Geometric Mechanics

A Lie-group integration method for constrained multibody systems is proposed in the paper and applied for numerical simulation of a satellite dynamics. Mathematical model of multibody system dynamics is shaped as DAE system of equations of index 1, while dynamics is evolving on Lie-group introduced as system 'state-space formulation'. The basis of the method is Munthe-Kaas algorithm for ODE on Liegroups, which is re-formulated and expanded to be applicable for the integration of constrained multibody dynamics in DAE index 1 form. The constraint violation stabilization algorithm at the generalized position and velocity level is introduced by using two different algorithms: a first one that operates directly on the 'state-space' manifold and, a second one, that uses Cartesian rotation vectors as local coordinates for the generalized positions. A numerical example of 'dual-spin' satellite that demonstrates the proposed integration procedure is described and discussed at the end of the paper.

Résumé – Nous exposons ici le modèle géométrique de l'Information Quantique, tel qu'introduit par Jean-Marie  Souriau. Ce dernier a élaboré le concept de quantification géométrique en introduisant la notion de variété quantique fibrée en cercle au-dessus d’une variété symplectique, laquelle est une orbite coadjointe. Il devient ainsi possible d’étendre la définition de la fonction hamiltonienne, permettant de caractériser les « états quantiques » comme des fonctions complexes définies sur le groupe 𝐺, et satisfaisant à deux inégalités précises. Pour ce faire, la construction de Gelfand Naimark-Segal est utilisée pour établir un lien entre l’état quantique, la représentation unitaire du groupe,  et le vecteur unitaire dans l’espace de Hilbert. Ces axiomes garantissent l’interprétation probabiliste de la mécanique  quantique, où l’état 𝑚 associe une loi de probabilité à toute « observable » du groupe 𝐺. Cela permet, dans le cadre  linéaire, de garantir les relations d'incertitude énoncées par Heisenberg. La convexité de l'espace des états produit des états quantiques dits « mixtes », dont l'existence s'avère indispensable pour la thermodynamique quantique, notamment  en ce qui concerne les états de Gibbs.
Abstract – We present here the geometric model of Quantum Information, as introduced by Jean-Marie Souriau. He developed the concept of geometric quantisation by introducing the notion of a quantum fibre bundle over a symplectic  manifold, which is a coadjoint orbit. This allows for the extension of the definition of the Hamiltonian function,  enabling the characterisation of "quantum states" as complex functions defined on the group 𝐺, satisfying two specific  inequalities. To achieve this, the Gelfand-Naimark-Segal construction is employed to establish a connection between  the quantum state, the unitary representation of the group, and the unit vector in the Hilbert space. These axioms ensure  the probabilistic interpretation of quantum mechanics, where the state 𝑚 associates a probability measure with each  "observable" of the group 𝐺. In the linear case, this guarantees the Heisenberg uncertainty relations. The convexity of  the state space produces so-called "mixed" quantum states, whose existence is essential for quantum thermodynamics,  particularly concerning Gibbs states.

The n-dimensional extension of the one dimensional Position-dependent mass (PDM) Lagrangians under the nonlocal point transformations by Mustafa [38] is introduced. The invariance of the n-dimensional PDM Euler-Lagrange equations is examined using two possible/different PDM Lagrangian settings. Under the nonlocal point transformation of Mustafa [38], we have shown that the PDM Euler-Lagrange invariance is only feasible for one particular PDM-Lagrangians settings. Namely, when each velocity component is deformed by some dimensionless scalar multiplier that renders the mass position-dependent. Two illustrative examples are used as reference Lagrangians for different PDM settings, the nonlinear n-dimensional PDM-oscillators and the nonlinear isotonic n-dimensional PDM-oscillators. Exact solvability is also indulged in the process.

