Related papers: Pure shape dynamics: General framework
We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm.…
We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two…
We provide the construction of a de Broglie-Bohm model of the $N$-body system within the framework of Pure Shape Dynamics. The equation of state of the curve in shape space is worked out, with the instantaneous shape being guided by a wave…
We demonstrate that a system of self-propelled particles (SPP) exhibits spontaneous symmetry breaking and self-organization in one dimension, in contrast with previous analytical predictions. To explain this surprising result we derive a…
Classical field theories together with the Lagrangian and Eulerian approaches to continuum mechanics are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, space…
Humans easily recognize object parts and their hierarchical structure by watching how they move; they can then predict how each part moves in the future. In this paper, we propose a novel formulation that simultaneously learns a…
We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be…
The basic problem of shape complementarity analysis appears fundamental to applications as diverse as mechanical design, assembly automation, robot motion planning, micro- and nano-fabrication, protein-ligand binding, and rational drug…
Physics relies on mathematical spaces carefully matched to the phenomena under study. Phase space in classical mechanics, Hilbert space in quantum theory, configuration spaces in field theory all provide representations in which physical…
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits…
The equivalence class of absolute configurations of a system under the group of similarity transformations $Sim(3)$ is called the shape of the system. The $Sim(3)$ invariant Lagrangian of the modified Newtonian theory ensures the existence…
Swimming in curved spacetimes is a phenomenon whereby free bodies in curved spacetimes are able to propel themselves by performing cyclic internal motions. When originally proposed, it was further suggested that, in the limit of fast…
The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to…
A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism…
We study how the spin-statistics theorem relates to the geometric structures on phase space that are introduced in quantisation procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the…
We consider the possibility that the basic space of physics is not spacetime, but configuration space. We illustrate this on the example with a system of gravitationally interacting point particles. It turns out that such system can be…
The manifold of pure quantum states is a complex projective space endowed with the unitary-invariant geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given…
We remind how relationality arises as the core insight of general-relativistic gauge field theories from the articulation of the generalised hole and point-coincidence arguments. Hence, a compelling case for a manifestly relational…
It is shown that space-time may be not only in a state which is described by Riemann geometry but also in states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase…