Symmetry-Based Selection of Gravitational Lagrangians via Noether Approach

Abstract

Starting from the Noether first theorem, we discuss a criterion for identifying physically consistent gravitational models. In particular, we demonstrate that applying the Lie derivative to a point-like Lagrangian makes it possible to determine the underlying symmetries, their associated generators, and the corresponding conserved quantities. This method is then generalized through its first prolongation and applied to several point-like Lagrangians, with special attention to $f\left(R\right)$ gravity in a cosmological setting. In each example, the existence of symmetries results in a simplification of the dynamical system, enabling the integration of the equations of motion and the derivation of exact solutions. It is worth noticing that the existence of symmetries is always related to physically consistent models.

1. Introduction

In 1918, Emmy Noether introduced her famous theorem [1], which establishes a direct correspondence between the symmetries of a Lagrangian and the existence of conserved quantities. Since its formulation, this result has played a fundamental role across many areas of theoretical physics. Typically, Noether’s theorem is used to derive constants of motion associated with transformation rules that leave the action invariant. Such conserved quantities are essential for the study of dynamical systems, as they significantly simplify their analysis and deepen our understanding of their evolution.

In this work, however, we adopt an inverse viewpoint. Rather than starting from a prescribed action and identifying its symmetries, we investigate how imposing symmetry requirements can guide the construction of viable theories even when the action is not specified in advance. From this perspective, symmetries serve as a selection principle that allows one to infer the functional form of the action. This strategy provides an alternative and powerful framework for exploring possible physical models, particularly within theories of gravity, where the fundamental action is often not uniquely determined a priori.

Modified and extended theories of gravity were originally proposed to overcome several conceptual and observational limitations of General Relativity (GR) across different energy scales. Indeed, a number of well-known issues challenge its status as a complete description of gravitational phenomena [2]. At the quantum level, GR faces serious difficulties when treated as a quantum field theory, most notably its non-renormalizability [3], as well as the appearance of singularities associated with unbounded tidal forces [4], which signal a breakdown of the classical theory [5,6].

At cosmological scales, the observed accelerated expansion of the Universe [7,8] cannot be explained within the standard cosmological framework without introducing a cosmological constant. The value predicted by Quantum Field Theory, however, exceeds the one inferred from the Friedmann equations by approximately 120 orders of magnitude [9,10]. The effective repulsive component responsible for this acceleration is commonly referred to as dark energy, whose presence is supported primarily by indirect observational evidence [11,12].

Further tensions arise at galactic and larger scales, where the observed dynamics of stars in the outer regions of galaxies (and more generally anomalies in large-scale structure formation) cannot be reconciled with visible matter alone [13,14]. These discrepancies are usually attributed to a hypothetical non-luminous component known as dark matter [15,16]. Taken together, these unresolved issues motivate the search for extended or alternative gravitational theories capable of addressing such phenomena within a unified framework.

To tackle these challenges, along with several others, consideration began to be given to gravitational frameworks that go beyond GR [17,18,19,20]. Within the proposed extensions, a prominent class is obtained by generalizing the Hilbert–Einstein action through the introduction of an arbitrary function of the Ricci scalar, R, giving rise to the so-called $f\left(R\right)$ gravity theories [21,22,23,24,25]. More broadly, the gravitational action can be modified in several directions, including non-minimal couplings between geometry and dynamical scalar fields [26], the inclusion of higher-order derivative terms [27,28,29], or the addition of further curvature invariants beyond the Ricci scalar [30,31,32,33,34].

In this work, we concentrate on the $f\left(R\right)$ generalization of GR, whose field equations (when derived within the metric formalism) are fourth-order in the metric components. A noteworthy feature of these equations is that their further geometric contributions can be formally recast as an effective stress–energy tensor. This interpretation allows the modified geometry itself to mimic dark energy–like behavior, potentially accounting for cosmic acceleration without the explicit introduction of a cosmological constant $\Lambda$. Within this framework, it has been shown that late-time acceleration and quintessence-type phenomena can naturally emerge for suitable choices of the function $f\left(R\right)$ [21,35,36].