The n-dimensional extension of the one dimensional Position-dependent mass (PDM) Lagrangians under the nonlocal point transformations by Mustafa [38] is introduced. The invariance of the n-dimensional PDM Euler-Lagrange equations is examined using two possible/different PDM Lagrangian settings. Under the nonlocal point transformation of Mustafa [38], we have shown that the PDM Euler-Lagrange invariance is only feasible for one particular PDM-Lagrangians settings. Namely, when each velocity component is deformed by some dimensionless scalar multiplier that renders the mass position-dependent. Two illustrative examples are used as reference Lagrangians for different PDM settings, the nonlinear n-dimensional PDM-oscillators and the nonlinear isotonic n-dimensional PDM-oscillators. Exact solvability is also indulged in the process.

In this paper we introduce the notion of geodesically complete Lie algebroid. We give a Riemannian distance on the connected base manifold of a Riemannian Lie algebroid. We also prove that the distance is equivalent to natural one if the base manifold was endowed with Riemannian metric. We obtain Hopf Rinow type theorem in the case of transitive Riemannian Lie algebroid, and give a characterization of the connected base manifold of a geodesically complete Lie algebroid.

We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. This extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino in and allows us to obtain the most general convexity statement available in the literature for momentum maps associated to a symplectic Lie group action.

Discrete and periodic contact switching is a key characteristic of steady-state legged locomotion. This paper introduces a framework for modeling and analyzing this contactswitching behavior through the framework of geometric mechanics on a toy robot model that can make continuous limb swings and discrete contact switches. The kinematics of this model form a hybrid shape-space and by extending the generalized Stokes' theorem to compute discrete curvature functions called stratified panels, we determine average locomotion generated by gaits spanning multiple contact modes. Using this tool, we also demonstrate the ability to optimize gaits based on the system's locomotion constraints and perform gait reduction on a complex gait spanning multiple contact modes to highlight the method's scalability to multilegged systems.

This is the second part of integrability analysis of cosmological models with scalar fields. Here, we study systems with conformal coupling, and show that apart from four cases, where explicit first integrals are known, the generic system is not integrable. We also comment on some chaotic properties of the system, and the issues of integrability restricted to the real domain.

The Eisenhart lift establishes a fascinating connection between non-relativistic and relativistic physics, providing a space-time geometric understanding of non-relativistic Newtonian mechanics. What is still little known, however, is the fact that there is a Hilbert space representation of classical mechanics (also called Koopman-von Neumann mechanics) that attempts to give classical mechanics the same mathematical structure that quantum mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN) mechanics using the Eisenhart toolkit. We then use a geometric view of KvN mechanics to find transformations that relate the harmonic oscillator, linear potential, and free particle in the context of KvN mechanics.

The Eisenhart lift establishes a fascinating connection between non-relativistic and relativistic physics, providing a space-time geometric understanding of non-relativistic Newtonian mechanics. What is still little known, however, is the fact that there is a Hilbert space representation of classical mechanics (also called Koopman-von Neumann mechanics) that attempts to give classical mechanics the same mathematical structure that quantum mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN) mechanics using the Eisenhart toolkit. We then use a geometric view of KvN mechanics to find transformations that relate the harmonic oscillator, linear potential, and free particle in the context of KvN mechanics.

Resumen. Interacciones no lineales de ondas de Alfvén existen tanto para plasmas en el espacio como en laboratorios, con efectos que van desde calentamiento hasta conducción de corriente. Un ejemplo de emisión de ondas de Alfvén en ingeniería aparece en amarras espaciales. Estos dispositivos emiten ondas en estructuras denominadas "Alas de Alfvén". La ecuación Derivada no lineal de Schrödinger (DNLS) posee la capacidad de describir la propagación de ondas de Alfvén de amplitud finita circularmente polarizadas tanto para plasmas fríos como calientes. En esta investigación, dicha ecuación es truncada con el objetivo de explorar el acoplamiento coherente, débilmente no lineal y cúbico de cuatro ondas cerca de resonancia (k 1 + k 2 = k 3 + k 4 ). La onda 1 que corresponde al vector de onda k 1 puede ser linealmente inestable y las tres restantes ondas 2, 3 y 4, correspondientes a k 2 , k 3 y k 4 respectivamente, son amortiguadas. Por medio de la utilización de este modelo se genera un flujo 5D formado por cuatro amplitudes y una fase relativa. En una serie de trabajos previos se ha analizado la transición dura hacia caos en flujos 3D ) y 4D (Elaskar et al., 2005; Sánchez-Arriaga et al., 2007). Se presenta en este artículo un análisis teórico-numérico del comportamiento del sistema cuando la tasa de crecimiento de la onda inestable es nula.

