86 problems

Let a Hamiltonian system possess a complete algebra of first integrals that provides noncommutative, or superintegrable, integrability. Mishchenko–Fomenko conjecture. Noncommutativ…

Let $V$ be a potential and let $E$ be an energy such that $V^{-1}((-infty,E])$ is homeomorphic to an $n$-dimensional disk. A brake orbit is a pendulum-like periodic solution of the…

Let a $\tau$-symmetric Hamiltonian perturbation of the Riemann--Hopf hierarchy be written in generalized standard form, with first Hamiltonian … Here $\mathcal{P}_k$ denotes the se…

Maximal superintegrability conjecture. The Hamiltonian system with the potentials in equation $$ is maximally superintegrable.

Let $u^i_t+V^i_j({\bf u})u^j_x=0$ be a system of hydrodynamic type that is Hamiltonian with respect to a first-order Hamiltonian operator of hydrodynamic type, and let $V^i_j$ be i…

Flat Darboux coordinate conjecture. These quantities are flat Darboux coordinates for the reduced Poisson structure. The source states that the associated result was explicitly pro…

Two-configuration conjecture. There are only two types of configurations of centers for such a vector field.

Let $H$ be a Hamiltonian of type III, with an integrable Hamiltonian density $h(v,w)$ of the type III form described in the source. Type III trivial-deformation conjecture. Every d…

Kupershmidt's integrability conjecture. The conservation laws $H_1,H_2,\dots$ of the Kupershmidt deformation should commute in an appropriate sense, so that the deformation is inte…

Let ${\cal F}^{(r)}(q,p)$ and ${\cal F}^{(2r-2)}(q,p)$, for $r=3,4,\ldots$, be perturbed harmonic oscillator Hamiltonians with potentials $V^{(r)}$ and $V^{(2r-2)}$ of degrees $r$…

Consider Hamiltonian systems of dimension $2n$ with $n>1$. In the preceding result, Theorem, no general integrators conserve both energy and volume for all Hamiltonians in…

Multiseparability conjecture. Any MS system should separate in multiple coordinate systems.

Let a Hamiltonian pair be obtained by the frozen-point construction from the Lie–Poisson structure associated with the extended $\mathfrak{bms}_3$ algebra. The frozen point is not…

Consider Einstein's field equations in the stationary axially symmetric case. A linear eigenvalue problem means a spectral problem of Lax type whose compatibility condition is the…

Let a conservative Hamiltonian system have an energy surface in its phase space. The ergodic hypothesis. A trajectory of the system eventually covers its energy surface uniformly.…

Consider Hamiltonians that are at most rational in all variables, including the phase-space variables and the independent variable. The Okamoto Hamiltonians form a distinguished cl…

The Okamoto Hamiltonians for the Painlevé equations are polynomial Hamiltonians whose Newton polygons have genus $1$; for the known cases, the areas are $2$ for…

Let $A_1$ be a first-order Hamiltonian operator of Ferapontov type for a first-order WDVV system, compatible with the third-order Hamiltonian operator supplied by the Hamiltonian t…

Symplecticity conjecture. The explicit method $\widetilde{\Phi}_h$ remains symplectic when the two weight vectors satisfy $\lambda=\xi$ and are freely selected.

Let $c$ denote the velocity and $\mathcal{H}$ the Hamiltonian along the tertiary branch of two-crested class $\mathrm{II}$ waves, which bifurcates from the primary branch at…

Let $c$ be the velocity and $\mathcal{H}$ the Hamiltonian along the secondary branch of class $\mathrm{II}$ waves, and let…

Natural Hamiltonian interval conjecture. In the $n$-layered case, this choice of intervals should yield a natural Hamiltonian formulation for the averaged problem. The procedure is…

Polynomiality conjecture. If $P$ and $R$ are compatible, meaning that their Schouten bracket vanishes, $[P,R]=0$, then every entry of $Q^{\alpha\beta}$ is a polynomial of degree tw…

A WDVV-type system is a nonlinear PDE system admitting the same bi-Hamiltonian structure as the WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous…

Jimbo–Miwa–Ueno differential conjecture. One has