Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians

@article{Chanu2018BlockSeparationOV,
  title={Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians},
  author={Claudia M. Chanu and Giovanni Rastelli},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2018}
}
  • C. ChanuG. Rastelli
  • Published 6 August 2018
  • Mathematics
  • Symmetry, Integrability and Geometry: Methods and Applications
We study twisted products $H=\alpha^rH_r$ of natural autonomous Hamiltonians $H_r$, each one depending on a separate set, called here separate $r$-block, of variables. We show that, when the twist functions $\alpha^r$ are a row of the inverse of a block-Stackel matrix, the dynamics of $H$ reduces to the dynamics of the $H_r$, modified by a scalar potential depending only on variables of the corresponding $r$-block. It is a kind of partial separation of variables. We characterize this block… 

Figures from this paper

Haantjes algebras of classical integrable systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

Painleve metrics are a class of Riemannian metrics which generalize the well-known separable metrics of Stackel to the case in which the additive separation of variables for the Hamilton-Jacobi

Higher Haantjes Brackets and Integrability

We propose a new, infinite class of brackets generalizing the Frölicher–Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular,

Haantjes algebras of classical integrable systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or

On the theory of polarization of generalized Nijenhuis torsions

. The theory of generalized Nijenhuis torsions, recently introduced, offers new powerful tools to detect the Frobenius integrability of operator fields on a differentiable manifold. In this work, we

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

  • 2022

References

SHOWING 1-10 OF 30 REFERENCES

SEPARATION OF SETS OF VARIABLES IN QUANTUM MECHANICS

Separation of the Schroedinger equation for molecular dynamics into sets of variables can sometimes be performed when separation into individual variables is neither possible nor, for certain

Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables

Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of

Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization

In previous papers we determined necessary and sufficient conditions for the existence of a class of natural Hamiltonians with non-trivial first integrals of arbitrarily high degree in the momenta.

Complex variables for separation of the Hamilton-Jacobi equation on real pseudo-Riemannian manifolds

In this paper the geometric theory of separation of variables for the time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on

Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation

It was established by Levi‐Civita that in n dimensions there exist n+1 types of coordinate systems in which the Hamilton–Jacobi equation is separable, n of which are in general nonorthogonal; the

Separable coordinates for four-dimensional Riemannian spaces

AbstractWe present a complete list of all separable coordinate systems for the equations $$\sum\limits_{i,j = 1}^4 {g^{ - 1/2} \partial _i (g^{1/2} g^{ij} \partial _j \Phi )} = E\Phi$$ and

Equivalence problem for the orthogonal webs on the 3-sphere

We solve the equivalence problem for the orthogonally separable webs on the 3-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved

Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation

The nonorthogonal separation of variables in the Hamilton–Jacobi equation corresponding to a natural Hamiltonian H=12gijpipj+V, with a metric tensor of any signature, is intrinsically characterized

Separable Systems of Stackel

so that the variables are separable, the solution being of the form 2Xi, where Xi is a function of xi alone. In 18932 he showed that when the quadratic differential form 2H 2dxi so determined is

Three and four-body systems in one dimension: Integrability, superintegrability and discrete symmetries

Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean