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Abstract. To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third or fourth order ordinary differential equations.

In this paper, we define a complete lift for semisprays. If S is a semispray on a manifold M, its complete lift is a new semispray Scon TM. The motivation for this lift is two-fold: First, geodesics for Sccorrespond to the Jacobi fields for S, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When S is a spray, we prove that the projective geometry of Scuniquely determines S. We also study how symmetries and constants of motions for S lift into symmetries and constants of motions for Sc.

We study the wave equation on a bounded domain M in $\mathbb{R}^m$ or on a compact Riemannian manifold with boundary. Assume that we do not know the coefficients of the wave equation but are given only the hyperbolic Robin-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. In this paper we show that at a fixed time $t_0$ a wave can be cut off outside a suitable set. That is, if $N\subset M$ is a union of balls in the travel time metric having centers at the boundary, then we can modify a given Robin boundary value of a wave such that at time $t_0$ the modified wave is arbitrarily close to the original wave inside N and arbitrarily small outside N. Also, at time $t_0$ the time derivative of the modified wave is arbitrarily small in all of M. We apply this result to construct a sequence of Robin boundary values so that at a time $t_0$ the corresponding waves converge to a delta distribution $\delta_{\widehat{x}}$ while the time derivative of the waves converge to zero. Such boundary values are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point $\widehat{x}$ where the wave focuses is known in travel time coordinates, and $\widehat{x}$ satisfies a certain geometrical condition

Abstract-In the first part of this work we show that, by working in Fourier space, the Bohren decomposition and the Helmholtz's decomposition can be combined into one decomposition. This yields a completely mathematical decomposition, which decomposes an arbitrary vector field on R 3 into three components. A key property of the decomposition is that it commutes both with the curl operator and with the time derivative. We can therefore apply this decomposition to Maxwell's equations without assuming anything about the media. As a result, we show that Maxwell's equations split into three completely uncoupled sets of equations. Further, when a medium is introduced, these decomposed Maxwell's equations either remain uncoupled, or become coupled depending on the complexity of the medium.In the second part of this work, we give a short introduction to contact geometry and then study its relation to electromagnetism. By studying examples, we show that the decomposed fields in the decomposed Maxwell's equations always seem to induce contact structures. For instance, for a plane wave, the decomposed fields are the right and left hand circulary polarized components, and each of these induce their own contact structure. Moreover, we show that each contact structure induces its own Carnot-Carathéodory metric, and the path traversed by the circulary polarized waves seem to coincide with the geodesics of these metrics.This article is an abridged version of the author's master's thesis written under the instruction of Doctor Kirsi Peltonen and under the supervision of Professor Erkki Somersalo.