The geometric nature of Euler fluids has been clearly identified and extensively studied in mathematics. However computational approaches to fluid mechanics, mostly derived from a numerical-analytic point of view, are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts. In contrast, we geometrically derive discrete equations of motion for fluid dynamics from first principles. Our approach uses a finite-dimensional Lie group to discretize the group of volume-preserving diffeomorphisms, and the discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion induce a structure-preserving time integrator with good longterm energy behavior, for which an exact discrete Kelvin circulation theorem holds. Possible extensions of our method to magnetohydrodynamics, viscous flows, optimal transport and a link to Brenier's generalized flows are also discussed. vi

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.

This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of freedom of interest for statistical mechanics. The first part of the paper concerns the applications of methods used in classical differential geometry to study the chaotic dynamics of Hamiltonian systems. Starting from the identity between the trajectories of a dynamical system and the geodesics in its configuration space, when equipped with a suitable metric, a geometric theory of chaotic dynamics can be developed, which sheds new light on the origin of chaos in Hamiltonian systems. In fact, it appears that chaos can be induced not only by negative curvatures, as was originally surmised, but also by positive curvatures, provided the curvatures are fluctuating along the geodesics. In the case of a system with a large number of degrees of freedom it is possible to approximate the chaotic instability behavior of the dynamics by means of a geometric model independent of the dynamics, which allows then an analytical estimate of the largest Lyapunov exponent in terms of the averages and fluctuations of the curvature of the configuration space of the system. In the second part of the paper the phenomenon of phase transitions is addressed and it is here that topology comes into play. In fact, when a system undergoes a phase transition, the fluctuations of the configuration-space curvature, when plotted as a function of either the temperature or the energy of the system, exhibit a singular behavior at the phase transition point, which can be qualitatively reproduced using geometric models. In these models the origin of the singular behavior of the curvature fluctuations appears to be caused by a topological transition in configuration space, which corresponds to the phase transition of the physical system. This leads us to put forward a Topological Hypothesis (TH). The content of the TH is that phase transitions would be related at a deeper level to a change in the topology of the configuration space of the system. We will illustrate this on a simple model, the mean-field XY model, where the TH can be checked directly and analytically. Since this model is of a rather special nature, namely a mean-field model with infinitely-ranged interactions, we discuss other more realistic (non mean-field-like) models, which can not be solved analytically, but which do supply direct supporting evidence for the TH via numerical simulations.

We consider the class of physical theories whose dynamics are given by natural equations, which are partial differential equations determined by a functor from the category of n manifolds, for some n, to the category of fiber bundles, satisfying certain further conditions. We show how the theory of natural equations clarifies several important foundational issues, including the status and meaning of minimal coupling, symmetries of theories, and background structure. We also state and prove a fundamental result about the initial value problem for natural equations.

We study the geometry of the phase space of a particle in a Yang -Mills-Higgs field in the context of the theory of Dirac structures. Several kno wn constructions are merged into the framework of coupling Dirac structures. Functorial pro perties of our constructions are discussed and examples are provided. Finally, application s t fibered symplectic groupoids are given.

Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and super-linear). In [Vit1, MVZ], Mather's α function is recovered from the homogenization of symplectic capacities. This allows the authors to extend the Mather functional to non convex cases. This article shows that the relation between invariant measures and the subdifferential of Mather's functional (which is the foundational statement of Mather) is preserved in the non convex case. We give applications in the context of the classical KAM theory to the existence of invariant measures with large rotation vector after the possible disappearance of some KAM tori.

