Hamiltonian Cubical Tensors

• Hamiltonian cubical tensors are a multilinear extension of Hamiltonian matrices that capture core algebraic and geometric properties in polynomial dynamical systems.
• They leverage the T-product framework and symplectic algebra to generalize spectral theory, normal forms, and stability criteria for higher-order interactions.
• Applications include efficient numerical methods for stability analysis, spectral decomposition, and the construction of polynomial invariants in complex dynamical systems.

Hamiltonian cubical tensors are a multilinear generalization of Hamiltonian matrices, capturing the fundamental algebraic and geometric properties of Hamiltonian systems in the context of higher-order tensors. These structures play a central role in the emerging theory of tensor-based polynomial Hamiltonian systems and in the extension of classical symplectic geometry to the algebra of third-order tensors via the T-product formalism. They enable direct analysis and synthesis of Hamiltonian properties, invariants, and stability criteria for polynomial dynamical systems with fundamentally multilinear interactions.

1. Definitions and Algebraic Characterizations

Let $n$ be even and $J \in \R^{n \times n}$ the canonical symplectic matrix,

$$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$

A $k$th-order $n$-dimensional cubical tensor is a multi-array $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$ (with $k$ modes). Hamiltonian cubical tensors are defined as follows (Cui et al., 27 Mar 2025):

A tensor $A \in \R^{[k, n]}$ is called Hamiltonian cubical (with respect to $J$) if for every nontrivial permutation $\sigma \in S_k$,

$J \in \R^{n \times n}$0

where $J \in \R^{n \times n}$1 denotes the mode permutation of $J \in \R^{n \times n}$2 according to $J \in \R^{n \times n}$3. For $J \in \R^{n \times n}$4 this recovers the classical Hamiltonian matrix condition $J \in \R^{n \times n}$5.

The following are equivalent for $J \in \R^{n \times n}$6:

1. $J \in \R^{n \times n}$7 is Hamiltonian cubical.
2. There exists a supersymmetric tensor $J \in \R^{n \times n}$8 (i.e., $J \in \R^{n \times n}$9 for all $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$0) such that $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$1.
3. The tensor $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$2 is supersymmetric.

This equivalence generalizes the classical result that every Hamiltonian matrix can be written as $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$3 with $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$4 symmetric (Cui et al., 27 Mar 2025).

2. Hamiltonian Cubical Tensors in the T-product Algebra

In the T-product framework, the theory is developed for third-order (“cubical”) tensors $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$5, represented via their block-circulant matricization and manipulated using the circular T-product. The T-product is defined so that $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$6, or equivalently via a diagonalization by discrete Fourier transform (DFT) along the third mode (Lopez-Moreno et al., 20 May 2026).

Within this formalism, the symplectic unit tensor $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$7 is defined so that each of its DFT frontal slices is $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$8. A tensor $$J = \begin{pmatrix} 0 & I_{n/2} \ - I_{n/2} & 0 \end{pmatrix}$$9 is T-Hamiltonian if

$k$0

where $k$1 denotes the T-conjugate transpose.

In the Fourier domain, this equates to each slice $k$2 satisfying the classical Hamiltonian matrix condition: $k$3 or, equivalently,

$k$4

Therefore, T-Hamiltonian tensors in this setting are precisely those whose DFT slices are Hamiltonian matrices (Lopez-Moreno et al., 20 May 2026).

3. Hamiltonian Polynomial Systems and Tensor-based Structure

A polynomial vector field on $k$5 admits the representation: $k$6 where $k$7 are tensors. A polynomial Hamiltonian is similarly expressed as

$k$8

with $k$9 supersymmetric cubical tensors. The multivariate gradient structure of $n$0 yields

$n$1

and the induced Hamiltonian flow is $n$2. The polynomial system is Hamiltonian with Hamiltonian $n$3 if and only if all system tensors $n$4 are Hamiltonian cubical. In this case,

$n$5

This criterion extends the matrix-based test for the Hamiltonian property and allows algorithmic recovery of the Hamiltonian structure from the polynomial flow (Cui et al., 27 Mar 2025).

