Related papers: Discrete dynamics versus analytic dynamics

A recent article in J. Chem. Phys. argues that the two algorithms, the velocity-Verlet, and position-Verlet integrators, commonly used in Molecular Dynamics (MD) simulations, are different \cite{Ni2024}. But not only are the two algorithms…

Computer simulation of the time evolution in a classical system is a standard numerical method, used in numerous scientific articles in Natural Science. Almost all the simulations are performed by discrete Molecular Dynamics (MD). The…

Almost all Molecular Dynamics (MD) simulations are discrete dynamics with Newton's algorithm first published in 1687, and much later by L. Verlet in 1967. Discrete Newtonian dynamics has the same qualities as Newton's classical analytic…

In 1687 Isaac Newton published PHILOSOPHI\AE \ NATURALIS PRINCIPIA MATHEMATICA, where the classical analytic dynamics was formulated. But Newton also formulated a discrete dynamics, which is the central difference algorithm, known as the…

Simulations of objects with classical dynamics are in fact a particular version of discrete dynamics since almost all the classical dynamics simulations in natural science are performed with the use of the simple ''Leapfrog" or ''Verlet"…

Some problems on variations are raised for classical discrete mechanics and field theory and the difference variational approach with variable step-length is proposed motivated by Lee's approach to discrete mechanics and the difference…

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This…

Modifying the discrete mechanics proposed by T.D. Lee, we construct a class of discrete classical Hamiltonian systems, in which time is one of the dynamical variables. This includes a toy model of time machines which can travel forward and…

We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the…

We study the classical dynamics of a particle in Snyder spacetime, adopting the formalism of constrained Hamiltonian systems introduced by Dirac. We show that the motion of a particle in a scalar potential is deformed with respect to…

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal…

Variational wave functions and Green's functions are two important paradigms for solving quantum Hamiltonians, each having their own advantages. Here we detail the Variational Discrete Action Theory (VDAT), which exploits the advantages of…

A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie…

Here we propose the Variational Discrete Action Theory (VDAT) to study the ground state properties of quantum many-body Hamiltonians. VDAT is a variational theory based on the sequential product density matrix (SPD) ansatz, characterized by…

Students encounter harmonic-oscillator models in many aspects of basic physics, within widely-varying theoretical contexts. Here we highlight the interconnections and varying points of view. We start with the classical mechanics of masses…

This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description…

We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in…

Most classical mechanical systems are based on dynamical variables whose values are real numbers. Energy conservation is then guaranteed if the dynamical equations are phrased in terms of a Hamiltonian function, which then leads to…

Often the analysis of time-dependent chemical and biophysical systems produces high-dimensional time-series data for which it can be difficult to interpret which individual features are most salient. While recent work from our group and…

Molecular dynamics is based on solving Newton's equations for many-particle systems that evolve along complex, highly fluctuating trajectories. The orbital instability and short-time complexity of Newtonian orbits is in sharp contrast to…