We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler--Poincar\'e and Lie--Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan--Schouten connections, this leads to an integro-differential Euler--Poincar\'e equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.
The heavy top is a classic example of Euler-Poincare reduction on a semidirect product group, demonstrating how an advected quantity-the direction of gravity-couples to rotational motion. While the description of relativistic spinning particles via Lie group configurations is well established in Hamiltonian (symplectic...
We explain how the'bar-cobar'construction for homotopy algebras reformulates the equations of motion of arbitrary gauge theories as gauge-covariant quadratic equations for an extended set of fields. The linear term of the new equations encodes the interactions of the original theory, while the quadratic term is univers...
Christoph Chiaffrino, Raji Ashenafi Mamade, Barton Zwiebach· 0 citations
We formulate the inverse problem in information geometry within a two-point tensorial framework and solve it for general metric-affine manifolds, without imposing any curvature or torsion constraints. The construction is explicit and starts directly from the given geometric data: the metric tensor is paired to the affi...
Florio M. Ciaglia, Giuseppe Marmo, M. Pacelli et al.· 0 citations
This book is a self-contained, tutorial introduction to topology, differential geometry, and general relativity for students and researchers in physics. Its distinguishing feature is a Mathematica-based approach that makes abstract constructions concrete through explicit, runnable notebooks and worked examples. A key c...
Balasubramanian Ananthanarayan, Souvik Bera, Subhasis Chakrabarty et al.· 0 citations
In this paper we propose a geometrically motivated formulation of gravity in an invariant Weyl--Integrable space-time (IWIST), in which the background geometry is not imposed a priori but is dynamically determined through the Palatini variational principle. For a scalar field non-minimally coupled to gravity action, th...
J. Aguilar, A. Bernal, M. Montes et al.· 0 citations
We initiate a symmetry descent procedure for M-theory engineered quantum field theories. Starting from a higher form BF theory supplemented by a cubic bulk topological interaction, we construct a gauge-invariant bulk-boundary system whose edge modes acquire generalized Maxwell--Chern--Simons dynamics after the introduc...
