Eisenhart Lift Variants: Extensions & Applications

• Variant of the Eisenhart lift is a family of constructions that extend the classical embedding of mechanical systems into higher-dimensional geodesic motion with varied signature and causal structures.
• This framework employs diverse techniques—including conformal, time-dependent, Bohlin, Herglotz, and Stäckel lifts—to map dynamics into new geometric settings.
• The approach offers practical insights into integrating quantum, higher-derivative, and field-theoretic extensions by reinterpreting conserved quantities and uncovering hidden symmetries.

A variant of the Eisenhart lift is a modification of the standard procedure that embeds a mechanical system into geodesic motion on a higher-dimensional manifold. In the recent literature, the phrase covers several distinct constructions: conformally generalized Eisenhart–Bargmann geometries with flatness constraints, time-dependent Bargmann lifts, timelike conformally flat lifts inspired by the Bohlin transformation, action-dependent Brinkmann metrics for Herglotz dynamics, higher-derivative and Koopman–von Neumann lifts, field-theoretic and minisuperspace lifts, and matrix-based generalizations such as the Stäckel lift (Dhasmana et al., 2021, Galajinsky, 24 Feb 2026, Cariglia et al., 2016, Bartczak et al., 29 Aug 2025, Kubů et al., 24 Sep 2025).

1. Standard framework and the meaning of “variant”

The classical reference point is the Eisenhart lift of a natural Hamiltonian system to a geodesic Hamiltonian on a higher-dimensional space. In the Lorentzian pp-wave or Bargmann form, one uses a Brinkmann metric such as

$g \defeq 2dvdt -2V(t,x)(dt)^2+ \sum_{i=1}^n (dx^i)^2,$

or, with vector potential and general kinetic metric,

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$

so that mechanical trajectories are projections of geodesics, and in the Lorentzian formulation the relevant lifts are null geodesics (Aazami, 2024, Cariglia et al., 2015). In the Riemannian formulation, one extra cyclic coordinate is added and the potential is absorbed into the metric coefficient of that direction, while the Lorentzian lift adds two coordinates and encodes the original dynamics by the null constraint $\mathcal H=0$ (Cariglia et al., 2015).

A useful taxonomy is already visible in the standard literature.

Variant	Defining change	Representative paper
Flatness-constrained conformal lift	conformally generalized Eisenhart metric; flat lifted spacetime	(Dhasmana et al., 2021)
Time-dependent Bargmann lift	explicit $t$-dependence in the lifted metric	(Cariglia et al., 2016)
Bohlin variant	timelike geodesics of a conformally flat Lorentzian metric	(Galajinsky, 24 Feb 2026)
Herglotz lift	Brinkmann metric depending on all $(n+2)$ coordinates	(Bartczak et al., 29 Aug 2025)
Generalized coupling lift	one extra coordinate for each coupling constant	(Cariglia et al., 2013)
Stäckel lift	lifted Stäckel matrix, possibly momentum-dependent	(Kubů et al., 24 Sep 2025)

This suggests that “variant” does not designate a single deformation of Eisenhart’s construction. It designates a family of extensions in which the lift may change the signature, the causal class of the geodesics, the conserved quantity used in the reduction, the dependence of the lifted metric on extra coordinates, or even the algebraic object being lifted.

2. Conformal, flatness-constrained, and time-dependent variants

A particularly sharp variant is the conformally generalized Eisenhart lift in one spatial dimension, where the ambient metric is taken in the form

$$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$

The decisive restriction is that the conformally extended Eisenhart metric be flat. Conformal flatness of

$ds^2=dx^2+2dt\,du-2Udt^2$

requires

$\frac{\partial^3 U}{\partial x^3}=0,$

and actual flatness of $ds^{*2}$ yields

$$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$

Writing $$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$0, the admissible potentials are exactly

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$1

The lifted flat spacetime can then be brought to Minkowski form, and the induced coordinate change downstairs maps one Schrödinger problem to another. In this formulation, the harmonic-oscillator/free-particle and linear-potential/free-particle correspondences arise from ambient coordinate transformations, while the quantum phase comes from the extra null coordinate (Dhasmana et al., 2021).

That same analysis also isolates an important limitation. For the harmonic oscillator, the transformation carries an amplitude factor $$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$2, so the map is non-unitary and only local in time; accordingly, it is not interpreted as an equivalence-principle transformation. By contrast, the linear-potential case includes the standard accelerated-frame realization of Einstein’s equivalence principle in nonrelativistic quantum mechanics (Dhasmana et al., 2021).