Besides modified gravity, late-time acceleration can also be modeled through non-equilibrium properties of the cosmic medium. In viscous cosmology, the large-scale matter content is described as an imperfect fluid endowed with bulk viscosity. In a homogeneous and isotropic cosmological background, bulk viscosity effectively shifts the thermodynamic pressure into an effective pressure of the form $p_{eff}=p-3H\zeta$, where H is the Hubble rate and $\zeta$ is the bulk-viscosity coefficient. The viscosity contribution, therefore, acts as a negative, generally time-dependent pressure term and can drive accelerated expansion even when the equilibrium equation of state would correspond to dust-like matter. A recent representative example is provided by Ref. [37], which investigates bulk-viscous matter in a four-dimensional Einstein–Gauss–Bonnet setting and constrained the model through BAO, Pantheon supernovae, and $H\left(z\right)$ data. The analysis shows that the cosmological dynamics can interpolate between a matter-dominated decelerated era and an asymptotic de Sitter accelerated phase, thus cosmic acceleration may arise either from geometric modifications of gravity or from effective negative-pressure contributions in the matter sector (viscosity), and the two mechanisms may lead to similar background expansion histories.

The use of Noether’s theorem in the context of extended and alternative theories of gravity has been widely explored in the literature (see Refs. [17,18,38] and references therein for a comprehensive overview). In these studies, Noether symmetries are often employed as a guiding principle to constrain or even determine the gravitational action itself. The procedure consists of identifying symmetry transformations of the gravitational Lagrangian and exploiting them to obtain the associated conserved quantities. The existence of such symmetries typically leads to a substantial simplification of the equations of motion, thereby facilitating the search for exact solutions.

Moreover, the conserved quantities provided by Noether’s theorem suggest suitable changes of variables that effectively reduce the dimensionality of the minisuperspace. This reduction plays a crucial role in lowering the mathematical complexity of the system and making the dynamics more manageable. In cosmological applications, for instance, it may correspond to a decrease in the number of independent degrees of freedom, allowing for a clearer analysis of the Universe’s evolution across different epochs. More generally, simplifying the dynamical structure of a system is central to many areas of physics and particularly valuable in gravitational theories, where the equations of motion are often highly intricate. As an example, in quantum cosmology, a reduced minisuperspace can greatly assist in finding solutions to the Wheeler–DeWitt equation [33], which encodes the quantum dynamics of the Universe.

The paper is structured as follows. In Section 2, we provide an overview of Noether’s first theorem and discuss how continuous symmetries are linked to conserved quantities. Building on this connection, we show that, in the presence of internal symmetries, one can outline a systematic procedure for introducing cyclic variables into the dynamical system. Section 3 reframes Noether’s theorem as a tool for selecting physically admissible models that admit symmetries. Within this framework, we apply the Noether symmetry approach to the most general canonical Lagrangian and derive a general condition on the symmetry generator, which is particularly relevant for applications to modified gravity theories. This condition offers a structured method for identifying the symmetries of a given Lagrangian and for constructing the corresponding conserved quantities. In Section 4, we then employ the Noether prescription to determine the functional form of the $f\left(R\right)$ gravitational action in a cosmological setting. Finally, Section 5 summarizes the main results and emphasizes the flexibility and effectiveness of Noether’s theorem in providing deep insights into the dynamics of physical systems, with special emphasis on modified theories of gravity.

2. The Noether First Theorem

We now turn to the statement and proof of Noether’s first theorem. In its formal formulation, the theorem establishes that every differentiable symmetry arising from local transformations gives rise to a corresponding conserved current, commonly referred to as the Noether current. While the connection between symmetries and conservation laws was already recognized before the twentieth century, Noether’s result provides the first general and systematic framework that links continuous symmetries to integrals of motion.

