27 problems

Symplectic camel conjecture. The property of the symplectic camel is the key to a better understanding of not only quantum mechanics, but also of classical phenomena.

Converse integrability conjecture. Every integrable spin system is H-integrable.

Conjecture. If $H$ is of Heisenberg type, then

For each manifold $M$ and each simple mechanical system with kinetic energy $T$ and potential energy $V$, let $\psi(M,T,V)$ be the homomorphism from the universal Lie algebra…

Generalization conjecture. Similar results are valid for the $n$-dimensional generalization of the Kovalevskaya top given in [4].

For a rotationally invariant scalar potential, the conserved angular momentum implies that the lowest-order bosonic motion takes place in a plane. Orthogonal-direction reduction co…

Equipartition duality. Among all such states, the phase-equipartitioned product state, unique when it exists, simultaneously (i) maximizes the entropy, (ii) minimizes every within-…

Calogero's conjecture. The classical Calogero–Moser system is completely integrable.

Let $a_i$ and $F_i$ denote the acceleration and impressed force associated with the $i$-th Cartesian degree of freedom of a mechanical system, with masses $m_i$, and define Gauss's…

Let $N$ be the order of a superintegrable system, let $L_z$ denote the angular momentum, and let $p_1,p_2$ be the Cartesian momenta. Let $V_1$ and $V_2$ be functions of the Cartesi…

A Hooke billiard is a mechanical billiard in $\mathbb{R}^2$ with Hooke potential $U=f r^2$, where $r$ is the distance from a fixed center. Let its reflection wall be smooth and con…

Tree-representation injectivity conjecture. The map $\Theta$ is injective.

Euclidean kernel conjecture. Consider all smooth potentials $V\in C^{\infty}(\mathbb{R}^d)$ with arbitrary $d$. Then

Euclidean freeness conjecture. The class of systems with Euclidean kinetic energy metrics is already free: the only linear identities satisfied by the Lie brackets of all such syst…

The Newton duality is a correspondence between dynamical systems associated with potentials, generalizing the original duality for power-law potentials. Universality conjecture for…

Let $x_i(t)\in\mathbb R^3$, $i=1,\ldots,n$, be the positions of $n$ point charges evolving under the repulsive Coulomb $n$-body equations, with all charges having the same sign. La…

One-to-one correspondence conjecture. A similar argument should show a one-to-one correspondence between $\Sigma_{x_i,x_f}$ and an appropriate subset of $\mathbb{Z}^2/\sim$ determi…

A mechanical system is described by constraints that may be nonlinear in the velocity. Existence conjecture. The existence of mechanical systems with nonlinear constraints in the v…

Consider the two-dimensional Tremblay–Turbiner–Winternitz Hamiltonian … with $0\leqslant\theta\leqslant\frac{\pi}{2k}$, $0\leqslant r<\infty$, angular frequency $\omega$, nonnegati…

Asymptotic restoration conjecture. The validity of the no-interaction theorem could be restored by introducing a suitable asymptotic approximation for the $N$-body system dynamics.

Constrained-dynamics conjecture. The limitations imposed by the Currie no-interaction theorem might be avoided by formulating the dynamics with constrained dynamics in a super-abun…

Let $E$ be a complex energy, and consider the tunneling probabilities of classical particles with complex energy and the corresponding quantum tunneling probabilities. Bender–Hook…

Let $\epsilon$ be the parameter governing the classical trajectory in the complex plane, and let the trajectory's turning points be the points at which it is deflected. Turning-poi…

Shift-vector conjecture. $Q^\mu$ is a LRL vector for arbitrarily large, composite relativistic systems with arbitrary internal interactions.

Super-integrability conjecture. The system is super-integrable for arbitrary integer $n$. For small integer values of $n$, the corresponding quantum system is known to be super-int…