Ermakov Mapping in Differential Systems

• Ermakov mapping is a framework that converts linear differential equations into nonlinear inverse‐cubic systems via invariant structures such as the Ermakov–Lewis invariant.
• It employs techniques like time rescaling, Möbius transformations, and Schwarzian derivatives to reformulate nonautonomous equations into autonomous Hamiltonian systems.
• The method enables reduction of complex PDEs and multidimensional models into integrable systems, unifying approaches in classical mechanics, quantum dynamics, and geometric formulations.

to=shell.run code to=shell.run code to=shell.run code to=shell.run code Ermakov mapping denotes a class of structurally invariant transformations that relate linear differential equations, Riccati- or Schwarzian-type equations, and nonlinear Ermakov, Ermakov–Pinney, or Painlevé–Ermakov systems. In the literature surveyed here, the term appears in several technically distinct but closely related senses: as the passage from a linear oscillator to an inverse-cubic amplitude equation; as a projective or Schwarzian construction generating Ermakov and Painlevé XXV–Ermakov solutions from a Schwarzian field; as a time-rescaling and coordinate-scaling transformation sending nonautonomous systems to autonomous Hamiltonian ones; and as a reduction of quantum, hydrodynamic, or PDE models to finite-dimensional Ermakov dynamics (Carillo et al., 2022, Guerrero et al., 2013, Mitsopoulos et al., 2021, Kumar, 31 Jan 2026, An et al., 2012). A common theme is that the nonlinear variable is not introduced ad hoc: it is produced by an invariant structure, typically an Ermakov–Lewis invariant, a Schwarzian derivative, a Liouville normalization, or a Sundman/Arnold-type transformation.

1. Core concept and canonical forms

At its most classical, the relevant nonlinear equation is the Ermakov–Pinney equation

$u''(z)=B(z)u(z)+\frac{I}{u(z)^3}, \tag{27}$

or, equivalently,

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$

This equation is paired with a linear oscillator,

$\eta''(z)=B(z)\eta(z), \tag{4}$

or

$\ddot{u}(t)+\omega^2(t)u(t)=0.$

The standard nonlinear superposition principle expresses the Ermakov variable as a quadratic form in two linearly independent linear solutions. In one formulation,

$$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$

where $W$ is the constant Wronskian of $y_1,y_2$ (Kumar, 31 Jan 2026). In another,

$$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$

with

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$

solves

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$

These formulas are the prototypical Ermakov mappings from linear solution spaces to nonlinear inverse-cubic dynamics (Kim et al., 2016).

The associated invariant is the Ermakov–Lewis invariant. In one standard form,

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$0

while in separated stationary Bohm–Madelung sectors it becomes

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$1

with $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$2 (Paliathanasis et al., 2022, Kumar, 31 Jan 2026). In multidimensional generalized Ermakov systems the invariant assumes Ray–Reid form, for example

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$3

A concise way to describe the mapping principle is therefore: one starts from a linear problem, constructs an amplitude, ratio, or scaled coordinate obeying an inverse-cubic equation, and obtains a conserved quantity that organizes the nonlinear dynamics.

2. Linearization, time rescaling, and autonomous reformulations

One major class of Ermakov mappings is built from changes of variables that convert nonautonomous equations into autonomous systems. For the two-dimensional generalized Ermakov system

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$4

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$5

the transformation

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$6

with $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$7 solving

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$8

maps the original nonautonomous system into the autonomous equations

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$9

The invariant becomes time-independent: $\eta''(z)=B(z)\eta(z), \tag{4}$0 This is the central nonautonomous-to-autonomous Ermakov mapping in the geometric treatment of generalized conservative systems (Mitsopoulos et al., 2021).

Within the conservative subclass, the autonomous system is identified with a Hamiltonian system possessing

$\eta''(z)=B(z)\eta(z), \tag{4}$1

provided

$\eta''(z)=B(z)\eta(z), \tag{4}$2

This embeds the generalized Ermakov equations into the known class of integrable two-dimensional autonomous conservative systems with Euclidean kinetic metric (Mitsopoulos et al., 2021).

