F. Calogero
We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems…
Various solutions are displayed and analyzed (both analytically and numerically) of arecently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling…
A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot - or, in some cases, only -- periodic solutions. Several examples (ODEs and PDEs) are exhibited.
This paper presents a more complete version than hitherto published of our explanation of a transition from regular to irregular motions and more generally of the nature of a certain kind of deterministic chaos. To this end we introduced a…
We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on…
A new solvable many-body problem of goldfish type is identified and used to revisit the connection among two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated.…
Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion ("acceleration equal force"), with linear and cubic forces, in N-dimensional space (N being an…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
New upper and lower limits are given for the number of S-wave bound states yielded by an attractive (monotonic) potential in the context of the Schrodinger or Klein-Gordon equation.
In a previous paper the \textit{real} evolution of the system of ODEs \ddot{z}_{n} + z_{n}=\sum\limits_{m = 1, m \ne n}^{N} g_{nm}{(z_{n} - z_{m})} ^{- 3}, z_{n} \equiv z_{n}(t), \qquad \dot {z}_{n} \equiv \frac{d z_{n}(t)}{dt}, \qquad n =…
The eigenvalues of the 3 off-diagonal matrices of rank $n$ with elements $1+i cot[(j-k)\pi/n], sin^{-2}[(j-k)\pi/n]$ and $sin^{-4}[(j-k)\pi /n], (j=1,2,...,n, k=1,2,...,n, j\neq k)$ are computed. The sums over $k$ from 1 to $n-1$ of…