This note discusses Routh reduction for hybrid time-dependent mechanical systems. We give general conditions on whether it is possible to reduce by symmetries a hybrid time-dependent Lagrangian system extending and unifying previous results for continuous-time systems. We illustrate the applicability of the method using the example of a billiard with moving walls.

Symmetries in modern physics are a fundamental theme under study that allows to appreciate the Particle Physics. In addition, gauge theories are tools that allow to do a description more complete. With this paper, we analyze a gauge symmetry that appears in complex holomorphic systems. A complex system can be reduced to different real systems, using different gauge conditions and several real systems are connected by gauge transformations in the complex space. We prove that the space of solutions of one system is related to another one using a gauge transformation. Gauge transformations are in some cases canonical transformations. However, in other cases are more general transformations that change the symplectic structure, but there is still a map between systems. We establish a construction extend of gauge transformations and show how to extend the analysis to the quantum case using path integrals by means of the Batalin-Fradkin-Vilkovisky theorem and with in the canonical formalism, where we show explicitly that solutions of the Schrödinger equation are gauge related.

We will further develop the study of the dissipation for a Hamilton-Poisson system introduced in [3]. We will give a tensorial form of this dissipation and show that it preserves the Hamiltonian function but not the Poisson geometry of the initial Hamilton-Poisson system. We will give precise results about asymptotic stabilizability of the stable equilibria of the initial Hamilton-Poisson system.

The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

Thermodynamics understanding by Geometric model were initiated by all precursors Carnot, Gibbs, Duhem, Reeb, Carathéodory. It is only recently that Symplectic Foliation Model introduced in the domain of Geometric statistical Mechanics has opened the door for a solid  bedrock, giving a geometric definition of Entropy as invariant Casimir function on Symplectic leaves (coadjoint orbit of the Lie Group acting on the system and interpreted as level sets of Entropy).

As observed by Georges Reeb "Thermodynamics has long accustomed mathematical physics [see DUHEM P.] to the consideration of completely integrable Pfaff forms: the elementary heat dQ [notation of thermodynamicists] representing the elementary heat yielded in an infinitesimal reversible modification is such a completely integrable form. This point does not seem to have been explored since then." Notion of foliation in thermodynamics appears in  C. Carathéodory paper where horizontal curves roughly correspond to adiabatic processes, performed in the language of Carnot cycles.  The properties of the couple of Poisson manifolds was  previously explored by C. Carathéodory in 1935, under the name of “function groups, polar to each other”. This seminal work of C. Caratheodory leads to the concept of a Poisson structure which was first defined independently by Lichnerowicz and  Kirillov.

A symplectic foliation model of Thermodynamics has been defined by Jean-Marie Souriau based on "Lie Groups Thermodynamics" model. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, a symplectic bifoliation structure is associated to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Libermann's foliations, clarified as dual to Poisson Gamma-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of  “Séminaire Sud-Rhodanien de Géométrie ». The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. We conclude with link to Cartan foliation and Edmond Fedida works on Cartan's mobile frame-based foliation.

In the first part, we will present the theme "Statistical learning on Lie groups" [1,2] which extends statistics and machine learning to Lie groups based on the theory of representations and cohomology of Lie algebra. From the work of Jean-Marie Souriau on " Lie Groups Thermodynamics" [4] initiated within the framework of symplectic models of statistical mechanics, new geometric statistical tools have been developed to define probability densities (Gibbs density of Maximum Entropy) on Lie Groups or homogeneous manifolds for supervised methods, and the extension of the Fisher metric of Information Geometry for unsupervised methods (in metric spaces).