A distinct, but structurally similar, mapping arises in the Arnold framework. For a linear second-order ODE

$\eta''(z)=B(z)\eta(z), \tag{4}$3

the Classical Arnold Transformation sends it to the free equation $\eta''(z)=B(z)\eta(z), \tag{4}$4 by

$\eta''(z)=B(z)\eta(z), \tag{4}$5

where $\eta''(z)=B(z)\eta(z), \tag{4}$6 are canonical homogeneous solutions (Guerrero et al., 2013). Composing two such maps yields the Arnold–Ermakov–Pinney transformation

$\eta''(z)=B(z)\eta(z), \tag{4}$7

with $\eta''(z)=B(z)\eta(z), \tag{4}$8 satisfying the generalized Ermakov–Pinney equation

$\eta''(z)=B(z)\eta(z), \tag{4}$9

when the target system is a constant-frequency oscillator (Guerrero et al., 2013). Here the mapping function itself is the Ermakov variable.

3. Schwarzian and projective formulations

A projective formulation of Ermakov mapping is developed through the Schwarzian derivative

$\ddot{u}(t)+\omega^2(t)u(t)=0.$0

If $\ddot{u}(t)+\omega^2(t)u(t)=0.$1 are linearly independent solutions of

$\ddot{u}(t)+\omega^2(t)u(t)=0.$2

then their quotient $\ddot{u}(t)+\omega^2(t)u(t)=0.$3 satisfies

$\ddot{u}(t)+\omega^2(t)u(t)=0.$4

This linear–Schwarzian correspondence is the backbone of the mapping structure in the Schwarzian treatment of Ermakov and Painlevé XXV–Ermakov equations (Carillo et al., 2022).

From a solution $\ddot{u}(t)+\omega^2(t)u(t)=0.$5 of the Schwarzian equation, one obtains in closed form: $\ddot{u}(t)+\omega^2(t)u(t)=0.$6 which solves the third-order linear equation

$\ddot{u}(t)+\omega^2(t)u(t)=0.$7

$\ddot{u}(t)+\omega^2(t)u(t)=0.$8

which solves the Ermakov equation

$\ddot{u}(t)+\omega^2(t)u(t)=0.$9

and

$$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$0

which solves the $$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$1 Painlevé XXV–Ermakov equation

$$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$2

This identifies the Schwarzian equation as a master equation for linear, Ermakov, and Painlevé XXV–Ermakov dynamics (Carillo et al., 2022).

The Möbius invariance of the Schwarzian,

$$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$3

lifts to full four-parameter solution families: $$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$4

$$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$5

$$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$6

Accordingly, the mapping is many-to-one at the level of $$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$7, but one-to-one at the level of Möbius-equivalence classes (Carillo et al., 2022).

The same work also provides a direct linear-to-Painlevé XXV–Ermakov map: $$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$8 when $$\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2}, \tag{5.1}$$9 solves

$W$0

When $W$1, the first integral

$W$2

and the substitution $W$3 recover the classical Ermakov equation

$W$4

This is the paper’s explicit formulation of the classical Ermakov mapping (Carillo et al., 2022).

4. Geometric, multidimensional, and Riemannian extensions

In higher dimensions, Ermakov mappings are closely tied to homothetic geometry and $W$5 symmetry. For the $W$6-dimensional Hamiltonian Ermakov system,

$W$7

the potential has the characteristic form $W$8, and the system always admits an $W$9 algebra of Noether symmetries generated by the homothetic structure of flat space (Paliathanasis et al., 2022). The corresponding Ermakov invariant is obtained as a Casimir-like combination of the Noether integrals; in two dimensions,

$y_1,y_2$0

The classification of generalized $y_1,y_2$1-dimensional Hamiltonian Ermakov systems with additional conservation laws is expressed in Cartesian form by potential families such as

$y_1,y_2$2

along with inverse-square hyperplane generalizations of the type described in equations (48)–(49) of the same work (Paliathanasis et al., 2022). A central conclusion is that Ermakov invariants and additional linear or quadratic first integrals are determined by Killing vectors, Killing tensors, and the homothetic vector of the background Euclidean metric.