4. Spectral Theory, T-eigenvalues, and Normal Forms

The spectral properties of Hamiltonian cubical tensors generalize those of classical Hamiltonian matrices. In the T-product algebra, the T-Jordan canonical form shows every third-order tensor is T-similar to a tensor whose DFT slices are in ordinary Jordan form, with diagonal entries termed T-eigenvalues (Lopez-Moreno et al., 20 May 2026).

For T-Hamiltonian tensors, the spectrum is symmetric under $n$6: every T-eigenvalue appears paired with its negative conjugate. For classical Hamiltonian tensors (mode-$n$7 cubical tensors or with $n$8), the sum of H-eigenvalues vanishes and the product equals $n$9, where $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$0 is the associated supersymmetric tensor (Cui et al., 27 Mar 2025).

A key structural result is the T-Williamson normal form: For a real symmetric positive-definite third-order tensor $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$1 ($$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$2 symmetric positive-definite for all $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$3), there exists a T-symplectic tensor $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$4 and T-diagonal $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$5 such that

$$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$6

In the Fourier domain, each $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$7 with $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$8. The construction relies on slice-wise classical Williamson factorization, DFT and inverse DFT (Lopez-Moreno et al., 20 May 2026).

5. Stability, Analytical Criteria, and Numerical Methods

For tensor-based polynomial Hamiltonian systems, Lyapunov stability of an equilibrium $$A \in \R^{[k, n]} = \R^{n \times n \times \cdots \times n}$$9 is determined by the Hessian of the Hamiltonian: $k$0 An equilibrium is stable if this matrix is sign-definite. For $k$1, this reduces to examining $k$2 (Cui et al., 27 Mar 2025).

Numerical computation of the T-Williamson form is efficient: it employs FFT on the third tensor mode, slice-wise classical Williamson factorization (cost per slice $k$3), and inverse FFT. Overall complexity is $k$4 for a tensor of size $k$5 (Lopez-Moreno et al., 20 May 2026). Numerical results confirm that the residuals of the defining relations are at machine precision and the runtime scales linearly in $k$6 and cubically in $k$7.

6. Illustrative Examples and Applications

Hamiltonian cubical tensors provide robust methods to identify and construct Hamiltonian structure in polynomial dynamical systems:

• For a cubic 2D system $k$8, $k$9, system tensors $A \in \R^{[k, n]}$0 can be directly checked to be Hamiltonian cubical, yielding $A \in \R^{[k, n]}$1 (Cui et al., 27 Mar 2025).
• In higher-dimensional examples, recovery of the polynomial Hamiltonian and stability analysis via the tensor-form Hessian are computationally tractable and more efficient than symbolic algebra as $A \in \R^{[k, n]}$2 increases.
• In the T-product framework, applications include Fourier-domain encoding of covariance-matrix families as arise in continuous-variable quantum dynamics, enabled by T-Hamiltonian and T-symplectic tensors (Lopez-Moreno et al., 20 May 2026).

These results link the theory of cubical tensors with polynomial first integrals and generalized Killing tensors, as appear in covariant algorithms for higher-order invariants in nonrelativistic Hamiltonian systems (Cariglia et al., 2014).

7. Connections and Extensions

The classical theory of Hamiltonian matrices is fully recovered as the $A \in \R^{[k, n]}$3 (matrix) case, while for higher-order ($A \in \R^{[k, n]}$4) tensors, Hamiltonian structure is captured by the interplay between symplectic algebra and tensor symmetries (“supersymmetry”). The block structure, spectral symmetry, and normal form results parallel those in symplectic geometry and linear algebra but require fundamentally multilinear generalizations.

Hamiltonian cubical tensors are essential in the analysis of integrability, conservation laws, and the construction of polynomial invariants in both finite and infinite-dimensional Hamiltonian dynamics. They also provide the algebraic backbone for constructive algorithms in numerical and symbolic computation of invariants and normal forms in modern tensor-based approaches to dynamical systems (Cui et al., 27 Mar 2025, Lopez-Moreno et al., 20 May 2026).