant of Korpelevich??s method for solving the variational inequality problem is extended from Euclidean spaces to constant curvature Hadamard manifolds. Under a pseudomonotone assumption on the underlying vector field, we prove that the sequence generated by the method converges to a solution of variational inequality, whenever it exists. Moreover, we give an example to show the effectiveness of our method.     

Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.     

Fundamental group and contractible closed geodesics

We prove the existence of a nonempty class of finitely presented groups with the following property: If the fundamental group of a compact Riemannian manifold M belongs to this class, then there exists a constant c(M) > 1 such that for any sufficiently large x the number of contractible closed geodesics on M of length not exceeding x is greater than c(M)^x. In order to prove this result, we give a lower bound for the number of contractible closed geodesics of length ≤ x on a compact Riemannian manifold M in terms of the resource-bounded Kolmogorov complexity of the word problem for π₁ (M), thus answering a question posed by Gromov. © 1996 John Wiley & Sons, Inc.     

On Topological Index of Solutions for Variational Inequalities on Riemannian Manifolds

In this paper, based on the fixed point index theory for a class of -multivalued maps on absolute neighbourhood retracts, we introduce the notion of index of solvability for a variational inequality on a Riemannian manifold involving a multivalued vector field. We describe the main properties of this topological characteristic and use it to justify the existence of a solution for a variational inequality problem. As application, the problem of optimization of a non-smooth functional on a Hadamard manifold is considered.     

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of M we assume that M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for     

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold—viewed as a parabolic boundary value problem for infinite cylinders—to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined by counting, modulo time shift, heat flow trajectories between geodesics of Morse index difference one. By Salamon and Weber (GAFA 16:1050–138, 2006) this heat flow homology is naturally isomorphic to Floer homology of the cotangent bundle for Hamiltonians given by kinetic plus potential energy.     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature and the structure of generalized solutions for the constant mean curvature equation

As is well known, no manifold of constant mean curvature 1 can exist in a domain strictly containing the unit ball. In part 1 of this paper, we shall consider the problem of estimating absolute apriori bounds for a manifold of constant mean curvature 1 in a ball of radius less than 1 without imposing boundary conditions or bounds of any sort in the higher dimensional case. It is also well-known that we may reduce the capillary problem in the absence of gravity to the variational problem for the functional indicated in the beginning of 2.6.2. In case that a generalized solution for exists, we consider the sets P and N where this generalized solution takes the value and respectively. Using the method in part 1, we shall, in part 2, try to characterize the geometry of the boundaries of P and N in the interior of the domain and ask whether they are spherical caps or not. Received May 27, 1997 / Accepted October 3, 1997     

Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.     

Existence and multiplicity of normal geodesics in Lorentzian manifolds

In this paper we shall prove some results pertaining to the existence and multiplicity of normal geodesics joining two given submanifolds of an orthogonal splitting Lorentzian manifold. To this aim, we look for critical points of an unbounded suitable functional by using a Saddle-Point Theorem and the relative category theory.     

Plane Curves and Geodesic Diffeomorphisms

In a pseudo-Riemannian manifold we can define anr-plane curve as a curve with vanishingr-th curvature. We show that every diffeomorphism that carriesr-plane curves intor-plane curves (for a fixedr) is a geodesic diffeomorphism, i.e. carries geodesics into geodesics.     

Homogeneous geodesics in solvable Lie groups

After the second author and J. Szenthe [10] proved that every homogeneous Riemannian manifold admits a homogeneous geodesic, several authors studied the set of all homogeneous geodesics in various homogeneous spaces. In this paper, we consider special examples of homogeneous spaces of solvable type of arbitrary odd dimension given in [1] and [7] and we show that their sets of homogeneous geodesics have an interesting structure, closely connected to the notion of Hadamard matrices.