In a 2nd part, TINNs (Thermodynamics-Informed Neural Networks) [3,5] will be discussed for AI-augmented engineering applications. The geometric structures of TINNs are studied by their metriplectic flow (also called GENERIC flow) modeling non-dissipative dynamics (1st thermodynamic principle of energy conservation) and dissipative dynamics (2nd thermodynamic principle of entropy production). The Thermodynamics of Lie Groups of Souriau makes it possible to geometrically characterize the metriplectic flow by a structure of “webs” composed of symplectic foliations and transversely Riemannian foliations. From the symmetries of the problem, the coadjoint orbits of the Lie group generate via the moment map (geometrization of Noether's theorem) the symplectic foliation (defined as the level sets of entropy, where entropy is an invriant Casimir function on these symplectic leaves). The metric on symplectic leaves is given by the Fisher metric. The dynamics along these symplectic leaves, given by the Poisson bracket, characterizes the non-dissipative dynamics. The dissipative dynamics is then given by the transverse Poisson structure and a metric flow  bracket, with an evolution from leaf to leaf constrained by the production of entropy. The transverse foliation is a Riemannian foliation whose metric is given by the dual of Fisher metric (Hessian of Entropy).

These machine-learning tools on Lie groups and TINNs are addressed in two European projects: a European network COST CaLISTA [6] and the European Marie-Curie action MSCA CaLIGOLA [7].

References:

[1] Barbaresco, F. (2022) Symplectic theory of heat and information geometry, chapter 4, Handbook of Statistics, Volume 46, Pages 107-143, Elsevier, https://doi.org/10.1016/bs.host.2022.02.003 https://www.sciencedirect.com/science/article/abs/pii/S0169716122000062 

[2] Barbaresco, F. (2023). Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_17  ; https://link.springer.com/chapter/10.1007/978-3-031-38299-4_17 

[3] Barbaresco, F. (2022) Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation. Entropy 2022, 24, 1626. https://doi.org/10.3390/e24111626

[4] Souriau, J.M. (1969). Structure des systèmes dynamiques. Dunod. http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm   

[5] Cueto E. and Chinesta F. (2022), Thermodynamics of Learning of Physical Phenomena, arXiv:2207.12749v1 [cs.LG] 26 Jul 2022

[6] European COST network CA21109 : CaLISTA  - Cartan geometry, Lie, Integrable Systems, quantum group Theories for Applications ; https://site.unibo.it/calista/en

[7] European HORIZON-MSCA-2021-SE-01-01 project CaLIGOLA - Cartan geometry, Lie and representation theory, Integrable Systems, quantum Groups and quantum computing towards the understanding of the geometry of deep Learning and its Applications; https://site.unibo.it/caligola/en

[8] Barbaresco, F.  (2021) Koszul lecture related to geometric and analytic mechanics, Souriau’s Lie group thermodynamics and information geometry. Info. Geo. 4, 245–262. https://doi.org/10.1007/s41884-020-00039-x

[9]  Barbaresco, F. (2021). Gaussian Distributions on the Space of Symmetric Positive Definite Matrices from Souriau’s Gibbs State for Siegel Domains by Coadjoint Orbit and Moment Map. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_28

[10] Barbaresco, F. (2021). Koszul Information Geometry, Liouville-Mineur Integrable Systems and Moser Isospectral Deformation Method for Hermitian Positive-Definite Matrices. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_39

[11] Barbaresco, F. (2021). Jean-Marie Souriau’s Symplectic Model of Statistical Physics: Seminal Papers on Lie Groups Thermodynamics - Quod Erat Demonstrandum. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_2

[12] 7th SEE Geometic Science of Information; https://conference-gsi.org/

Using contact geometry we give a new characterization of a simple but important class of thermodynamical systems which naturally satisfy the first law of thermodynamics (total energy preservation) and the second law (increase of entropy). We completely clarify its qualitative dynamics, the underlying geometrical structures and we show how to use discrete gradient methods.

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.