Let us consider an infinitesimal transformation acting on the spacetime coordinates $x^a$ and on the fields ${\varphi }^i$, and examine its effect on the Lagrangian density $\mathcal{L}({\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i,x^a)$:

$$\left\{\begin{array}{c}\mathcal{L}({\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i,x^a)\to \mathcal{L}({\tilde{\varphi }}^i,{\mathcal{\partial }}_a{\tilde{\varphi }}^i,{\tilde{x}}^a)\hfill \\ {\tilde{x}}^a=x^a+\delta x^a\hfill \\ {\tilde{\varphi }}^i\left({\tilde{x}}^a\right)={\varphi }^i\left(x^a\right)+\tilde{\delta }{\varphi }^i\left(x^a\right).\hfill \end{array}\right.$$

(1)

Here, the symbol ∼ indicates the transformed variables, and ${\mathcal{\partial }}_a$ denotes differentiation with respect to the coordinates $x^a$. Furthermore, we distinguish between two types of variations: $\tilde{\delta }$ represents a transformation acting simultaneously on the coordinates and the fields, whereas $\delta$ is reserved for variations affecting only the fields. In particular, we define:

$$\left\{\begin{array}{c}\tilde{\delta }{\varphi }^i\left(x^a\right)={\tilde{\varphi }}^i\left({\tilde{x}}^a\right)-{\varphi }^i\left(x^a\right)\hfill \\ \delta {\varphi }^i\left(x^a\right)={\tilde{\varphi }}^i\left(x^a\right)-{\varphi }^i\left(x^a\right).\hfill \end{array}\right.$$

(2)

By inserting the second relation in Equation (1) into the first expression in Equation (2) and retaining only terms up to first order, one finds

$$\tilde{\delta }{\varphi }^i\left(x^a\right)=\delta {\varphi }^i\left(x^a\right)+{\mathcal{\partial }}_b{\varphi }^i\left(x^a\right)\delta x^b\to \tilde{\delta }=\delta +\delta x^a{\mathcal{\partial }}_a.$$

(3)

If the transformation (1) leaves the equations of motion unchanged, the variational principle in four spacetime dimensions can be rewritten in the form:

$$\tilde{\delta }S=0=\int \tilde{\delta }\mathcal{L}{d}^{4}x+\int \mathcal{L}\delta d^{4}x,$$

(4)

where S denotes the action of the system and $d^{4}x$ represents the spacetime volume element. It is worth emphasizing that requiring the Euler–Lagrange equations to be invariant does not necessarily imply invariance of the Lagrangian density itself; consequently, the condition $\tilde{\delta }\mathcal{L}=0$ corresponds to a particular, more restrictive situation within the general framework expressed in Equation (4). The second integral on the right-hand side can then be computed by making use of the relation:

$$\delta d^{4}x=d^4\tilde{x}-d^{4}x=\left(\left|{\mathcal{\partial }}_b{\tilde{x}}^a\right|-1\right)d^{4}x=\left(\left|{\delta }_b^a+{\mathcal{\partial }}_b\delta x^a\right|-1\right)d^{4}x,$$

(5)

where the vertical bars indicate the determinant of a rank-two matrix. This determinant can be evaluated by exploiting standard matrix properties. In particular, for an n-dimensional matrix M with eigenvalues ${\lambda }_i$, the following identities hold:

$$\begin{array}{ccc}& & \mathrm{Tr}M=\sum _i{\lambda }_i,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & \left|M\right|=\prod _i{\lambda }_i.\hfill \end{array}$$

(7)

Making use of Equation (6), one finds that the inequality $\mathrm{Tr}M\ge {n\left|M\right|}^{1/n}$ is satisfied for strictly positive values of n. When working at first order, this condition reduces to the simpler relation $\mathrm{Tr}M=\left|M\right|$. It is worth noting that this equality is automatically fulfilled by any diagonal matrix whose entries are all equal to unity. In light of these considerations, Equation (5) can therefore be recast as

$$\delta d^{4}x={\mathcal{\partial }}_a\delta x^a{d}^{4}x.$$

(8)

