Gilberto Silva-Ortigoza

We pose and solve a classical inverse problem suitable for advanced mechanics courses. Given a two-dimensional orthogonal coordinate system in the plane, we ask how to construct a scalar potential such that, on a chosen energy shell, one set of coordinate curves becomes an exact set of mechanical trajectories. More precisely, if a particle is launched tangent to one of the prescribed curves with the chosen energy, then the resulting natural motion remains on that curve. Working directly with the scale factors of the coordinate system, we obtain a simple closed-form expression for the potential in terms of the scale factor along the prescribed direction of motion. The construction is elementary, constructive, and visual. In parabolic coordinates, imposing central symmetry forces the remaining free function to be constant and recovers an inverse-radius potential of Kepler form. In polar and confocal elliptic–hyperbolic coordinates, the same method produces inverse-square-type and more exotic potentials tied to the geometry of the coordinates. A short geometric coda, based on the Jacobi metric and geodesic curvature, recasts the same condition in coordinate-free form. We conclude with concrete suggestions for classroom use, including analytic derivations, numerical projects, and links to the optics–mechanics analogy.

The paraxial wave equation is formally equivalent to the Schrödinger equation, enabling quantum-mechanical concepts to be applied to classical optical fields. Within this framework, we investigate the Bohmian dynamics of superpositions of Laguerre–Gaussian beams. We derive the velocity field and quantum potential and analyze the resulting trajectory dynamics. Interference between modes with different orbital angular momentum generates vortex formation, trajectory exclusion from nodal regions, and spiral motion around phase singularities. Remarkably, despite the complexity of the Bohmian trajectories, the vortex cores follow nearly linear paths along propagation. The dynamics are governed by the amplitude ratio and OAM indices, while the relative phase only induces global rearrangements. These results establish a coherent trajectory-based description of structured light and highlight vortex cores as robust topological features.

In this work, we show that the Madelung–Bohm trajectories determined by any solution, Ψ( ξ , ζ ), of the paraxial wave equation in an arbitrary two-dimensional optical medium, where ζ is the propagation coordinate and ξ the transversal one, are given by the contours of the function G ( ξ , ζ )≡∫|Ψ( ξ , ζ )| ² d ξ . Furthermore, we remark that an analogous result follows directly for the one-dimensional Schrödinger equation for a particle evolving in an arbitrary classical potential, where ξ gives the position of the particle at time ζ . We apply this result to several examples in free and quadratic optical media. They include the Airy beam, Hermite–Gaussian beams, and some of their superpositions. Additionally, this work provides an alternative derivation of Hermite–Gaussian (HG) beams in free space by employing the quantum potential equations and the framework of quantum trajectories. By analyzing known trajectories, we demonstrate that the fundamental parameters characterizing these beams—the Gouy phase, beam radius, and wavefront curvature—emerge naturally within the quantum potential formalism.

We study classical motion in the inverse-square cylindrical potential U ( ρ ) ∝ − 1/ ρ ² . After the two-body reduction, a single dimensionless parameter η —combining interaction strength and axial angular momentum—fully sets the dynamics. This splits the motion into three regimes. For η < 1 the motion is unbound: trajectories have a unique closest approach and an analytic deflection that depends only on the impact parameter. At the threshold η = 1 the effective potential vanishes: generic paths are scale-invariant spirals with constant radial speed, while the special case E = 0 gives neutrally stable circles. For η > 1 the motion is ‘fall to the centre’ in finite time: for E < 0 there is a single turning point (an apocentre) followed by plunge, whereas for E > 0 capture occurs from infinity without any turning point.

In the present work we use a point transformation defined in the extended configuration space to show that, in both classical and quantum formalisms, the evolution of a particle under the influence of a constant gravitational field can be related to the evolution of the harmonic oscillator. That is, in the classical mechanics regime, we show that this transformation allows us to establish the connection in the Newtonian, Lagrangian, Hamilton-Jacobi and Hamiltonian formalisms. In the quantum mechanics regime, the transformation determines a relationship between the wave functions, solutions to the corresponding Schrödinger equations, the propagators and the Wigner functions. Furthermore, by using the quantum potential approach to quantum mechanics, we show that the classical point transformation, in a natural manner, induces a quantum point transformation in the corresponding extended quantum configuration space, that determines the connection between the corresponding Madelung-Bohm trajectories. We find that the quantum transformation has the same functional form as the classical one even though the quantum or Madelung-Bohm trajectories are in general totally different from the classical ones. Then, we show that the general results can also be applied to the corresponding paraxial wave equations, and by using the stationary solutions of the quantum Harmonic oscillator and the Airy solution obtained by Berry and Balazs, as two applications, we determine the evolution of the Hermite-Gaussian and Airy beams in free space, in a linear and in a quadratic optical media. Finally, by using the point transformation, we construct two metrics whose geodesics correspond precisely to the classical trajectories for the harmonic oscillator and a particle in a uniform gravitational field.

In this work, we use the quantum potential approach to quantum mechanics to show that the Madelung–Bohm trajectories for a particle in a constant gravitational field can be related to those of a free particle by means of a quantum point transformation defined in the extended configuration space. We find that the point transformation also gives a connection between the corresponding quantum Hamiltonians determined by the solutions of the corresponding Schrödinger equations for these two problems. We show that the Madelung–Bohm trajectories determined by the stationary solutions to the Schrödinger equation for the particle in a constant gravitational field are straight ones, while the corresponding ones for the free particle are parabolic trajectories. The Airy beam is one example of this type of solution. We study the properties of a solution to the Schrödinger equation for a free particle with phase singularities (zeroes), and we find that the corresponding solution to the Schrödinger equation for the particle in a constant gravitational field also has zeroes at the same spacetime points. However, the Madelung–Bohm trajectories determined by the two solutions are totally different. Furthermore, we remark that similar results can be directly obtained for the paraxial wave equation.