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Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation

This paper introduces the Stäckel lift as a unified geometric framework extending Eisenhart lifts to construct new classes of integrable Hamiltonian systems, demonstrating their connection to non-trivial symplectic-Haantjes structures, magnetic systems in cylindrical coordinates, and momentum-dependent metrics relevant to modified gravity and Finsler geometries.

Original authors: Ondřej Kubů, Piergiulio Tempesta

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ondřej Kubů, Piergiulio Tempesta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a complex, moving machine—a clockwork toy, perhaps—that you want to understand better. In physics, these machines are called "dynamical systems," and they are often described by equations that tell us how they move over time. Sometimes, these equations are incredibly hard to solve.

This paper introduces a clever new "trick" to make these difficult machines easier to study. The authors call this trick the Stäckel Lift.

Here is the breakdown of what they did, using simple analogies:

1. The Old Trick: The Eisenhart Lift

For a long time, physicists used a method called the Eisenhart Lift.

The Analogy: Imagine you are watching a ball roll across a bumpy floor (a 2D surface). It's hard to predict exactly where it will go because of the bumps (gravity/potential).
The Trick: The Eisenhart method says, "Let's pretend the ball is actually rolling on a giant, curved slide in a 3D room." By adding an extra dimension (a new coordinate), the "bumps" on the floor turn into the natural slope of the slide. The ball now just rolls in a straight line (a geodesic) on this new, higher-dimensional slide.
The Result: The complicated motion on the floor becomes a simple, straight-line motion in the 3D room. This makes the math much easier.

2. The New Trick: The Stäckel Lift

The authors of this paper say, "The Eisenhart trick is great, but it's a bit rigid. We can make it much more flexible." They created the Stäckel Lift.

The Analogy: Think of the old Eisenhart method as a specific type of elevator that only goes up one floor. The new Stäckel Lift is like a modular construction kit. You can take your original machine (the 2D ball), and instead of just adding one floor, you can add as many new "rooms" (dimensions) as you want.
How it works: They use a mathematical tool called a "Stäckel matrix" (think of it as a blueprint or a recipe card). They take the blueprint for the original machine and add a new row and column to it. This new blueprint allows them to build a bigger, more complex machine that is still perfectly predictable (integrable).
The Magic: Even though the new machine is bigger and more complex, it keeps the "separability" of the original. This means you can still solve the puzzle by breaking it into smaller, independent pieces, just like you could with the original small machine.

3. The Hidden Geometry: Haantjes Tensors

The paper also discovers something fascinating about the "shape" of these new machines.

The Analogy: Imagine the machine has a hidden internal skeleton that keeps it from falling apart. In math, this skeleton is called a Haantjes structure.
The Discovery: The authors prove that when they use their new Stäckel Lift, this hidden skeleton appears automatically.
The Twist: In the old Eisenhart method, this skeleton was simple and didn't change based on how fast the parts were moving. But with the new Stäckel Lift, the skeleton changes shape depending on the speed (momentum) of the machine. It's like a suit of armor that reshapes itself based on how fast you are running. This is a brand-new type of geometric structure that the authors have identified.

4. What Can You Build With This?

The authors show that this "construction kit" can build several interesting things:

Gravitational Waves (Platonic Waves): They can build models of spacetime that look like ripples in a pond (gravitational waves). Specifically, they create "Platonic waves," which are a special kind of ripple where the wave fronts are flat, but the space itself is curved. These are useful for testing theories about gravity that go beyond Einstein's original ideas.
Magnetic Systems: They show how to use this method to understand charged particles moving in magnetic fields, specifically those trapped in cylindrical shapes (like a tube). They found a way to organize the math for these systems that was previously missing.
Exotic Systems: They can even build systems where the rules of motion depend on the speed in weird, non-standard ways (like having terms with $e^p$ in the equations), creating "transcendental" systems that were hard to study before.

Summary

In short, the authors have invented a universal adapter for physics problems.

Input: A difficult, solvable physics problem.
Process: Apply the Stäckel Lift (add dimensions using a flexible blueprint).
Output: A bigger, more complex problem that is still solvable, often revealing new geometric shapes (like gravitational waves) or new types of mathematical structures (momentum-dependent skeletons) that were hidden before.

They aren't just solving one problem; they are providing a factory to generate infinite new, solvable problems, including models for gravity and magnetic fields, all while uncovering a new layer of geometric beauty in how these systems are structured.

Technical Summary: Stäckel and Eisenhart Lifts, Haantjes Geometry and Gravitation

Problem Statement
The paper addresses the geometrization of dynamics, a program inaugurated by Eisenhart in 1928, which describes mechanical systems as geodesic flows in higher-dimensional spaces. While the classical Eisenhart lift (both Riemannian and Lorentzian versions) successfully embeds mechanical systems with scalar and vector potentials into higher-dimensional pseudo-Riemannian manifolds, the authors seek a more unified and flexible framework. Specifically, they aim to:

Generalize the Eisenhart lift to construct new classes of integrable Hamiltonian systems.
Clarify the relationship between these geometric lifts and Haantjes geometry, particularly regarding the structure of symplectic-Haantjes manifolds.
Explore the generation of momentum-dependent metrics and non-geodesic flows that may have applications in modified gravity and magnetic systems.