A second major line of development introduces explicit time dependence already in the lifted Bargmann geometry. For

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$3

the lifted metric is

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$4

Projected null geodesics reproduce the original time-dependent equations of motion, including friction terms of the form $$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$5. The crucial structural statement is that the time reparametrization

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$6

corresponds to a conformal rescaling of the Bargmann metric, so different choices of clock define conformally related lifts with the same null curves (Cariglia et al., 2016).

A broader conformal point of view is developed for pp-wave lifts. Because lightlike geodesics are conformally invariant, any Hamiltonian system can be unfolded into a conformal class of non-diffeomorphic ODEs with solutions in common. The same conformal class also preserves the null-geodesic conjugate-point structure, so accumulation behavior of nearby trajectories is shared across the family (Aazami, 2024).

3. Timelike, coupling-based, Jacobi, and Stäckel generalizations

The Bohlin variant departs from the Bargmann/null scheme in a specific way. Instead of the standard Eisenhart–Duval metric, it uses the conformally flat Lorentzian metric

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$7

and the original conservative dynamics is recovered from timelike geodesics satisfying

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$8

The reduced equations give

$$ds^2= h_{ij}\,dq^i dq^j + 2\,du\bigl(dv - V\,du + A_i\,dq^i\bigr),$$9

so the potential itself determines the reparametrization between mechanical time and proper time. This geometry is explicitly not of Brinkmann/Bargmann type: the vector $\mathcal H=0$0 is not covariantly constant, the geodesic congruence is timelike rather than null, and the spacetime is not in the Kundt class (Galajinsky, 24 Feb 2026).

The Bohlin construction is also used to generate conformally flat Lorentzian metrics with hidden symmetries. For $\mathcal H=0$1, the lifted metric is a patch of $\mathcal H=0$2-dimensional anti de Sitter space, and the one-dimensional conformal mechanics potential $\mathcal H=0$3 uplifts to $\mathcal H=0$4. For Calogero models, the timelike uplift produces explicit irreducible rank-3 and rank-4 Killing tensors (Galajinsky, 24 Feb 2026).

A different generalization replaces the single auxiliary coordinate of the standard Riemannian lift by one extra coordinate for each coupling constant. For the non-periodic Toda chain,

$\mathcal H=0$5

the generalized lift introduces $\mathcal H=0$6 extra coordinates $\mathcal H=0$7 and the free geodesic Hamiltonian

$\mathcal H=0$8

The couplings become conserved higher-dimensional momenta,

$\mathcal H=0$9

and the ordinary Eisenhart coordinate is recovered by

$t$0

The paper identifies this as an inverse Kaluza–Klein reduction and interprets the symmetric-space realization $t$1 as a generalized Eisenhart lift (Cariglia et al., 2013).

The Jacobi-Eisenhart metric addresses a different problem: the ordinary Jacobi metric requires a fixed-energy shell and therefore fails for non-autonomous systems. After the Eisenhart-Duval lift with extra coordinate $t$2, the metric

$t$3

has a cyclic direction, so $t$4 is conserved. The resulting time-dependent Jacobi-Eisenhart metric is

$t$5

or, in the non-relativistic limit,

$t$6

Here the conserved momentum shell replaces the fixed-energy shell of the ordinary Jacobi-Maupertuis construction (Chanda et al., 2016).

The most expansive algebraic generalization in the supplied literature is the Stäckel lift. Instead of lifting a single Hamiltonian, it lifts an entire Stäckel matrix $t$7 to an $t$8 matrix $t$9, whose last column may depend explicitly on momenta. The lifted commuting Hamiltonians are defined by

$(n+2)$0

This construction is stated to extend the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. It naturally yields a symplectic-Haantjes structure, and explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries (Kubů et al., 24 Sep 2025).

4. Action-dependent, higher-derivative, and Koopman–von Neumann extensions

A genuine extension beyond ordinary Lagrangian mechanics is obtained by replacing the standard variational principle with Herglotz’s action-dependent formalism. The lifted metric is the fully general Brinkmann metric

$(n+2)$1

The null constraint gives

$(n+2)$2

so the extra coordinate $(n+2)$3 becomes the action variable of a Herglotz system. In this general case $(n+2)$4 is not affine, because $(n+2)$5 whenever the metric depends on $(n+2)$6. Dynamical symmetries lift to conformal Killing vectors rather than ordinary Killing vectors, and the corresponding conserved quantity reproduces the generalized Herglotz Noether law with its exponential integrating factor (Bartczak et al., 29 Aug 2025).