Recent technologies permit matching intermediaries to engage in unprecedented levels of targeting. Yet, regulators fear that the welfare gains of such targeting be hindered by the high degree of price customization practiced by matching intermediaries, whereby prices finely depend on the characteristics of the matching partners. To shed light on this debate, we develop a matching model in which agents’ preferences are both vertically and horizontally differentiated. Mirroring current practices, we show how platforms maximize profits by offering menus of matching plans defined by (a) a baseline configuration, (b) a baseline price, and (c) a collection of nonlinear tariffs for customization. We illustrate how, under such plans, prices are linked to structural elasticities, and derive primitive conditions under which market power distortions increase with the targeting level of a match. We then study the effects on targeting and consumer welfare of uniform-pricing regulation mandating ...

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the Riemann-flat condition. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether regularity singularities (points where the curvature is in L ∞ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to C 1,1 by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to C 1,1 locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme. 1 The space C 0,1 denotes the space of Lipschitz continuous functions, and C 1,1 the space of functions with Lipschitz continuous derivatives. A function is bounded in C 0,1 if and only if the function and its weak derivatives are bounded in L ∞ , c.f. [7], Chapter 5.8. 2 In geometry, the regularity of a Riemannian or Lorentzian metric is defined to be the regularity of the metric components in each coordinate system of a given atlas. This definition neglects the possibility that the metric might be more regular in particular coordinate systems of this atlas. 3 Since the writing of this paper the Riemann-flat condition has become the starting point for authors' further developments. In fact, we abandoned the Nash embedding idea and used the Riemann-flat condition to derive the Regularity Transformation equations, an elliptic system of PDE's equivalent to the Riemann-flat condition, c.f. .

The Chaplygin sleigh is a mechanical system subject to one linear nonholonomic constraint enforcing the plane motion. We solve equations of motion and study symmetries and conservation laws for this system after deriving general equations of nonholonomic symmetries of the constraint Lagrangian. Our considerations are based on an efficient geometrical theory on fibred manifolds first presented and developed by Olga Rossi (Krupkova). The obtained results are thoroughly discussed from the point of view of physics.

The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constraint variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constraint variationality is equivalent with the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with the recently found constraint Helmholtz conditions this result completes basic geometric properties of constraint variational systems. A few examples of constraint variational systems are discussed.

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from "Characteristic Functions", was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of "Information Geometry" theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean "Moment map" by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan "Inner Product". Interpreting Legendre transform as Fourier transform in (Min,+) algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus) Fourier (Min,+) o Log o Laplace (+,X) .

Having embedded minimal degrees below 0', it is natural to try the embed other uppersemilattices as initial segments of ^[0,0']. We prove such embedding theorems in this chapter. In the first four sections, we present a detailed proof of the embeddability of an arbitrary finite lattice as an initial segment of ^[0,0']. Extensions of this result to other usls or to embeddings below degrees other than 0' are discussed in Sec. 5. These results are applied to prove theorems about Q) and [0,0'].

Geometry and analysis of shape space. The core of shape space in one of its simplest forms is the orbit space of the action of the group of diffeomorphisms of the circle S1 (the reparametrization group) on the space of immersions of the circle into the plane R2. The aim is to find good Riemannian metrics which allow applications in pattern recognition and visualization. The contributions of the PI were obtained mainly in collaboration with David Mumford. In the paper [M107] via the Hamiltonian approach many metrics were investigated, together with their conserved quantities (one of them is the reparameterization momentum) and their sectional curvatures. Recall from a result of the predecessor of this project, that the L2-metric on the space of immersions has zero geodesic distance on the orbit space under the reparameterization group. These metrics come in 3 flavors: Some are derived from the the L2-metric by multiplying it with a function of the length of the curve (a conformal cha...

This work is licensed under a Creative Commons Attribution 4.0 International License. Users are free to use this algorithm for academic and practical purposes, provided proper credit is given to David Aranovsky

We introduce a novel approach to resolving challenges in multilinear interpolation using God’s Ratio (ℵ), defined as √𝟓/𝟐, a scaling constant arising from geometric necessity. ℵ offers inherent proportionality and infinite stability, making it a natural foundation for addressing restricted weak-type bounds, especially for 𝐪 ≤ 𝟏, where traditional methods relying on duality fail. This paper demonstrates how ℵ redefines kernel design, simplifies interpolation, and provides robust boundedness properties for multilinear operators and their adjoints. Numerical tests and theoretical results illustrate the efficacy of ℵ in stabilizing weak 𝐋𝐪 norms and enabling precise interpolation across challenging parameter ranges.

