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Reeb flows are a rich, ubiquitous class of dynamical systems arising in symplectic geometry, which include billiard systems, many-body orbital systems, geodesic flows and many Hamiltonian flows. Convexity hypotheses play an important, albeit mysterious, role in the study of these flows. In this thesis, we discuss several new results in the study of convexity in symplectic geometry and Reeb dynamics.In Chapter 1, we resolve a longstanding open problem on the intrinsic characterization of Reeb flows arising from Hamiltonian flows on the convex boundaries. Namely, we prove that dynamically convex Reeb flows, introduced by Hofer-Wysocki-Zehnder, are not all convex. Our proof uses a novel relation between Riemannian geometry and Reeb dynamics, and uses constructions of Abbondandolo-Bramham-Hryniewicz-Salomao.In Chapter 2, we describe a powerful new framework for computationally modelling Reeb dynamics on the boundaries of convex polytopes. We apply this framework to provide new evidence and examples relating to the Viterbo conjecture, a major open problem in Reeb dynamics and quantitative symplectic geometry.In Chapter 3, we study convex toric domains and toric surfaces. A longstanding conjecture in toric geometry states that the Gromov width is monotonic under inclusion of moment polytopes of closed toric varieties. We use methods from toric geometry and ECH to prove a generalization of this conjecture in dimension 4.

In this thesis I prove two isomorphisms in the local symplectic homology of a non iterated, isolated Reeb orbit. The isomorphisms are in $S^1$-equivariant and nonequivariant symplectic homology relating the local Floer homology group of the orbit to that of the return map. The $S^1$-equivariant symplectic homology isomorphism can be stated succinctly as the local $S^1$-equivariant symplectic homology of an uniterated isolated Reeb orbit is isomorphic to the local Hamiltonian Floer homology of the return map. An application of this result gives the equivalence of two different definitions of a Reeb orbit being a symplectically degenerate maximum.

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

This work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology. We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields.

In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by $$d_\mathrm{CBM}$$ . Explicitly, we consider the following two set-ups, either on a contact manifold $$W \times S^1$$ where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of $$d_\mathrm{CBM}$$ are different in these two cases. In the former case the inputs are (contact) star-shaped domains of $$W \times S^1$$ which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance $$d_\mathrm{CBM}$$ defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of $$d_\mathrm{CBM}$$ . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

The ride quality of the railway vehicle is not only affected by the wheel-rail contact geometry but also by the yaw damper. In order to explore this variation law, an equivalent parameter model of the yaw damper was established based on the internal characteristics of the yaw damper, which is both accurate and efficient. Then, considering the influence of wheel wear and wheel-rail contact geometry, ride quality of the railway vehicle under different parameters of yaw damper and wheel-rail contact parameters was analysed. The results show that the wheel-rail contact points are scattered on the wheel profile after the wheel wears out, and the equivalent conicity also tends to increase with the increasing operating mileage. The distribution of ride quality space is sensitive to the change of equivalent conicity. In the low equivalent conicity area, the expansion rate of excellent ride quality space is faster. In the high equivalent conicity area, the expansion rate of qualified ride quality space is faster. Appropriate additional stiffness which is oil stiffness in parallel with structural damping in the equivalent parameter model of the yaw damper can improve the vehicle ride quality. The lateral ride quality is influenced obviously with the condition of the damping of the yaw damper being less than 440 kN·s·m−1. Properly reducing the joint stiffness of the yaw damper could reduce the influence of characteristic parameters of the yaw damper and equivalent conicity of the wheel-rail contact on vehicle lateral ride quality. The optimized characteristic parameters of the yaw damper are used in the actual vehicle test, and the ride quality is effectively improved.

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and scattering symplectic gluings between strong convex symplectic fillings of a contact manifold. By giving an explicit computation of the Poisson cohomology of a scattering symplectic manifold, we also introduce a new method of computing Poisson cohomology., Final version accepted for publication in the Pacific Journal of Mathematics (PJM)

In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by Kronheimer and Mrowka. We then compute the sutured monopole Floer homology of some special balanced sutured manifolds, using tools closely related to contact geometry. For application, we obtain an exact triangle for the oriented skein relation in monopole theory and derive a connected sum formula for sutured monopole Floer homology., Comment: 34 pages, 11 figures. Correct typos in the new version. Comments are welcomed!