Higher-derivative dynamics require a different modification. For Ostrogradsky Hamiltonians, a direct Eisenhart-type homogenization leads to a degenerate metric because the Hamiltonian contains terms linear in momenta. One proposed remedy enlarges phase space with $(n+2)$7 additional canonical pairs and yields an ultrahyperbolic metric whose geodesics reproduce a deformation of the original higher-derivative equations. A second route performs a canonical transformation and then applies a more conventional Eisenhart lift to a pseudo-Euclidean ordinary Hamiltonian. The construction works cleanly only for restricted classes of potentials, notably sums of homogeneous functions or systems depending only on even-order derivatives; the Pais–Uhlenbeck oscillator is the main example (Galajinsky et al., 2016).

Koopman–von Neumann mechanics leads to another distinct variant. The KvN Hamiltonian on the enlarged $(n+2)$8 space,

$(n+2)$9

lifts to the ultrahyperbolic metric

$$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$0

The massless Klein–Gordon equation in this geometry reduces, after the null ansatz $$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$1, to the KvN evolution equation. This construction makes the hidden KvN variable $$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$2 into a genuine coordinate of the lifted space. It also reproduces the familiar free/linear/harmonic correspondences: the linear-potential map is unitary, while the harmonic/free correspondence acquires a time-dependent amplitude factor and is therefore not unitary (Sen et al., 2022).

5. Field-theoretic, scalar-field, and minisuperspace variants

For homogeneous scalar fields with target-space metric $$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$3, the Riemannian-type field-theory lift adds one fictitious scalar $$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$4 and rewrites

$$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$5

as

$$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$6

The enlarged field-space metric is

$$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$7

and the original equations are recovered when the conserved quantity associated with $$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$8 is normalized so that $$ds^{*2}=\Omega(x,t)\left(dx^2+2dt\,du-2U\,dt^2\right).$$9. In the full spacetime-dependent theory, a scalar lift is insufficient in general; the construction instead introduces a fictitious vector field $ds^2=dx^2+2dt\,du-2Udt^2$0 and a mixed kinetic tensor $ds^2=dx^2+2dt\,du-2Udt^2$1, with the important structural statement that $ds^2=dx^2+2dt\,du-2Udt^2$2 (Finn et al., 2018).

A closely related but technically different field-theoretic formulation starts from a Polyakov-type scalar action and constructs both Riemannian and Lorentzian lifts using cyclic scalar fields. In the Riemannian case one requires

$ds^2=dx^2+2dt\,du-2Udt^2$3

so that the momentum of the extra field is a genuine constant. In the Lorentzian version, two extra scalar fields $ds^2=dx^2+2dt\,du-2Udt^2$4 are introduced, with $ds^2=dx^2+2dt\,du-2Udt^2$5 and $ds^2=dx^2+2dt\,du-2Udt^2$6. If no cyclic base coordinate exists, the paper proposes a “Double Lift,” adding an extra base-space coordinate as well (Chanda et al., 2019).

The FLRW scalar-field adaptation of the Riemannian-type lift begins from an auxiliary vector field $ds^2=dx^2+2dt\,du-2Udt^2$7 in the underlying field theory and, after homogeneity reduction, produces the minisuperspace coordinate

$ds^2=dx^2+2dt\,du-2Udt^2$8

The lifted minisuperspace metric for $ds^2=dx^2+2dt\,du-2Udt^2$9 is

$\frac{\partial^3 U}{\partial x^3}=0,$0

so the dynamics becomes null geodesic motion on a three-dimensional pseudo-Riemannian field-space. The paper classifies Killing vectors, conformal Killing vectors, and Killing tensors for several exponential-type potentials, including

$\frac{\partial^3 U}{\partial x^3}=0,$1

and shows that coordinates adapted to the additional KV or CKV give complete integration of the equations (Chiba et al., 2024).