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In this paper, the notion of an operator \gamma on a supra topological space (X,\mu ) is studied and then utilized to analyze supra \gamma -open sets. The notions of {\mu }_{\gamma }-g.closed sets on the subspace are introduced and investigated. Furthermore, some new {\mu }_{\gamma }-separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator \gamma on \mu are presented and investigated. Finally, we give some properties of {S}_{(\gamma ,\beta )}-closed graph and strongly {S}_{(\gamma ,\beta )}-closed graph.

The metriplectic formalism couples Poisson brackets of the Hamiltonian description with metric brackets for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a preselected equilibrium state. Phenomena such as friction, electric resistivity, thermal conductivity and collisions in kinetic theories are well represented in this framework. In this paper we present an application of the metriplectic formalism of interest for the theory of control: a suitable torque is applied to a free rigid body, which is expressed through a metriplectic extension of its "natural" Poisson algebra. On practical grounds, the effect is to drive the body to align its angular velocity to rotation about a stable principal axis of inertia, while conserving its kinetic energy in the process. On theoretical grounds, this example shows how the non-Hamiltonian part of a metriplectic system may include convergence to a limit cycle, the first example of a non-zero dimensional attractor in this formalism. The method suggests a way to extend metriplectic dynamics to systems with general attractors, e.g. chaotic ones, with the hope of representing biophysical, geophysical and ecological models.

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton's equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction. Contents arXiv:2001.01143v2 [math.SG] 12 Dec 2020 6.1. Fully compressible fluids 28 6.2. Compressible magnetohydrodynamics 31 6.3. Relativistic inviscid Burgers' equation 33 7. Fisher-Rao geometry 35 7.1. Newton's equations on Diff(M ) 35 7.2. Riemannian submersion over densities 36 7.3. Newton's equations on Dens(M ) 37 8. Fisher-Rao examples 38 8.1. The µCH equation and Fisher-Rao geodesics 38 8.2. The infinite-dimensional Neumann problem 39 8.3. The Klein-Gordon equation 41 9. Geometric properties of the Madelung transform 41 9.1. Madelung transform as a symplectomorphism 42 9.2. Examples: linear and nonlinear Schrödinger equations 42 9.3. The Madelung and Hopf-Cole transforms 44 9.4. Example: Schrödinger's smoke 45 9.5. Madelung transform as a Kähler morphism 47 10. Casimirs in hydrodynamics 48 10.1. Casimirs for ideal fluids 48 10.2. Casimirs for barotropic fluids 49 10.3. Casimirs for magnetohydrodynamics 50 Appendix A. Symplectic and Poisson reductions 52 A.1. Symplectic reduction 52 A.2. Reduction and momentum map for semidirect product groups 54 Appendix B. Tame Fréchet manifolds 56 B.1. Tame Fréchet structures on diffeomorphism groups 56 B.2. Short-time existence of compressible Euler equations 58 References 59

Preface
The subject ‘Analytical Mechanics’ represents a very interesting exercise of human imagination and creativity. The basic foundational ideas were developed concurrently with the development of Newton’s ‘point mechanics’ in England. Enriched by the contributions of some legendary mathematicians like Euler, it got its final shape in the hands of Lagrange. The subject is avoided in the curriculum of many academic institutions (mostly engineering colleges and small universities), hoping for an exposure to Newtonian mechanics only. But this omission puts the students in a difficult situation when they intend to pursue work in areas ‘involving mechanics’. Newtonian approach is convenient in relatively simpler situations, but for studying more complicated system the most suitable approach is provided by analytical mechanics.

