Port Hamiltonian system

We describe the essential spectrum of a large class of $N$-particle discrete Schr\"odinger operators $H(K),$ $K\in {\mathbb T}^d,$ $d\ge1,$ associated with the Hamiltonian of the $N$-particles moving on the lattice $\mathbb Z^d$ and interacting via short-range pair potentials. We also show the existence of an $N$-particle bound state below the essential spectrum provided that the particles are purely attractive or purely repulsive and any two-particle subsystem has a bound state below its essential spectrum.

We consider the family $\hat h_\mu:=\hat\varDelta\hat \varDelta - \mu \hat v,$ $\mu\in\mathbb{R}, $ of discrete Schrodinger-type operators in $d$-dimensional lattice $\mathbb{Z}^d$, where $\hat \varDelta$ is the discrete Laplacian and $\hat v$ is of rank-one. We prove that there exist coupling constant thresholds $\mu_o,\mu^o\ge0$ such that for any $\mu\in[-\mu^o,\mu_o]$ the discrete spectrum of $\hat h_\mu$ is empty and for any $\mu\in \mathbb{R}\setminus[-\mu^o,\mu_o]$ the discrete spectrum of $\hat h_\mu$ is a singleton $\{e(\mu)\},$ and $e(\mu) \mu_o$ and $e(\mu)>4d^2$ for $\mu<-\mu^o.$ Moreover, we study the asymptotics of $e(\mu)$ as $\mu\to\mu_o$ and $\mu\to -\mu^o$ as well as $\mu\to\pm\infty.$ The asymptotics highly depend on $d$ and $\hat v.$

I wish to acknowledge the contribution of Prof. Niko Sauer, for valuable discussion on the work, Dr. Adewale Adedipe (Neurosurgeon) for giving the expository materials that gave me a leadway to understanding the neurosurgical part of my work, Ms.

We study the Schr ¨odinger operators H λμ (K) with K ∈ T 2 being the fixed quasimomentum of a pair of particles, associated with a system of two arbitrary particles on a two-dimensional lattice Z 2 with on-site and nearest-neighbor interactions of strengths λ ∈ R and μ ∈ R, respectively. We divide the (λ, μ)-plane of parameters λ and μ into connected components, such that in each component, the Schr ¨odinger operator H λμ (0) has a fixed number of eigenvalues. These eigenvalues are located both below the bottom of the essential spectrum and above its top. Additionally, we establish a sharp lower bound for the number of isolated eigenvalues of H λμ (K) within each connected component.

We are analyzing several types of dynamical systems which are both integrable and important for physical applications. The first type are the so-called peakon systems that appear in the singular solutions of the Camassa-Holm equation describing special types of water waves. The second type are Toda chain systems, that describe molecule interactions. Their complexifications model soliton interactions in the adiabatic approximation. We analyze the algebraic aspects of the Toda chains and describe their real Hamiltonian forms.

In his paper Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Denis Sullivan proves that a closed manifold supports a symplectic structure if and only if it admits a distribution of cones of bivectors satisfying two conditions. We prove a similar result for contact structures. It relies on a suitable variant of the symplectization process that produces a S 1 -invariant nondegenerate 2-form on the closed manifold S 1 ×M that is closed for a twisted differential.

In his paper Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Denis Sullivan proves that a closed manifold supports a symplectic structure if and only if it admits a distribution of cones of bivectors satisfying two conditions. We prove a similar result for contact structures. It relies on a suitable variant of the symplectization process that produces a S 1 -invariant nondegenerate 2-form on the closed manifold S 1 ×M that is closed for a twisted differential.

Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and ge...

We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height h. For chaotic orbits below the periodic islands, the survival probability for the particles to reach h is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when h reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when h reaches the position of the lowest KAM island.

Some dynamical properties for an ensemble of non-interacting classical particles along chaotic orbits and transport properties over the chaotic sea for the problem of a step and time dependent potential well are considered. The dynamics of each particle is described by a two-dimensional, nonlinear and area preserving mapping for the variables energy and time. The phase space is of mixed-type and contains periodic islands, a set of invariant KAM curves and chaotic seas. The chaotic orbits are characterized by the use of Lyapunov exponents. Transport over the chaotic sea is considered and scaling exponents are obtained. A sticky region around a chain of periodic islands produces local and temporarily trapping of the dynamics and discussions of the rearrangement of the phase space are made.

