Related papers: Newton's graphical method as a canonical transform…
We consider a minimal fractional deformation of Newtonian gravity characterized by a single parameter $\alpha$. In the limit $\alpha \to 1$, the theory reduces to standard Newtonian gravity. Previous works showed that the $\Lambda$CDM…
Most elementary numerical schemes found useful for solving classical trajectory problems are {\it canonical transformations}. This fact should be make more widely known among teachers of computational physics and Hamiltonian mechanics. From…
Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in…
In this paper Hamiltonian system of time dependent periodic Newton equations is studied. It is shown that for dimensions $3$ and higher the following rigidity results holds true: If all the orbits in a neighborhood of infinity are action…
We extend the canonical formalism for the motion of $N$-particles in lineal gravity to include charges. Under suitable coordinate conditions and boundary conditions the determining equation of the Hamiltonian (a kind of transcendental…
In book II of Newton's "Principia Mathematica" of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of…
We derive an action whose equations of motion contain the Poisson equation of Newtonian gravity. The construction requires a new notion of Newton--Cartan geometry based on an underlying symmetry algebra that differs from the usual Bargmann…
Attempts to merge Einsteinian gravity with Newtonian run into inconsistencies because in Newton's gravity time is absolute and the speed of gravity is infinite. Such an assumption was in a focus of attention of scientists in 19th century…
Using arbitrary symplectic structures and parametrization invariant actions, we develop a formalism, based on Dirac's quantization procedure, that allows us to consider theories with both space-space as well as space-time noncommutativity.…
Hamiltonian formulation of the string with dynamical geometry and two-dimensional gravity with torsion is given. Canonical Hamiltonian equals to the linear combination of first class constraints satisfying closed algebra. It is the…
The Hamilton-Jacobi equation in the sense of Poincar\'e, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction. We illustrate our…
In classical mechanics, we can describe the dynamics of a given system using either the Lagrangian formalism or the Hamiltonian formalism, the choice of either one being determined by whether one wants to deal with a second degree…
Hamilton's principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton's principle has a subtle pitfall that often…
The Bopp's shifts will be generalized through symplectic formalism. A special procedure, like a "diagonalization", which drives the completely deformed symplectic matrix to the standard symplectic form was found as suggested by…
In this paper the theory of time-dependent and time-independent canonical transformations is considered from a geometric perspective. Both the geometric formalism and the coordinate based approach are described in detail. In particular,…
We study the modification of Newton's second law, upto first order in the deformation parameter $a$, in the $\kappa$-space-time. We derive the deformed Hamiltonian, expressed in terms of the commutative phase space variables, describing the…
With the advent of relativistic mechanics, the Lorentz transformation replaced the Galilean transformation based on classical Newtonian mechanics among inertial frames at high uniform velocities, but both transformations are based on…
Since Newton's time, deterministic causality has been considered a crucial prerequisite in any fundamental theory in physics. In contrast, the present work investigates stochastic dynamical models for motion in one spatial dimension, in…
Under natural assumptions on the thermodynamic properties of space and time with the holographic principle we reproduce a MOND-like behaviour of gravity on particular scales of mass and length, where Newtonian gravity requires a…
Canonical formalism of the rank-three tensor model has recently been proposed, in which "local" time is consistently incorporated by a set of first class constraints. By brute-force analysis, this paper shows that there exist only two forms…