Methodology
The authors introduce the Stäckel lift, a procedure rooted in generalized Stäckel geometry. The core methodology involves:

Lifting the Stäckel Matrix: Starting with an $n$ -dimensional integrable Hamiltonian system defined on a cotangent bundle $T^*Q$ with an associated $n \times n$ Stäckel matrix $S^{(n)}$ , the authors construct an $(n+1) \times (n+1)$ "lifting matrix" $L^{(n+1)}$ .
Momentum Dependence: Unlike standard Stäckel matrices which depend only on configuration coordinates, the lifted matrix $L^{(n+1)}$ includes an additional row and column containing functions that depend explicitly on a new pair of conjugate variables $(q_{n+1}, p_{n+1})$ . This results in a generalized Stäckel matrix that depends on the new momentum $p_{n+1}$ .
Hamiltonian Construction: The lifted Hamiltonian system is defined via the inverse of the lifting matrix acting on a vector of arbitrary functions $f_i(q_i, p_i)$ . This procedure generates a set of $n+1$ Hamiltonians in involution.
Haantjes Analysis: The authors analyze the resulting systems using the theory of $\omega H$ (symplectic-Haantjes) manifolds. They investigate the properties of the Haantjes operators associated with the lifted Hamiltonians, specifically checking for projectability along the fibers of the extended cotangent bundle.
Iterative and Specific Applications: The method is applied iteratively to generate higher-dimensional models. Specific applications include the reconstruction of Eisenhart lifts, the analysis of Smorodinsky-Winternitz systems, the construction of Platonic-wave geometries, and the treatment of magnetic systems separable in cylindrical coordinates.

Key Contributions and Results

Unification of Eisenhart and Stäckel Lifts (Theorem 1):
The authors prove that the classical Riemannian Eisenhart lift is a specific instance of the Stäckel lift. By choosing a specific form for the lifting matrix (where the last row and column are constant or depend only on configuration variables), the standard Eisenhart geodesic Hamiltonian is recovered. This establishes the Stäckel lift as a generalization that encompasses the Eisenhart framework.
Non-Projectable Haantjes Structures (Theorem 2):
A central result is the proof that Stäckel lifts, particularly those with momentum-dependent coefficients, endow the extended phase space $T^*\tilde{Q}$ with a non-trivial symplectic-Haantjes structure.
- Crucially, when the lifting matrix depends on momenta, the associated Haantjes operators are non-projectable. They cannot be reduced to operators on the configuration space with vanishing Haantjes torsion. This distinguishes momentum-dependent lifts from purely geodesic lifts and represents a genuinely new class of geometric structures.
Generation of Platonic Waves and Modified Gravity Models:
The paper demonstrates that explicitly momentum-dependent lifting matrices generate Platonic-wave geometries. These are spacetimes admitting a null Killing vector field that is parallel with respect to a conformally related metric.
- The authors provide an explicit example of an integrable Platonic wave with flat wave fronts but non-constant Ricci scalar curvature.
- They note that while these geometries are not necessarily solutions to standard Einstein equations, they serve as integrable models that could potentially arise in modified gravity theories (e.g., $f(R)$ gravity).
Non-Geodesic Flows and Higher-Order Integrals:
The Stäckel lift is shown to generate families of non-geodesic integrable Hamiltonian systems. By choosing specific functions in the lifting matrix, the authors construct systems with:
- Higher-order integrals of motion (transcendental in momenta).
- Linear terms in momenta (associated with gyroscopic or dissipative dynamics).
- These systems are not restricted to geodesic motion in a pseudo-Riemannian space but define more general dynamical flows.
Application to Cylindrical Magnetic Systems:
The framework is applied to magnetic systems separable in cylindrical coordinates. The authors construct Stäckel matrices for three distinct cases of such systems.
- They highlight a subtle hierarchy: Case 3 (where only the $z$ -coordinate is cyclic) allows a straightforward Stäckel formulation that preserves dynamics on level sets.
- Cases 1 and 2 require incorporating the geometric coefficient $1/r^2$ into the change of basis. This modification alters the Hamiltonian itself rather than just reparametrizing the dynamics, demonstrating that the Stäckel framework can embed these systems but may require modifying the energy function to maintain separability.

Significance and Claims
The paper claims to establish the Stäckel lift as a unified geometric framework for constructing integrable Hamiltonian systems of dimension $n+k$ from lower-dimensional seed systems.

Structural Insight: It reframes the Eisenhart lift not as an ad hoc embedding but as a consequence of generalized Stäckel geometry.
Geometric Novelty: The discovery of non-projectable Haantjes structures in momentum-dependent lifts suggests a broader class of integrable systems where the separation of variables is not tied to Killing tensors on the configuration space.
Physical Relevance: The construction provides a systematic method for generating integrable spacetime models (Platonic waves) and Hamilton/Finsler spaces with momentum-dependent metrics. These are relevant to fields such as deformed special relativity, quantum gravity phenomenology, and the study of magnetic traps.
Versatility: The ability to generate non-geodesic flows, higher-order integrals, and systems with linear momentum terms expands the scope of integrable system theory beyond traditional geodesic motion.

The authors maintain a modest stance regarding the physical realization of their examples, noting that while the constructed Platonic waves are integrable and possess interesting geometric properties, their status as solutions to specific modified gravity field equations remains an open question for future work. Similarly, the classification of non-projectable Haantjes structures and the quantum analogues of the higher-order systems are identified as directions for future research.

1. The Old Trick: The Eisenhart Lift

2. The New Trick: The Stäckel Lift

3. The Hidden Geometry: Haantjes Tensors

4. What Can You Build With This?

Summary

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