Related papers: Neural Canonical Transformation with Symplectic Fl…
Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameterisations, but also generate the dynamics of the system. In the first part…
We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The…
This paper is focused on the development of the notions of canonical and canonoid transformations within the framework of Hamiltonian Mechanics on locally conformal symplectic manifolds. Both, time-independent and time-dependent dynamics…
Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and…
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a…
Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose,…
This work builds on an existing model of discrete canonical evolution and applies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $ and linear…
Integrable systems have provided various insights into physical phenomena and mathematics. The way of constructing many-body integrable systems is limited to few ansatzes for the Lax pair, except for highly inventive findings of conserved…
Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the…
Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic…
We observe that, within the effective generating function formalism for the implementation of canonical transformations within wave mechanics, non-trivial canonical transformations which leave invariant the form of the Hamilton function of…
The traditional method of teaching canonical transformations involves the introduction of generating functions of various types. This method obscures the underlying structure of the Hamiltonian least-action principle, and can make a…
We show that, when applied to any non-canonical Hamiltonian system, any integrator that is symplectic for canonical Hamiltonian problems is actually conjugate symplectic for the non-canonical structure. This result is useful because it…
In this paper we present canonical and canonoid transformations considered as global geometrical objects for Hamiltonian systems. Under the mathematical formalisms of symplectic, cosymplectic, contact and cocontact geometry, the canonoid…
The solution of problems in physics is often facilitated by a change of variables. In this work we present neural transformations to learn symmetries of Hamiltonian mechanical systems. Maintaining the Hamiltonian structure requires novel…
Introducing a Clebsch-like parameterization, we have formulated a canonical Hamiltonian system on a symplectic leaf of reduced magnetohydrodynamics. An interesting structure of the equations is in that the Lorentz-force, which is a…
Lithium is a typical quantum solid, characterized by cubic structures at ambient pressure. As the pressure increases, it forms more complex structures and undergoes a metal-to-semiconductor transformation, complicating theoretical and…
The problem of diagonalization of Hamiltonians of N-dimensional boson systems by means of time-dependent canonical transformations (CT) is considered, the case of quadratic Hamiltonians being treated in greater detail. The unitary generator…
Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable…
The article deals with the subject of solving the problem of canonical-map synthesis for Hamiltonian systems. For this purpose, the controlling-function method has been developed that allows appropriate changes of the variables in terms of…