Related papers: Study of chaos in rotating galaxies using extended…
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 talk provides a critical assessment of collisionless galactic dynamics, focusing on the interpretation and limitations of the collisionless Boltzmann equation and the physical mechanisms associated with collisionless relaxation.…
We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior to the pericenter of the parabolic orbit. The map…
In this work we revisit the geometric approach to chaos in Hamiltonian dynamics, by means of the Jacobi-Levi-Civita equation (JLCE). We inspect numerically two low-dimensional dynamical systems; show that, along chaotic orbits, the…
We show how to systematically apply the Faddeev-Jackiw symplectic method to General Relativity (GR) and to GR extensions. This provides a new coherent frame for Hamiltonian analyses of gravitational theories. The emphasis is on the…
To solve some problems of celestial mechanics and astrophysics, three new models of an elliptical galaxy (EG) have been created, which are in good agreement with modern understanding of the structure of such galaxies. Based on these models,…
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of…
Aims. This paper investigates the chaotic rotation of an oblate satellite in the context of chaos control. Methods. A model of planar oscillations, described with the Beletskii equation, was investigated. The Hamiltonian formalism was…
Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious…
We use so-called geometrical approach in description of transition from regular motion to chaotic in Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems is coexistence of…
Polynomial chaos based methods enable the efficient computation of output variability in the presence of input uncertainty in complex models. Consequently, they have been used extensively for propagating uncertainty through a wide variety…
A new approach is developed to integrate numerically the equations of motion for systems of interacting rigid polyatomic molecules. With the aid of a leapfrog framework, we directly involve principal angular velocities into the integration,…
The symplectic Wisdom-Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in…
We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic…
We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their…
In this note, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [Sakamoto-van~der Schaft, IEEE Transactions on Automatic Control, 2008]. Our algorithm…
Optimization tasks are crucial in statistical machine learning. Recently, there has been great interest in leveraging tools from dynamical systems to derive accelerated and robust optimization methods via suitable discretizations of…
Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the…
Self-gravitating systems of $N$ particles are chaotic. We wonder how chaotic the Galaxy is, and what the consequences are. We therefore simulate the dynamical evolution of a galaxy-scale distribution of point masses in order to measure the…
Symplectic integrators with long-term preservation of integrals of motion are introduced for the guiding-center model of plasma particles in toroidal magnetic fields of general topology. An efficient transformation to canonical coordinates…