We solve an optimal control problem for thermodynamic processes in an ideal gas. The thermodynamic state is given by a Legendrian manifold in a contact space. Pontryagin’s maximum principle is used to find an optimal trajectory (thermodynamic process) on this manifold that maximizes the work of the gas. In the case of ideal gases, it is shown that the corresponding Hamiltonian system is completely integrable and its quadrature-based solution is given. : contact geometry, thermodynamics, optimal control, Hamiltonian systems, integrability

The main content of this thesis is two papers. The first paper defines a canonical $\Q$-grading of the embedded contact homology chain complex under certain conditions. ECH is an invariant of 3-manifolds isomorphic to the Seiberg-Witten Floer cohomology as defined by Kronheimer and Mrowka. The underlying chain complex comes with a relative $\Z/p$-grading, or, an absolute $\Z/p$-grading after fixing a reference generator. However, when $p=0$, we show that one can actually specify an absolute grading of the chain complex without having to choose a reference generator. Explicitly, that grading set is a subset of $\Q$ isomorphic to $\Z$. As a consequence, $ECH(Y,\lambda)$ is canonically $\Q$-graded whenever $H_1(Y)$ is torsion. The second paper is an application of ECH capacities to the problem of embedding ellipsoids into irrational ellipsoids. More specifically, we analyze lattice point counting functions to deduce that the existence of an accumulation point of an infinite staircase is impossible for all irrational targets minus a set of irrationals in bijection with $\N$, for which the question has yet to be resolved.

We improve the previous result “A complete Ricci soliton whose metric g is K-contact and the soliton vector field X is strictly contact, is compact Sasakian Einstein” and show that, if a complete Ricci soliton (M, g, X) whose metric g is a contact metric and the soliton vector field X is strictly contact, then X is an infinitesimal automorphism and g is Einstein. Finally, for a Ricci soliton with X as the Reeb vector field, we show that (M, g) is compact Einstein and and Sasakian.

M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

Liouville domains have become central objects in symplectic and contact geometry. However, the auxiliary data they involve --- namely, Liouville forms --- and the non-compactness of their completions generate some inconvenience. The notion of ideal Liouville domains is designed to suppress these awkward aspects and to let symplectic structures play the leading role. Ideal Liouville domains are compact manifolds with boundary whose interior carries a symplectic form satisfying some tameness condition along the boundary. Their definition and their basic properties are presented in the first part of these notes, while the second part discusses their relevance in contact geometry., Comment: 22 pages

This paper studies the instantaneous second-order total freedom of two smooth objects initially in contact at a single point. The analytical determination of the set of physically allowed second-order motions is formulated in the Euclidean space using the screw theoretic concepts of twist and twist-derivative. Mathematical expression that characterizes the nature of second-order motion for an arbitrary contact geometry is derived in terms of twist coordinates, twist-derivative coordinates and the principle normal curvatures of the two surfaces at the point of contact. It is shown that the characteristic of this expression is equivalent to that of the derivative of reciprocal product of the instantaneous twist and contact normal line. The efficacy of the theory developed is demonstrated through two illustrative examples.

A two-parametric geometry optimization problem for a symmetrical punch with compound curvature in frictionless contact with and an elastic half-plane is considered. An almost constant pressure distribution is sought for by means of least-square method. Another sub-optimal solution, which equalizes the three local maximum peaks (one at the center, and two at either side of the contact interval), is constructed using an asymptotic modeling approach and compared with the exact solution known in the literature.

We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of $$ \eta $$ -Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field $$\xi $$ . Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided $$m\ne 1$$ . We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is $$ \eta $$ -Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.

