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Recently Abrahams and Cook devised a method of estimating the total radiated energy resulting from collisions of distant black holes by applying Newtonian evolution to the holes up to the point where a common apparent horizon forms around…
The mathematical theory underlying Hamiltonian mechanics is called symplectic geometry. So symplectic geometry arose from the roots of mechanics and is seen as one of the most valuable links between physics and mathematics today. Symplectic…
Within the context of Newton's theory of gravitation, restricted to point-like test particles and central bodies, stable circular orbits in ordinary space are related to stable circular paths on a massless, unmovable, undeformable…
The geometric foundations of General Relativity are revisited, with particular attention to its gauge invariance, as a key to understanding the true nature of spacetime. Beyond the common image of spacetime as a deformable 'fabric' filling…
Newton's Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently. He figured out what kind of central force exerted…
This paper investigates the classical and quantum elementary systems with Newton-Hoooke symmetry. A complete classification is given by explicit computation. In addition, we present an application example of quantization using the Moyal…
This work concerns a study of the quantum mechanical extension of the work of Horwitz et al. [1] on the stability of classical Hamiltonian systems by geometrical methods. Simulations are carried out for several important examples, these…
The rarely used Hamilton-Jacobi equation has been utilized as an elegant way to find the trajectories of mechanical systems and to derive symplectic maps. Further, the exact solution in kick approximation of Hamilton's equations of motion…
Newton's theorem of revolving orbits states that one can multiply the angular speed of a Keplerian orbit by a factor $k$ by applying a radial inverse cubed force proportional to $(1-k^2)$. In this paper we derive an extension of this…
Aristotle has split physics at the sphere of the moon; above this sphere there is no change except eternal spherical motion, below are two different kinds of motion: Natural motion (without specific cause) and enforced motion. In modern…
In 1834-1835, Hamilton published two papers that revolutionized classical mechanics. In these papers, he introduced the Hamilton-Jacobi equation, Hamilton's equations of motion and the principle of least action. These three formulations of…
Newton's method is the most widespread high-order method, demanding the gradient and the Hessian of the objective function. However, one of the main disadvantages of Newtons method is its lack of global convergence and high iteration cost.…
We study the behaviour of Yukawa and Newtonian gravitational forces in a cubic box with fully periodic boundaries commonly encountered in N-body simulations of the structure formation. Placing a single gravitating body at the origin of…
We derive the dynamics of several rigid bodies of arbitrary shape in a 2-dimensional inviscid and incompressible fluid, whose vorticity field is given by point vortices. We adopt the idea of Vankerschaver et al. (2009) to derive the…
It is widely acknowledged that the Galilean Relativity Principle, according to which the laws of classical systems are the same in all inertial frames in relative motion, has played an important role in the development of modern physics. It…
This paper expounds the modern theory of symplectic reduction in finite-dimensional Hamiltonian mechanics. This theory generalizes the well-known connection between continuous symmetries and conserved quantities, i.e. Noether's theorem. It…
The equations of motion for a Lagrangian mainly refer to the acceleration equations, which can be obtained by the Euler--Lagrange equations. In the post-Newtonian Lagrangian form of general relativity, the Lagrangian systems can only…
The relationships between port-Hamiltonian systems modeling and the notion of monotonicity are explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to maximal cyclically monotone relations, together…
Reliable studies of the long-term dynamics of planetary systems require numerical integrators that are accurate and fast. The challenge is often formidable because the chaotic nature of many systems requires relative numerical error bounds…
It is known that Einstein's conceptual base for his theory of relativity was the philosophy formulated by Immanuel Kant. Things appear differently to observers in different frames. However, Kant's Ding-an-Sich leads to the existence of the…