[CONTINUA]

In a first part, we will present pioneering THALES Sensors/Radars algorithms: Geometric Matrix CFAR based on Jean-Louis Koszul’s Information Geometry and its extension for STAP, Complex-Valued
Convolutional Neural Networks and Covariance-Matrix-Valued HPDNet for Micro-Doppler ATDR, Lie Group-based Convolutional Equivariant Neural Network from Geometric Deep Learning for Doppler clutter map, IEKF (Invariant Extended Kalman Filter) Frenet-Serret Tracker based on Lie Groups for hyper-maneuvering targets, Tracker parameters tuning by Deep Learning and finally, Multi-Agent Reinforcement Learning for Radar Task Scheduling and Active-Track/TWS collaborative Resources Management.
In a second part, we will present Avant-Garde tools using statistics on Lie Groups for different sensors applications (detection, tracking and recognition). From French Jean-Marie Souriau’s Symplectic Model of Statistical Physics and Russian Kirillov’s Representation Theory of Lie Groups, we will introduce Gaussian statistical density for Lie Groups defined as Maximum Entropy Gibbs density on coadjoint orbits though moment map. This Symplectic model of Information gives new geometric
foundation for Entropy, defined purely geometrically (and no longer axiomatically) as Casimir Invariant Function in Coadjoint Representation. We will conclude with new perspectives opened by this new Symplectic Theory of Heat and Information.

The quantum mechanical harmonic oscillator Hamiltonian H = (t 2 − ∂ 2 t)/2 generates a one-parameter unitary group W (θ) = e iθH in L 2 (R) which rotates the time-frequency plane. In particular, W (π/2) is the Fourier transform. When W (θ) is applied to any frame of Gabor wavelets, the result is another such frame with identical frame bounds. Thus each Gabor frame gives rise to a one-parameter family of frames, which we call a deformation of the original. For example, beginning with the usual tight frame F of Gabor wavelets generated by a compactly supported window g(t) and parameterized by a regular lattice in the time-frequency plane, one obtains a family {F θ : 0 ≤ θ < 2π} of frames generated by the non-compactly supported windows g θ = W (θ)g, parameterized by rotated versions of the original lattice. This gives a method for constructing tight frames of Gabor wavelets for which neither the window nor its Fourier transform have compact support. When θ = π/2, F θ is the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family {F θ } therefore interpolates these two familiar examples.

This paper formulates an optimal control problem for a system of rigid bodies that are neutrally buoyant, connected by ball joints, and immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in three-dimensional space, and each joint has three rotational degrees of freedom. We assume that internal control moments are applied at each joint. We present a computational procedure for numerically solving this optimal control problem, based on a geometric numerical integrator referred to as a Lie group variational integrator. This computational approach preserves the Hamiltonian structure of the controlled system and the Lie group configuration manifold of the connected rigid bodies, thereby finding complex optimal maneuvers of connected rigid bodies accurately and efficiently. This is illustrated by numerical computations.

This paper treats the geometric formulation of optimal control problems for rigid bodies and it presents computational procedures based on this geometric formulation that can be used for numerical solution of these optimal control problems. The dynamics of each rigid body is viewed as evolving on a configuration manifold that is a Lie group. Discrete time dynamics of each rigid body are developed that evolve on the configuration manifold according to a discrete version of Hamilton's principle so that the computations preserve geometric features of the dynamics and guarantee evolution on the configuration manifold; these discrete-time dynamics are referred to as Lie group variational integrators. Rigid body optimal control problems are formulated as discrete-time optimization problems for discrete Lagrangian/Hamiltonian dynamics, to which standard numerical optimization algorithms can be applied. This general approach is illustrated by presentation of results for several different optimal control problems for a single rigid body and for multiple interacting rigid bodies. Although the emphasis in this work is on computational results, the philosophy and the approach follow the work on optimal control problems for rigid body kinematics by Roger Brockett and his colleagues in the 1970s. This version of the paper provides an overview of the concepts and contributions; the final version of the paper will include additional details on geometric optimal control of rigid bodies.

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.