We study a long-recognised but under-appreciated symmetry called "dynamical similarity" and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton's principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology., 1 figure. Version accepted to Synthese. Substantive improvements and clarifications

Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of V&times, V*&times, R. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds of V&times, R by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants.

Given a surface $S$ in a 3D contact sub-Riemannian manifold $M$, we investigate the metric structure induced on $S$ by $M$, in the sense of length spaces. First, we define a coefficient $\widehat K$ at characteristic points that determines locally the characteristic foliation of $S$. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points., 24 pages, 15 figures

Deep brain stimulation (DBS) involves activation of targeted brain tissue through implantable electrodes to treat neurological disorders. In this study, two novel electrode designs, recessed flat-contact and recessed curvature-contact models were developed where the electrode contacts were recessed to a specified depth to improve directional selectivity. Furthermore, the contact geometry was also modified for the recessed curvature-contact model in order to obtain a hemispherical configuration that will help increase current steering and reduce the propensity of tissue damage. The predicted tissue damage produced by these models were compared to the standard array model using the Shannon tissue damage model criteria. Furthermore, the volume of tissue activated by each of the electrode models was analyzed, and the radial projection relative to the total projection of each geometry was determined as a measure of directional selectivity. Based on the trends observed in the current density, tissue damage, and volume of tissue activated (VTA) analyses, it is inferred that the recessed contact electrode geometries help improve directional selectivity and safety of DBS.

This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics., 48 pages

This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $\beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $\beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry., Comment: 15 pages. v2 fixes some attribution issues, updates bibliography and corrects some typos. v3: major strengthening of the main theorem (Theorem 1.1), plus small corrections

Riemannian and contact geometry formalisms are used to study the fundamental equation of electromagnetic radiation-like systems, obeying a Stefan–Boltzmann’s-like law. The vanishing of metric determinant is used for classifying what kind of systems cannot represent a possible generalization of blackbody-like systems. In addition, thermodynamic curvature scalar R is evaluated for a thermodynamic metric, giving R = 0 , which validates the non-interaction hypothesis stating that the scalar curvature vanishes for non-interacting systems.

A metamodel considering material plasticity is presented for computationally efficient prediction of wheel–rail normal contact in railway switches and crossings (S&C). The metamodel is inspired by the contact theory of Hertz, and for a given material, it computes the size of the contact patch and the maximum contact pressure as a function of the normal force and the local curvatures of the bodies in contact. The model is calibrated based on finite element (FE) simulations with an elasto-plastic material model and is demonstrated for rail steel grade R350HT. The error of simplifying the contact geometry is discussed and quantified. For a moderate difference in contact curvature between wheel and rail, the metamodel is able to accurately predict the size of the contact patch and the maximum contact pressure. The accuracy is worse when there is a small difference in contact curvature, where the influence of variation in curvature within the contact patch becomes more significant. However, it is shown that such conditions lead to contact stresses that contribute less to accumulated plastic deformation. The metamodel allows for a vast reduction of computational effort compared to the original FE model as it is given in analytical form.

Embedded contact homology (ECH) capacities were defined by Hutchings and provide a family of obstructions to embeddings of four-dimensional symplectic manifolds. In Part I of this thesis, we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This was joint work with Daniel Cristofaro-Gardiner and Michael Hutchings. In Part II of this thesis, we construct topological absolute gradings in Heegaard Floer homology and bordered Floer homology that satify all of the expected properties. This was joint work with Yang Huang. We also show that the isomorphism between Heegaard Floer homology and ECH preserves the absolute gradings.

Let Pfaffian system ${\omega}$ define an intrinsically nonlinear control system which is invariant under a Lie group of symmetries $G$. Using the contact geometry of Brunovsky normal forms and symmetry reduction, this paper solves the problem of constructing subsystems ${\alpha}\subset{\omega}$ such that ${\alpha}$ defines a static feedback linearizable control system. A method for representing the trajectories of ${\omega}$ from those of ${\alpha}$ using reduction by a distinguished class $G$ of Lie symmetries is described. A control system will often have a number of inequivalent linearizable subsystems depending upon the subgroup structure of $G$. This can be used to obtain a variety of representations of the system trajectories. In particular, if $G$ is solvable, the construction of trajectories can be reduced to quadrature. It is shown that the identification of linearizable subsystems in any given problem can be carried out algorithmically once the explicit Lie algebra of $G$ is known. All the const...