A further generalization replaces Euclidean space by a Riemannian manifold admitting a gradient homothetic vector. In adapted coordinates the metric is

$y_1,y_2$3

and the autonomous Hamiltonian Riemannian Kepler–Ermakov Lagrangian is

$y_1,y_2$4

with

$y_1,y_2$5

for the polynomial $y_1,y_2$6 representation, or

$y_1,y_2$7

for the exponential one (Tsamparlis et al., 2012). The Riemannian Ermakov invariant is

$y_1,y_2$8

and the radial equation becomes an Ermakov–Pinney equation in $y_1,y_2$9. In this setting, the “mapping” is from a dynamical system on a manifold with gradient homothety to a radial inverse-cubic equation plus angular dynamics encoded by $$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$0 and $$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$1.

This geometric picture is used to embed specific cosmological models into Riemannian Kepler–Ermakov systems. In particular, a scalar-field cosmology with exponential potential and stiff fluid, and the $$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$2 model

$$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$3

are shown to reduce to autonomous Hamiltonian Riemannian Kepler–Ermakov systems that are Liouville integrable via Noether integrals (Tsamparlis et al., 2012). A plausible implication is that Ermakov mapping, in this geometric sense, functions as a symmetry-adapted coordinate reduction for minisuperspace dynamics.

5. Quantum-mechanical realizations

In quantum mechanics, Ermakov mapping appears both in time-dependent and stationary settings. For the one-dimensional harmonic oscillator

$$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$4

writing $$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$5 and imposing a Gaussian density leads to the width equation

$$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$6

This reduces the full Schrödinger dynamics of Gaussian states to a single Ermakov equation for the width $$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$7, with $$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$8 interpreted as the rms displacement or wave-packet width (Tsekov, 2010). Dissipative and thermal extensions preserve the inverse-cubic quantum term while adding friction, thermal, or radiative contributions, for example

$$x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2}, \tag{3}$$9

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$0

and

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$1

Here the mapping is from a PDE for $AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$2 to a nonlinear ODE for a single amplitude variable (Tsekov, 2010).

A stationary analogue appears in Bohm–Madelung quantum mechanics for diagonal, separable Hamiltonians. After separation and Liouville normalization, each sector satisfies a linear normal-form equation

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$3

while the stationary continuity constraint implies the nonlinear companion

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$4

with $AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$5 (Kumar, 31 Jan 2026). The corresponding Ermakov–Lewis invariant is coordinate-constant: $AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$6 The mapping here is from a stationary Schrödinger sector to a linear Sturm–Liouville problem plus a nonlinear Ermakov–Pinney amplitude equation.

The Quantum Arnold Transformation provides another quantum Ermakov mapping. The Quantum Arnold Transformation

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$7

maps solutions of a time-dependent quadratic Schrödinger equation to free-particle solutions, while the Quantum Arnold–Ermakov–Pinney transformation

$AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$8

maps between two quadratic systems, with $AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2}, \tag{7}$9 solving a generalized Ermakov–Pinney equation (Guerrero et al., 2013).

The same theme appears in time-dependent oscillator invariants. If

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$0

then $$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$1 satisfies

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$2

and the Lewis–Riesenfeld invariant is

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$3

In this formulation, choosing $$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$4 and then reconstructing $$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$5 yields an inverse Ermakov mapping from amplitude profiles to exactly solvable time-dependent oscillators (Kim et al., 2016).

A photonic realization implements the same structure experimentally. In a semi-infinite waveguide array, the lattice Hamiltonian is mapped to

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$6

with

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$7

After displacement and squeezing transformations, the dynamics are governed by the Ermakov–Lewis invariant and the generalized Ermakov equation for $$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$8 (Lara et al., 2014). This is an optical realization of an Ermakov-mapped time-dependent harmonic oscillator.

