7 problems

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…

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

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

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…