Contact geometry has been successfully employed for the geometric formulation and control of systems containing thermodynamic components. In this paper we elaborate on the geometric theory of symplectization of contact manifolds in order to lift contact control systems to Hamiltonian control systems with a Hamiltonian that is homogeneous in the co-state variables. This provides a new view on contact control systems as used in thermodynamics, and offers possibilities for unifying the theories of contact control systems, Hamiltonian input-output systems and port-Hamiltonian systems.

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and $$(\kappa ,\mu )$$ -contact manifolds. First, we prove that a complete K-contact metric satisfying $$\mathcal {L}^{*}_g(\lambda )=0$$ is Einstein and is isometric to a unit sphere $$S^{2n+1}$$ . Next, we prove that if a non-Sasakian $$(\kappa ,\mu )$$ -contact metric satisfies $$\mathcal {L}^{*}_g(\lambda )=0$$ , then $$ M^{3} $$ is flat, and for $$n > 1$$ , $$M^{2n+1}$$ is locally isometric to the product of a Euclidean space $$E^{n+1}$$ and a sphere $$S^n(4)$$ of constant curvature $$+\,4$$ .

This contribution reports on an ongoing research project aimed in developing a unified theoretical framework for the description of interconnected thermo-mechanical systems with a particular emphasis on thermodynamic engines. We analyse from the geometrical viewpoint the structure of thermodynamic and mechanical interconnection and propose an approach to the unified description of thermo-mechanical systems. The theoretical results are illustrated by a physical example.

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spin c structures induce contact structures with distinct Ozsvath–Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca–Matic–Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks.

Mitsumatsu [9] constructed a leafwise symplectic structure of the Lawson foliation of S5. Following his construction, we improve a previous result of the author [11] on convergence of contact structure to codimension one foliation, and give a sufficient condition for convergence to foliation with leafwise symplectic structure. As an application, we show that the product S4 × S1 admits infinitely many codimension one leafwise symplectic foliations.

In this work we show that a Legendre transformation is nothing but a mere change of contact polarization from the point of view of contact geometry. Then, we construct a set of Riemannian and pseudo-Riemannian metrics on a contact manifold by introducing almost contact and para-contact structures and we analyze their isometries. We show that it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent i.e. the Legendre transformations are the corresponding isometries and, on the other, that it induces a Hessian metric into the corresponding Legendre submanifolds. This second property is motivated by the well known Riemannian structures of the geometric description of thermodynamics which are based on Hessian metrics on the space of equilibrium states and whose properties are related to the fluctuations of the system. We find that to define a Riemannian structure with such properties it is necessary to abandon the idea of an associated metric to an almost contact or para-contact structure. We find that even extending the contact metric structure of the thermodynamic phase space the thermodynamic desiderata cannot be fulfilled., Comment: 25 pages, revised version, accepted in Journal of Physics A: Mathematical and Theoretical

An important question of railway vehicle dynamics is to understand which parameters affect the transformation between sub- and supercritical bifurcation in the stability assessment. Recent studies report effects of the equivalent conicity, the wheelset guidance stiffness and other parameters. Performing parameter variations using a typical bogie model with linear suspension components but considering all nonlinearities of wheel/rail contact geometry, this paper shows that the type of bifurcation depends predominantly on the nonlinearity of the wheel/rail contact geometry, while the effects of the equivalent conicity, the creep forces and the wheelset guidance stiffness are less important.

