Newton’s laws of motion in the form of a Riccati equation

Abstract

We discuss two applications of Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential V (r) = kr ǫ . For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.

A Riccati model for Denmark Strait overflow variability

“…(1) is generally solved numerically for arbitrary initial conditions and source terms. This equation is known in many disciplines of physics [e.g., Nowakowski and Rosu, 2002] and has analytical solutions for a variety of forcing functions [e.g., Polyanin and Zaitsev, 2003]. An analytical solution exists for switching on a constant source at time t = 0.…”

Few sketches on connections between the Riccati and Ermakov–Milne–Pinney equations

“…Interestingly, the Riccati transformation (7) is rather exclusive in the following meaning. Consider (2) and suggest the generalized Riccati transformation that partly resembles the transformation that was applied to the general Riccati equation (1) in 17. Substituting y ( x ) and y ′( x ) derived from (8) into (2), we obtain By means of ψ = φ e , (9) is converted into φ″ + c φ = 0, which is trivially transformed to (2), simply derived by applying the Riccati transformation (7).…”

“…The Schrödinger equation that describes a quantum particle of mass m moving in the one‐dimensional potential U ( x ) is the second‐order linear differential equation 15 The Riccati differential equation, either (1) or (2), is widely used in many areas of quantum mechanics (see e.g., Refs. 16–24 for current works): viz., in the WKB approximation 19, supersymmetric (SUSY) quantum mechanics 20, 21, and quantum chemistry (see Refs. 22–24 and references therein).…”

Unusual properties of some localized states and the quantum-classical correspondence

“…In the first case, having exact solutions of the Schrödinger equation, we can study interrelations between the classical and quantum mechanics and its classical limit in the most interesting region of energies. On the other hand, the E = 0 states can still have and have already found various applications, for example, in the cold-atom collisions [15,16], in the construction of some vortex lattices [4], in the description of some modes in the AharonovBohm solenoids [17] and even in quantum cosmology [18].…”