Related papers: Safe domain and elementary geometry
We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems; a natural setting for this is a generalized version of convex toric domain which we also study, where there…
One-dimensional motion of Sommerfeld sphere in the case of potential barrier is numerically investigated. The effect of classical tunneling is found out - Sommerfeld sphere overcomes the barrier and finds itself in the forbidden, from…
A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account. The quadratic law for the resistance force is used. An analytic approach applies for the…
Discrete symmetries are commonplace in field theoretical models but pose a severe problem for cosmology since they lead to the formation of domain walls during spontaneous symmetry breaking in the early universe. However if one of the…
Geometrical formulation of classical mechanics with forces that are not necessarily potential-generated is presented. It is shown that a natural geometrical "playground" for a mechanical system of point particles lacking Lagrangian and/or…
Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic…
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle…
A simple position probability density formulation is presented for the motion of a particle in a spherically symmetric potential. The approach provides an alternative to Newtonian methods for presentation in an elementary course, and…
The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of the two other bodies, the primaries, which evolve in Keplerian ellipses. A trajectory is called oscillatory if…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
The nature of boundedness of orbits of a particle moving in a central force field is investigated. General conditions for circular orbits and their stability are discussed. In a bounded central field orbit, a particle moves clockwise or…
Assuming the hoop conjecture in classical general relativity and quantum mechanics, any observer who attempts to perform an experiment in an arbitrarily small region will be stymied by the formation of a black hole within the spatial domain…
In this study, we examine the domain wall within the framework of a cosmological harmonic oscillator. We investigate the interaction between the domain wall and a periodic background field, which can induce perturbations in the oscillatory…
To solve the quantum-mechanical problem the procedure of mapping onto linear space $W$ of generators of the (sub)group violated by given classical trajectory is formulated. The formalism is illustrated by the plane H-atom model. The problem…
Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical…
The effective classical/quantum dynamics of a particle constrained on a closed line embedded in a higher dimensional configuration space is analyzed. By considering explicit examples it is shown how different reduction mechanisms produce…
Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate…
Harmonic oscillator and the Kepler problem are superintegrable systems which admit more integrals of motion than degrees of freedom and all these integrals are polynomials in momenta. We present superintegrable deformations of the…
The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of…