In order to simulate the Ondes Martenot, a classic electronic musical instrument, we aim to model its circuit using Port-Hamiltonian Systems (PHS). PHS have proven to be a powerful formalism to provide models of analog electronic circuits for audio applications, as they guarantee the stability of simulations, even in the case of non-linear systems. However, some systems cannot be converted directly into PHS because their architecture cause what are called realizability conflicts. The Ondes Martenot circuit is one of those systems. In this paper, a method is introduced to resolve such conflicts automatically: problematic components are replaced by equivalent components without altering the overall structure nor the content of the modeled physical system.

In this paper optimal control for hybrid systems will be discussed. While defining hybrid systems as causal and consistent dynamical systems, a general formulation for an optimal hybrid control problem is proposed. The main contribution of this paper shows how necessary conditions can be derived from the maximum principle and the Bellman principle. An illustrative example shows how optimal hybrid control via a set of Hamiltonian systems and using dynamic programming can be achieved. However, as in the classical case, difficulties related to numerical solutions exist and are increased by the discontinuous aspect of the problem. Looking for efficient algorithms remains a difficult and open problem which is not the purpose of this contribution.

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the three-dimensional half space. It is shown that in this case the Born series converges regardless the value of the impedance term in the Robin boundary condition.

In this paper, we consider a new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type, denoted by W (α) n,λ (δ, ζ; ρ; µ). We obtain several summation formulae, a recurrence relation, two difference operator formulas, two derivative operator formulas, an implicit summation formula, and a symmetric property for these polynomials. Also, we provide a representation of the degenerate differential operator on the degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. Moreover, we define the degenerate unified Hermite-based Apostol-Stirling polynomials of the second kind and derive some properties of these newly established polynomials. In addition, we prove multifarious correlations, including the new polynomials. Furthermore, we list the first few degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type for some special cases and present data visualizations of zeros forming 2D and 3D structures. Finally, we provide a table covering approximate solutions for the zeros of W (α) n,3 (δ, 4; 3; 2).

We investigate the phenomenon we call "collapse of the hypotenuse," where orthogonal contributions do not yield the expected diagonal length. Two distinct mechanisms are identified. First, in semi-Riemannian geometry, orthogonal nonzero vectors may sum to a null vector, algebraically collapsing the hypotenuse. Second, in curved or topologically nontrivial Riemannian manifolds, geodesic concatenations can lead to shorter or degenerate diagonals. Examples are given from Minkowski space, the 2-sphere, the flat torus, and the hyperbolic plane. We propose this collapse as a unifying theme illustrating how Euclidean intuitions about Pythagoras fail outside flat space.

We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a TASEP-like process with arbitrary boundary conditions. Using the known solution of matrix models, this method allows to find the large size asymptotic expansion of plane partitions, to ALL orders. It also allows to describe several universal regimes. On the algebraic geometry point of view, this gives the Gromov-Witten invariants of C 3 with branes, i.e. the topological vertex, in terms of the symplectic invariants of the mirror's spectral curve.

There is the possibility that on a Planck-scale the universe is not a fourdimensional manifold but only a flat spacetime, consisting only of a permanent timelike dimension and periodic caused spacelike dimensions by the quantum potential of this timelike dimension. In this case the quantumstates of spacelike dimensions form an orthogonal system. For this model a twodimensional Riccitensor can be constructed for every dimension and four of them can be formed to a fourdimensional structure but without intercoupling. A possible intercoupling then would lead to description of classical GRT-structure. This model also can be described mathematically by a special hypocycloid, a deltoid, where the points of the curve mapping exactly on the outer circle of the rotation process. These are the resonance-points of the timelike with the spacelike potentials, causing the dimensions as triple eigenvalue of signature. And so a form of ON/OFF-operators are introduced, which shows similiar structures as spin-operators and can possibly be used to describe the forming of the dimensions.

We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin-orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an excellent measure of agreement between analytical predictions and the proposed fast algorithm.

