arXiv research

Abstract: Counterexamples found for lifting Hamiltonian and contact isotopies.

The Eisenhart lift connects Hamiltonian systems to geodesics in pp-wave spacetimes.

Defines lift of partial cohomological field theories and finds new bi-Hamiltonian structures.

Geometrically reduces Hamiltonian systems using particular integrals.

Defines new canonical lifts for field theories, analyzing Klein-Gordon, Polyakov string, and Einstein-Cartan gravity.

Unified approach to constructing integrable systems using Stäckel lifts.

The paper provides a geometric framework for understanding non-equilibrium thermodynamics.

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the actio…

The paper extends Vlasov kinetic theory to time-dependent dynamics using cosymplectic and cocontact manifolds.

Starting from a Lie algebroid ${\cal A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle ${\cal R}$ over the cotangent bundle $T^*V$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is the Hamiltonian algebroid ${\cal A}^H$ over $…

Given a submersion $π:Q \to M$ with an Ehresmann connection $\mathcal{H}$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a gener…

This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to a…

New integrators preserve geometric structure in Hamiltonian systems.

Starting with a Lie algebroid ${\cal A}$ over a space $M$ we lift its action to the canonical transformations on the affine bundle ${\cal R}$ over the cotangent bundle $T^*M$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is a Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ …

Develops integrators for contact Hamiltonian systems preserving geometric structure.

Lie algebroids relate to constrained systems and their BFV/BFV formulation.

Study $G_2$-manifolds from symplectic $SU(3)$-manifolds with $T^2$-symmetry.

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

Starting from a bundle E over R, the dual of the first jet bundle, which is a co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1,1) tensor…

Study shows bi-Hamiltonian structure for polygon evolutions in centro-affine space.

This work improves convergence guarantees for unadjusted HMC in KL and Rényi divergences.

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian act…

Constructs a weak Poisson bracket and lifts it to differential forms.

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…

Symmetries of bundle gerbes modeled using multiplicative vector fields.

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let $L\to X$ be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class $c\sb 1…

We give the characterization of Arnol'd-Mather type for stable singular Legendre immersions. The most important building block of the theory is providing a module structure on the space of infinitesimal integral deformations by means of the notion of natural liftings of differential systems and of contact Hamiltonian v…

Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.

Explains how Reeb dynamics connects to topology.

Paper bridges quantum and classical mechanics for open systems.

Generalizations of Conway-Gordon theorems for complete graphs with new key results.

Quantum propagation studied for Berezin-Toeplitz operators.

Geometrically transforms nonconservative dynamics to linearize Kepler and Manev systems.

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculu…

The paper explores scaling symmetries in symplectic geometry and their applications to central configurations.

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $α$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions …

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a des…

An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. Specialising to 4-dimensional target, we find that surf…

Develops a new deformation theory for Dirac structures.

Unified geometric framework for integrability of conservative and dissipative systems.

Contact reductions explained through symplectic reductions.

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the B…

The phase space of relativistic particle mechanics is defined as the 1st jet space of motions regarded as timelike 1-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally on the odd-dimensional phase space a generalized contact structure. In the paper infinitesimal sy…

The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M de…

Develops a gradient flow for Muon optimizer, a method for optimization.

New framework models non-conservative stochastic processes without energy conservation constraints.

This article announces a series of articles aiming at introducing the concept of symplectic spinors into symplectic topology resp. the concept of Frobenius structures. We will give lower bounds for the number of fixed points of a Hamiltonian diffeomorphism on the cotangent bundle over a compact manifold $M$ by defining…