A cosmological Eisenhart–Duval variant instead modifies the Bargmann spacetime itself by inserting a cosmic scale factor: $\frac{\partial^3 U}{\partial x^3}=0,$2 With a null-dust stress tensor $\frac{\partial^3 U}{\partial x^3}=0,$3, Einstein’s equations reduce to the Ermakov–Milne–Pinney equation

$\frac{\partial^3 U}{\partial x^3}=0,$4

After a coordinate transformation, the metric becomes the Bargmann metric of a time-dependent isotropic oscillator. The same framework also encodes the Ermakov–Lewis invariant, the Friedmann equations, and the Dmitriev–Zel’dovich equations (Cariglia et al., 2018).

A minisuperspace-specific finite-dimensional lift appears in Rosen-Lagrangian cosmology with

$\frac{\partial^3 U}{\partial x^3}=0,$5

The lifted Lagrangian is

$\frac{\partial^3 U}{\partial x^3}=0,$6

with metric

$\frac{\partial^3 U}{\partial x^3}=0,$7

The extra coordinate $\frac{\partial^3 U}{\partial x^3}=0,$8 is cyclic, the Hamiltonian constraint is preserved, and the paper derives a conformal factor

$\frac{\partial^3 U}{\partial x^3}=0,$9

from the conformal Killing equations. In the quantum theory, the lifted Wheeler–DeWitt equation reduces to a Bessel equation in the flat, cosmological-constant-dominated regime (Kaewkhao et al., 31 May 2026).

6. Symmetry, quantization, and physical admissibility

The quantum extension of the one-extra-coordinate Eisenhart lift uses the lifted Hamiltonian

$ds^{*2}$0

so the stationary Schrödinger equation becomes

$ds^{*2}$1

After separation $ds^{*2}$2, the reduced equation is

$ds^{*2}$3

The original Schrödinger problem is recovered in the sector $ds^{*2}$4. Upon compactifying the fictitious direction, $ds^{*2}$5 is quantized and becomes a new conserved quantum number. In the field-theory extension, the analogue of the lifted momentum is the conserved quantum charge $ds^{*2}$6, and different $ds^{*2}$7-values label disjoint Fock spaces (Finn et al., 2020).

There are also further geometric reinterpretations of the lift itself. In the brane construction, the standard Eisenhart–Duval metric

$ds^{*2}$8

is realized as the induced worldvolume metric of an $ds^{*2}$9-brane embedded in flat $$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$0. When $$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$1 is harmonic in the configuration variables, the Eisenhart–Duval metric is Ricci-flat and the brane is minimal, in the sense that it extremizes its spacetime volume (Gibbons, 2020).

A distinct issue is physical admissibility of the lifted spacetime. For the 4D Eisenhart lift of two-dimensional mechanics on a curved background,

$$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$2

the Einstein tensor is determined by

$$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$3

A practical sufficient condition for the weak energy condition is

$$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$4

Within the Darboux–Koenigs family, only type I can yield Eisenhart lifts satisfying the weak energy condition, although the paper also presents physically viable metrics with hidden symmetries, including spherical and separable examples (Fordy et al., 2019).

Several recurring misconceptions are corrected by these variants. A variant of the Eisenhart lift is not necessarily Bargmann or null-geodesic based: the Bohlin construction uses timelike geodesics and is explicitly not of Brinkmann/Bargmann type (Galajinsky, 24 Feb 2026). A lifted time variable is not necessarily affine: in the Herglotz extension, $$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$5 fails to be affine whenever the metric depends on the action coordinate $$\frac{\partial \Omega}{\partial x}=0, \qquad \frac{\partial^2 U}{\partial x^2} = \frac1{\Omega^2} \left[ \frac{\Omega}{2}\frac{\partial^2\Omega}{\partial t^2} -\frac34\left(\frac{\partial\Omega}{\partial t}\right)^2 \right].$$6 (Bartczak et al., 29 Aug 2025). A free/oscillator correspondence is not automatically an equivalence-principle statement: in the flatness-constrained conformal lift, the harmonic-oscillator map is non-unitary and only local in time (Dhasmana et al., 2021). And a lifted geometry need not remain an ordinary configuration-space metric: momentum-dependent Stäckel lifts naturally lead to Hamilton and Finsler geometries rather than standard Riemannian or Lorentzian metrics on configuration space (Kubů et al., 24 Sep 2025).

Taken together, these constructions show that the phrase “variant of the Eisenhart lift” denotes a broad research program rather than a single formula. The common invariant is the geometric encoding of dynamics in a higher-dimensional structure; the variable elements are the underlying variational principle, the reduction data, the signature and causal class of the lift, and the geometric category in which integrability and symmetry are expressed.