Belinski, Khalatnikov and Lifshitz (BKL) pioneered the study of the statistical properties of the never-ending oscillatory behavior (among successive Kasner epochs) of the geometry near a spacelike singularity. We show how the use of a "cosmological billiard" description allows one to refine and deepen the understanding of these statistical properties. Contrary to previous treatments, we do not quotient the dynamics by its discrete symmetry group (of order 6), thereby uncovering new phenomena, such as correlations between the successive billiard corners in which the oscillations take place. Starting from the general integral invariants of Hamiltonian systems, we show how to construct invariant measures for various projections of the cosmological-billiard dynamics. In particular, we exhibit, for the first time, a (non-normalizable) invariant measure on the "Kasner circle" which parametrizes the exponents of successive Kasner epochs. Finally, we discuss the relation between: (i) the unquotiented dynamics of the Bianchi IX (a, b, c or mixmaster) model; (ii) its quotienting by the group of permutations of (a, b, c); and (iii) the billiard dynamics that arose in recent studies suggesting the hidden presence of Kac-Moody symmetries in cosmological billiards.

This work focuses on topics related to Hamiltonian stochastic differential equations with Lévy noise. We first show that the phase flow of the stochastic system preserves symplectic structure, and propose a stochastic version of Hamilton's principle by the corresponding formulation of the stochastic action integral and the Euler-Lagrange equation. Based on these properties, we further investigate the effective behaviour of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with Lévy noise. We establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero. Furthermore, we obtain the estimation for the rate of this convergence. Finally, we present an example to illustrate these results.

Analytical perturbations of the Euler top are considered. The perturbations are based on the Poisson structure for such a dynamical system, in such a way that the Casimir invariants of the system remain invariant for the perturbed flow. By means of the Poincaré-Pontryagin theory, the existence of limit cycles on the invariant Casimir surfaces for the perturbed system is investigated up to first order of perturbation, providing sharp bounds for their number. Examples are given.

What is the least surface area of a shape that tiles R d under translations by Z d ? Any such shape must have volume 1 and hence surface area at least that of the volume-1 ball, namely Ω( √ d). Our main result is a construction with surface area O( √ d), matching the lower bound up to a constant factor of 2 p 2π/e ≈ 3. The best previous tile known was only slightly better than the cube, having surface area on the order of d. We generalize this to give a construction that tiles R d by translations of any full rank discrete lattice Λ with surface area 2π ‚ ‚ V -1 ‚ ‚ fb , where V is the matrix of basis vectors of Λ, and • fb denotes the Frobenius norm. We show that our bounds are optimal within constant factors for rectangular lattices. Our proof is via a random tessellation process, following recent ideas of Raz [11] in the discrete setting. Our construction gives an almost optimal noise-resistant rounding scheme to round points in R d to rectangular lattice points.

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will ¢nd situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.

We generalize the sufficient condition for the stability of relative periodic orbits in symmetric Hamiltonian systems presented in [J.-P. Ortega, T.S. Ratiu, J. Geom. Phys. 32 (1999) 131-1591 to the case in which these orbits have non-trivial symmetry. We also describe a block diagonalization, similar in philosophy to the one presented in [J.-P. Ortega, T.S. Ratiu, Nonlinearity 12 (1999) 693-7201 for relative equilibria, that facilitates the use of this result in particular examples and shows the relation between the stability of the relative periodic orbit and the orbital stability of the associated singular reduced periodic orbit.

The presence of symmetries in a Hamiltonian system usually implies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden-Weinstein reduction procedure takes advantage of this and associates to the original system a new Hamiltonian system with fewer degrees of freedom. However, in a large number of situations, this standard approach does not work or is not efficient enough, in the sense that it does not use all the information encoded in the symmetry of the system. In this work, a new momentum map will be defined that is capable of overcoming most of the problems encountered in the traditional approach.

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W * -algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.

In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev. A central problem in control theory is that of stabilizing nonlinear systems. Very often the systems one is interested in are mechanical systems, which, in the absence of dissipation, are Hamiltonian in nature. Further, the equilibria we wish to stabilize are relative equilibria-equilibria modulo a group action, usually the rotation group. Recently two distinct but related methods have been developed to analyze the stability of the relative equilibria of Hamiltonian systems. The first, the "Energy-Casimir" method, originally goes back to Arnold [2] and was developed and formalized in Holm, Marsden, Ratiu and Weinstein [8] and Krishnaprasad

We show that Euler equations on the Lie algebra so(4) near a sin- gular adjoint orbit can be represented as a perturbed Hamiltonian system whose unperturbed part is a completely integrable system, non-degenerate in the sense of Russmann. To make this property visible we apply a "twisting" mapping defined as the composition of a Floquet-Lyapunov type transforma- tion and the time-1 flow of a time-dependent dynamical system associated with the Euler equations.

