Related papers: Lambert's theorem and projective dynamics
Lambert's theorem (1761) on the elapsed time along a Keplerian arc drew the attention of several prestigious mathematicians. In particular, they tried to give simple and transparent proofs of it (see our timeline \S 9). We give two new…
We give simple proofs of some simple statements concerning the Lambert problem. We first restate and reprove the known existence and uniqueness results for the Keplerian arc. We also prove in some cases that the elapsed time is a convex…
We conjecture a time dependent Lambda(t), in terms of the Gaussian curvature of the causal horizon, which is nonvanishing even in Minkowski space due to the lack of informations beyond the light cone. Using the Heisenberg Principle, the…
We generalize the curved $N$-body problem to spheres and hyperbolic spheres whose curvature $\kappa$ varies in time. Unlike in the particular case when the curvature is constant, the equations of motion are non-autonomous. We first briefly…
We present algebraic derivation of the result of Schr\"{o}dinger [1] for the spectrum of hydrogen atom in the space with constant curvature.
In this paper we clarify and generalise previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as…
The orbital boundary value problem, also known as Lambert Problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to…
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…
Lambert's problem is a classical boundary value problem in analytical mechanics. It arises when trying to determine the energy required to place a particle, subject to a central gravitational potential, in a "free fall" trajectory…
The Lambert problem is to determine the gravitational orbit between two points that has a specified time of flight, allowing the second point to be a moving target such as a satellite. After a review of gravitational orbits, a solution of…
After some more than four centuries from the formulation and publication (in Astronomia Nova) of the Kepler's Equation, which relates the eccentric (and, intermediately, the true) anomaly of the planetary trajectories to the uniformly…
Under broad conditions, we prove that the probability amplitudes in the quantum mechanics are either always constant in time or changing continuously in any interval of time.
Perlick's classification of (3+1)-dimensional spherically symmetric and static spacetimes (\cal M,\eta=-1/V dt^2+g) for which the classical Bertrand theorem holds [Perlick V Class. Quantum Grav. 9 (1992) 1009] is revisited. For any Bertrand…
We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type…
We study the motion of a particle in a 3-dimensional lattice in the presence of a Coulomb potential, but we demonstrate semiclassicaly that the trajectories will always remain in a plane which can be taken as a rectangular lattice. The…
Reasonable parametrizations of the current Hubble data set of the expansion rate of our homogeneous and isotropic universe, after suitable smoothing of these data, strongly suggests that the area of the apparent horizon increases…
The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating…
The Lambert problem consists in connecting two given points in a given lapse of time under the gravitational influence of a fixed center. While this problem is very classical, we are concerned here with situations where friction forces act…
Quantum mechanics rests on the assumption that time is a classical variable. As such, classical time is assumed to be measurable with infinite accuracy. However, all real clocks are subject to quantum fluctuations, which leads to the…
This work addresses the Hamiltonian dynamics of the Kepler problem in a deformed phase space, by considering the equatorial orbit. The recursion operators are constructed and used to compute the integrals of motion. The same investigation…