# 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} }

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…

## 6 Citations

### Haantjes algebras of classical integrable systems

- MathematicsAnnali di Matematica Pura ed Applicata (1923 -)
- 2021

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

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2019

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

- MathematicsCommunications in Mathematical Physics
- 2021

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

- MathematicsAnnali di Matematica Pura ed Applicata (1923 -)
- 2021

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

- Mathematics
- 2022

. 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…

## References

SHOWING 1-10 OF 30 REFERENCES

### SEPARATION OF SETS OF VARIABLES IN QUANTUM MECHANICS

- Mathematics
- 1964

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

- Mathematics
- 2012

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

- Mathematics
- 2012

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

- Mathematics
- 2006

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

- Mathematics
- 1975

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

- Mathematics
- 1978

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

- Mathematics
- 2011

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

- Mathematics
- 1997

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

- Mathematics
- 1934

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

- Mathematics
- 2011

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…