A periodic array of rf SQUIDs in an alternating magnetic field acts as an inherently nonlinear magnetic metamaterial, due to the nonlinearity of the Josephson element and the resonant properties of the SQUIDs themselves. Neighboring SQUIDs are weakly coupled due to magnetic ...

In this paper we provide a review of Shape Dynamics, a new theory of gravity which overlaps with General Relativity in places but is built from fewer and more fundamental principles. Shape Dynamics is based on a different symmetry group to General Relativity and uses this to implement spatial and temporal relationalism as well as successfully satisfying the Mach-Poincar´e Principle.

This paper tackles the problem of the origin of dynamic chaos in Hamiltonian systems, with a special emphasis on the self-gravitating N-body systems. A Riemannian approach is adopted. The relationship between dynamic instability and curvature properties of the con6guration space manifold is the main concern of this paper. Dynamic instability is studied with the aid of the Jacobi -Levi-Civita (JLC) equa- tion for the geodesic spread. We point out that the approximations introduced so far to make the JLC equation handy, that is, to obtain a scalar equation describing the dynamical instability, are still too severe. In order to assess the validity limits of these approximations, the aid of numerical simulations is essential. For this reason, our analysis is supported by the numerical study of the dynamics of 10 and 100 gravitationally interacting point masses. The self-gravitating S-body systems provide an illuminat- ing ex@mple of the relevant difference between geodesic Bows of abstract ergodic theory and geodesic Qows of physical interest. In fact, even though they correspond to manifolds of almost everywhere nega- tive scalar curvature, this does not determine by itself the degree of chaos of these systems. %'e show that the quantities determining the instability of nearby trajectories do not simply coincide with scalar or Ricci curvature; rather they involve also other ingredients that are ultimately responsible for the ex- istence of two different mechanisms to make chaos. The quantities mentioned enter a Hill equation and give instability either when they are negative orwhen positivebecause of parametric resonance. From numerical computations the e (energy density) dependence of the dynamic instability exponents is found to be -e . Our paper aims at warning about the possibility of misleading conclusions that might be drawn from the geometric approach if the existence of the problems discussed here is ignored. Finally, we briefly discuss the relationship of the Riemannian geometric description of chaos with Lyapunov exponents in the special case of gravitational N-body systems.

This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. The stability of dynamics, related to curvature properties of the configuration space manifold, is investigated through the Jacobi -Levi-Civita equation (JLC) for geodesic spread. The case of two-degrees-of-freedom Hamiltonians is considered in general and is applied to the Henon-Heiles model. The detailed qualita- tive information provided by Poincare sections are compared with the results of geometric investigation; a complete agreement is found. The solutions of the JLC equation are also in quantitative agreement with the solutions of the tangent dynamics equation. It is shown here that chaos in the Henon-Heiles model stems from parametric instability due to positive curvature fluctuations along the geodesics (dynamical motions) of configuration space manifold. This mechanism is apparently the most relevantand in many cases uniquesource of chaoticity in physically meaningful Hamiltonians. Hence a major difference with the geometric description of chaos in abstract ergodic theory is found; chaotic Hamiltonian flows of physics have nothing to do with Anosov flows defined on negative curva- ture manifolds. Even in the case of fully developed Hamiltonian chaos, hyperbolicity is not necessarily involved. FinaBy, the paper deals with the problem of finding other criteria for the onset of chaos based on purely geometric tools and independently of the numerical knowledge of the trajectories.