Let $\mathrm S^3$ be the unit sphere of $\mathbb C^2$ with its standard Cauchy-Riemann (CR) structure. This paper investigates the CR geometry of curves in $\mathrm S^3$ which are transversal to the contact distribution, using the local CR invariants of $\mathrm S^3$. More specifically, the focus is on the CR geometry of transversal knots. Four global invariants of transversal knots are considered: the phase anomaly, the CR spin, the Maslov index, and the CR self-linking number. The interplay between these invariants and the Bennequin number of a knot are discussed. Next, the simplest CR invariant variational problem for generic transversal curves is considered and its closed critical curves are studied., Comment: 44 pages, 7 figures. This is an extended version of a talk presented at the Conference "Geometry, Differential Equations and Analysis" in memory of A. V. Pogorelov for his 100th birthday anniversary, June 17-21, 2019, Kharkiv, Ukraine

We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz' variational principle. First we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle. Then we discuss the backward error analysis of our variational integrators, including the construction of a modified Lagrangian. Surprisingly, this construction presents some interesting simplifications compared to the corresponding construction for symplectic systems. Throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues.

In this chapter, more characteristic curves are derived on the premises of “Plucker conoid”, constructed at a point of a smooth regular part surface. At the beginning main properties of the surface of “Plucker conoid” are briefly outlined. This includes but not limited to basics, analytical representation, and local properties along with auxiliary formulae. This analysis is followed by analytical description of local geometry of a smooth regular part surface. Ultimately, expressions for two more characteristic curves are derived. These newly introduced characteristic curves are referred to as Plucker curvature indicatrix and \( An R(P_1) \)-indicatrix of a part surface. The performed analysis makes it possible derivation of equations for two more planar characteristic curves for analytical description of the contact geometry of two smooth regular part surfaces at a point of their contact. One of the newly derived characteristic curves is referred to as “\( An_{R}(P_1/P_2) \)-relative indicatrix of the first kind” of two contacting part surfaces. Another one in a curve inverse to the characteristic curve \( An_{R}(P_1/P_2) \). This second characteristic curve is referred to as “\( An_{k}(P_1/P_2) \)-relative indicatrix of the second kind”. Main properties of both the characteristic curves are briefly discussed in this section of the monograph.

Feasible kinds of contact of two smooth regular part surfaces in the first order of tangency are discussed in this chapter. This analysis begins with investigation of a possibility of implementation of the indicatrix of conformity for the purposes if identification of actual kind of contact of two smooth regular part surfaces. Then, impact of accuracy of the computation of the parameters of the indicatrix of conformity at point of contact of two parts surfaces is investigated. Ultimately, a classification of all possible kinds of contact of two smooth regular part surfaces in the first order of tangency is developed.

In Chap. 5 a novel kind of characteristic curve for the purposes of analytical description of the contact geometry of two smooth regular part surfaces in the first order of tangency is discussed in detail. The discussion begins with preliminary remarks and follows with introduction and with derivation of an equation of the indicatrix of conformity at point of contact of two parts surfaces. Then, directions of extremum degree of conformity of two part surfaces in contact are specified and are described analytically. This analysis is followed by determination and by derivation of corresponding equations of asymptotes of the indicatrix of conformity. Capabilities of the indicatrix of conformity R point of contact of two smooth regular part surfaces in the first order of tangency are compared with the corresponding capabilities of “Dupin indicatrix” of the surface of relative curvature. Important properties of the indicatrix of conformity of two smooth regular part surfaces are outlined. Ultimately, the converse indicatrix of conformity at point of contact of two regular part surfaces in the first order of tangency is introduced and is briefly discussed as an alternative to the regular indicatrix of conformity.

The analytical modelling of coupled slip and sliding contact response of two elastic spheres is presented for the kinematically imposed sphere centre relative motion trajectories. One sphere is assumed as a fixed, the other translating along a specified trajectory and remaining in contact condition. Two cases are considered, the first is corresponding to a linear trajectory with the contact engagement in the combined slip-sliding mode, the other is related to the contact initiation by normal loading and subsequent motion along an inclined linear trajectory. The formulae and diagrams of the evolution of driving force along the sliding path in terms of main contact geometry parameters were analytically specified. Further extensions and applications of the analysis can be envisaged in the creation of the translation controlled apparatus for the measurements of friction and restitution coefficients for the pair of spherical grains.