6. PDE reductions, Painlevé hybridization, and nonlocal transforms

Ermakov mapping also operates at the level of nonlinear PDEs and nonlocal transformations. In one direction, generalized Sundman transformations

$$\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}. \tag{1}$$9

are used to map generalized Liénard-type equations

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$00

to the linear oscillator

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$01

The pullback of the oscillator energy gives the invariant

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$02

Specializations of the resulting mapped equations include dissipative generalized Ermakov–Pinney systems such as

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$03

with invariant

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$04

Coupled Liénard systems are similarly mapped to coupled dissipative Ermakov–Milne–Pinney equations with coupled Ermakov-type invariants (Guha et al., 2019).

In integrable PDE theory, Ermakov–Painlevé reductions provide another extension. The temporally modulated extended mKdV equation

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$05

admits the similarity ansatz

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$06

and for

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$07

the reduction becomes

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$08

With $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$09 and an additional scaling, this is brought to the canonical Ermakov–Painlevé II equation

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$10

This PDE-to-ODE reduction is explicitly described as an Ermakov mapping in the moving-boundary analysis of extended mKdV equations (Rogers et al., 5 Nov 2025).

A closely related $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$11-dimensional construction starts from the temporally modulated modified Kadomtsev–Petviashvili-type equation

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$12

and, under

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$13

reduces it to the same Ermakov–Painlevé II structure (Rogers et al., 17 Mar 2026). In that setting, involutory transformations

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$14

with $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$15 satisfying the classical Ermakov equation

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$16

generate a broad class of temporally modulated PDEs that preserve the Ermakov–Painlevé II reduction.

A hydrodynamic example is supplied by the $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$17-dimensional anisotropic non-isothermal magnetogasdynamic system. Under an elliptic vortex ansatz for the density and a linear ansatz for the velocity, the PDEs reduce to a finite-dimensional dynamical subsystem; after the introduction of a scaling variable $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$18, the semi-axes $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$19 of the elliptical cross-section satisfy an Ermakov–Ray–Reid system (An et al., 2012). The Hamiltonian of the semi-axis dynamics,

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$20

is the relevant Ermakov-type invariant. In this context, the mapping is from a $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$21-dimensional plasma model to a Hamiltonian Ermakov–Ray–Reid subsystem.

7. Invariants, symmetries, and the role of solution generation

Across these formulations, the central organizing objects are invariants and symmetry groups. In classical Ermakov systems the invariant is quadratic in a Wronskian-like combination; in generalized multidimensional systems it is tied to $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$22 or to homothetic algebras; in the Schwarzian setting it is tied to Möbius invariance; in quantum mechanics it becomes a Lewis–Riesenfeld invariant operator; and in Painlevé hybridizations it is converted into a Painlevé parameter (Carillo et al., 2022, Paliathanasis et al., 2022, Guerrero et al., 2013, Kumar, 31 Jan 2026, Rogers et al., 2017).

Bäcklund transformations are especially important in the Schwarzian and Painlevé-based settings. In the Schwarzian approach, one family is Wronskian-based: if $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$23 solve

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$24

then their Wronskian

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$25

is also a solution, and this induces Bäcklund transformations for corresponding Ermakov equations through $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$26 (Carillo et al., 2022). A second family is Möbius/Schwarzian-based: if

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$27

then

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$28

maps one Ermakov solution to another.

In Ermakov–Painlevé IV systems, the invariant

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$29

enters the scalar reduction

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$30

through the identification $\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$31 (Rogers et al., 2017). The map

$\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.$32

is therefore an Ermakov mapping from a coupled Ermakov system to Painlevé IV, and the Bäcklund transformations of Painlevé IV generate new coupled Ermakov solutions.

Taken together, these constructions show that Ermakov mapping is less a single transformation than a family of structurally related correspondences. What persists across all variants is the passage from a linear or symmetry-reduced description to a nonlinear amplitude equation with inverse-power structure, together with an invariant that survives the transformation and controls reconstruction of the original variables.