This paper concerns the study of the strong stochasticity threshold (SST) in Hamiltonian systems with many degrees of freedom, and more specifically the stability problem of this threshold in the thermodynamic limit (N -+ oo). The investigation is based on a recently proposed differential geometrical description of Hamiltonian chaos. The mathematical framework is given by Eisenhart's formulation of Newtonian mechanics in a suitably enlarged configuration space-time: a Riemannian manifold equipped with an aKne metric. Using the Jacobi -Levi-Civita equation for geodesic spread, w'e estabilish a relation between curvature properties of the ambient manifold and the stabilityor instabilityof the dynamics. The use of the Eisenhart metric makes it clearly evident that a dominating source of chaoticity in Hamiltonian flows of physical interest is represented by parametric resonance induced by curvature fluctuations along the geodesics and not by negativeness of some curvature property. Here only Ricci curvature is involved because the scalar curvature vanishes identically with this metric. Thus a geometric quantity relevant to the study of chaos is the degree of bumpiness of the ambient manifold, i.e. , the integral of the Ricci curvature carried over the whole manifold with the constraint of energy constancy. This quantity, considered as a function of the energy density c, clearly marks the SST. A simple sufFicient criterion is given to identify integrable systems and is here applied to a chain of linear oscillators as well as to the Toda lattice. In the case of the Fermi-Pasta-Ulam P model we have worked out analytically the e dependence of the degree of bumpiness of the ambient manifold. This allowed us to prove that the SST is stable in the N ~oo limit. An operational definition of the critical energy density is also provided and shown to yield predictions in excellent agreement with previous results based on Lyapunov exponents.

This paper is concerned with the existence of homoclinic orbits of multi-bump type in the second order Hamiltonian system with superquadratic potential q -L(t)q + Wq(q) = 0, t ∈ R, (HS) where q = (q 1 , . . . , q ) is a symmetric matrix for all t.

Optical pulse dynamics in dispersion-managed fiber lines is studied using a combination of a Lagrangian approach and Hamiltonian averaging. By making self-similar transform in the Lagrangian and assuming in the leading order a bell-shaped pulse dynamics, we reduce the original system to a nonautonomous Hamiltonian system with two variables. Subsequent Hamiltonian averaging gives a function of two variables whose extrema correspond to periodic pulses. To describe a fine structure of the pulse tails, we further develop Hamiltonian averaging using the complete set of Gauss-Hermite functions and also applying averaging in the spectral domain. ͓S1063-651X͑99͒50504-7͔

Bu calismada tanim kumesi (domeyn) degiskeni ile degisen siradan dogrusal sistemlerin analitik olarak cozulebilir olmasi icin yeni bir siniflandirma tanimi yapilmistir. Onerilen yeni dinamik sistem sinifi icin uygun donusumlerin elde edilmesi ve cozulmesi yontemi aciklanmistir. Tanim kumesi degiskeni ile degisen (genellikle zamanla degisen) siradan dogrusal sistemlerin genel bir analitik cozum yontemiyoktur. Her ne kadar zamanla degisen birinci derece siradan dogrusal sistemlere ait denklemlerin cozulebilmesi gorece olasi olsa da ikinci ve daha ust dereceli sistemlerde matematiksel modelin ancak belli ve sinirli ozelliklere sahip olmasi durumunda cozumleri bulunabilir. Bu tur sistemlerin cozulebilir olmasi, uygun donusumler uygulanarak sistemlerin tanim kumesi degiskeni ile degismeyen sabit katsayili sistemlere donusebiliyor olmasi ile iliskilidir. Bu gune kadar yapilan modelleme ve cozumleme calismalarinda genel olarak ozdegerleri belli ozellikler tasiyan zamanla degisen siradan si...

La théorie des catastrophes. IV. Déploiements universels et leurs catastrophes Annales de l'I. H. P., section A, tome 24, n o 3 (1976), p. 261-300 <http © Gauthier-Villars, 1976, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 261 La théorie des catastrophes. IV. Déploiements universels et leurs catastrophes (1)

We consider a concrete model of the planar elliptic restricted three-body problem, viewed as a perturbation of the planar circular restricted three-body problem, with the eccentricity ε of the orbits of massive bodies as the perturbation parameter. For every non-zero value of ε up to the physical value of the eccentricity we obtain: (i) The existence of orbits along which the energy drifts by an amount independent of ε, with explicit estimates for the Lebesgue measure of the set of such orbits. The time required by such orbits to drift is O(1/ε); (ii) The existence of orbits along which the energy makes chaotic excursions; (iii) Explicit estimates for the Hausdorff dimension of the set of such chaotic orbits; (iv) The existence of orbits, with initial conditions in some suitable sets of positive Lebesgue measure, along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as ε tends to 0. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion, for appropriate sets of initial conditions. Our results provide answers to some conjectures of Arnold and of Chirikov. All the quantitative estimates that appear in our results are explicit. The arguments are based on topological methods and their implementation into a computer assisted proof.

We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in . This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an 'outer dynamics', given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an 'inner dynamics', given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems (roughly, they amount to checking that Melnikov-type integrals do not vanish and that some manifolds are transversal). As an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity. We verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough). The code of these

The Bekenstein-Hawking entropy, S = A/(4G), is a cornerstone of black hole thermodynamics, yet its statistical mechanical origin remains elusive in canonical quantum gravity due to two fundamental challenges: the problem of time, arising from the non-dynamical Hamiltonian constraint, and the indefinite DeWitt supermetric, which renders path integrals ill-defined. Shape Dynamics (SD) addresses these issues by reformulating general relativity (GR) as a conformally invariant theory on shape space, S = Riem(Σ)/(Diff × Weyl), where Riem(Σ) is the space of Riemannian 3-metrics on a 3-manifold Σ, quotiented by diffeomorphisms and Weyl transformations. In SD, the Hamiltonian constraint is traded for Weyl invariance, and dynamics are governed by relational evolution in the Constant Mean Curvature (CMC) gauge, where the mean extrinsic curvature K = τ(t) fixes time reparametrization. This framework yields a positive-definite effective metric on shape space, facilitating a well-defined statistical ensemble. We present the first explicit construction and semiclassical evaluation of the functional measure D µ S on shape space for the Bianchi IX minisuperspace, a simplified gravitational model serving as a proof of concept for black hole spacetimes. Using Faddeev-Popov determinants to fix the CMC gauge, zeta/heat-kernel regularization to handle divergences, and contour deformation in the complex ψ-plane to stabilize the path integral against the DeWitt supermetric's indefiniteness, we derive the measure D µ S mini = (e 3α /τ)dβ + dβ-. Macrostates are defined via the horizon area functional, and the microcanonical volume Ω(A) yields the Bekenstein-Hawking entropy S = A/(4G) with logarithmic corrections in a Euclidean Schwarzschild background. Numerical solutions to the Lichnerowicz-York equation are proposed to validate the embedding map M : S → G , a necessary step to ensure equivalence to GR, though this remains future work. This work establishes Shape Dynamics as a promising framework for deriving black hole entropy in simplified models, with potential extensions to realistic spacetimes. Future numerical and analytical studies, particularly through gravitational wave observations, could probe the predicted logarithmic corrections and distinguish SD from approaches like loop quantum gravity or string theory, offering insights into quantum gravity phenomenology.

This chapter, "Lagrangian mechanics," introduces the new formulation of mechanics that began with the Brachistochrone problem. It starts by providing a mathematical background on manifolds and metric tensors. The chapter then addresses the problem of finding the shortest path between two points on a flat plane (a geodesic) and extends this concept to the Brachistochrone problem, which seeks the path that minimizes the time for an object to roll between two points. It goes on to define a new operator for variations in path and introduces Hamilton's variational principle. The text discusses the Euler-Lagrange equations for mechanical systems and how they lead to the conservation of energy. Finally, the chapter connects the idea of symmetry with conservation laws through Noether's theorem and discusses important symmetries like translational, rotational, and time-translational symmetry, along with the concept of constraints.

This chapter discusses linear algebra concepts essential to quantum mechanics, focusing on the mathematical notation and framework. It reviews vector spaces, bases, and array representation for vectors and operators. The document also defines the inner product, which leads to the concepts of the norm and Hilbert spaces. Finally, it connects these mathematical tools to the fundamental postulates of quantum mechanics, defining observables as linear Hermitian operators and their possible values as eigenvalues, while also explaining the relationship between commuting operators and the Heisenberg Uncertainty Principle using Dirac's bra-ket notation.

This chapter discusses Hamiltonian mechanics, which is a powerful extension of the Lagrangian formalism. This new framework analyzes dynamic systems by shifting the focus from generalized coordinates and velocities to generalized coordinates and conjugate momenta, which together comprise the "phase space". The text explains how Hamiltonian mechanics can simplify equations of motion and provide deeper insight into the structure of a system's phase space.