By exploiting Equation (3), the variation of the Lagrangian appearing in the first term of Equation (4) can be written as

$$\begin{array}{ccc}& \tilde{\delta }\mathcal{L}& ={\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\varphi }^i}}\tilde{\delta }{\varphi }^i+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\tilde{\delta }{\mathcal{\partial }}_a{\varphi }^i+{\mathcal{\partial }}_a\mathcal{L}\delta x^a\hfill \\ & & ={\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\varphi }^i}}\delta {\varphi }^i+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\varphi }^i}}{\mathcal{\partial }}_a{\varphi }^i\delta x^a+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\mathcal{\partial }}_a{\varphi }^i+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}{\mathcal{\partial }}_b{\mathcal{\partial }}_a{\varphi }^i\delta x^b+{\mathcal{\partial }}_a\mathcal{L}\delta x^a\hfill \\ & & =\left({\mathcal{\partial }}_a{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}{\mathcal{\partial }}_a\delta {\varphi }^i\right)+\left({\mathcal{\partial }}_a\mathcal{L}+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\varphi }^i}}{\mathcal{\partial }}_a{\varphi }^i+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_b{\varphi }^i\right)}}{\mathcal{\partial }}_b{\mathcal{\partial }}_a{\varphi }^i\right)\delta x^a\hfill \\ & & ={\displaystyle \frac{d}{d{x}^a}}\left({\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i\right)+{\displaystyle \frac{d\mathcal{L}}{d{x}^a}}\delta x^a,\hfill \end{array}$$

(9)

where in the last equality we used the Euler–Lagrange equation to rewrite the first term in the first bracket.

By merging Equations (8) and (9), from Equation (4) one gets:

$$\begin{array}{ccc}& & {\displaystyle \frac{d}{d{x}^a}}\left({\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i\right)+{\displaystyle \frac{d\mathcal{L}}{d{x}^a}}\delta x^a+\mathcal{L}{\displaystyle \frac{d\delta x^a}{d{x}^a}}=\hfill \\ & & ={\displaystyle \frac{d}{d{x}^a}}\left({\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i\right)+{\displaystyle \frac{d\left(\mathcal{L}\delta x^a\right)}{d{x}^a}}=\hfill \\ & & ={\mathcal{\partial }}_a\left({\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i+\mathcal{L}\delta x^a\right)={\mathcal{\partial }}_a{j}^a=0,\hfill \end{array}$$

(10)

with $j^a$ being defined as the Noether Current, namely

$${\displaystyle j^a=\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}\tilde{\delta }{\varphi }^i-\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}{\mathcal{\partial }}_b{\varphi }^i\delta x^b+\mathcal{L}\delta x^a-g^a,}$$

(11)

with $g^a$ being a vector field with null divergence. This outcome shows that whenever the transformation (1) leaves the action invariant, a corresponding set of conserved quantities necessarily follows, thus completing the proof of the original claim. Furthermore, upon inserting Equation (3) into Equation (11), the Noether current may be written in the equivalent form:

$${\displaystyle j^a=\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}\delta {\varphi }^i+\mathcal{L}\delta x^a-g^a.}$$

(12)

Equations (11) and (12) are fully equivalent, and the choice between them is therefore a matter of practical convenience. The symmetry generator corresponding to the transformations (1) is denoted by $X^{\left[1\right]}$ and is referred to as the first prolongation of the Noether vector, as it depends only on the generalized coordinates and their first derivatives. Its explicit form is given by:

$$X^{\left[1\right]}=\delta x^a{\mathcal{\partial }}_a+\delta {\varphi }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\varphi }^i}}+({\mathcal{\partial }}_a\delta {\varphi }^i-{\mathcal{\partial }}_a{\varphi }^i{\mathcal{\partial }}_b\delta x^b){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}.$$

(13)

As discussed in greater detail in Section 3, one can readily verify that imposing the following condition yields the same conserved quantity as that given in Equation (11):

$$X^{\left[1\right]}\mathcal{L}+{\mathcal{\partial }}_a\delta x^a\mathcal{L}={\mathcal{\partial }}_a{g}^a.$$

(14)

which implies that the condition above may be regarded as an equivalent statement of Noether’s theorem.

Internal Symmetries

We now focus on a particular instance of the previous proof of Noether’s theorem by assuming that the symmetry transformation is an on-shell internal symmetry, meaning that it does not involve any variation of the spacetime coordinates. Under this assumption, the transformations (1) reduce to

$$\left\{\begin{array}{c}\delta x^a=0\hfill \\ \delta {\varphi }^a={\tilde{\varphi }}^i-{\varphi }^i\hfill \\ \tilde{\mathcal{L}}({\tilde{x}}^a,{\tilde{\varphi }}^i,{\mathcal{\partial }}_a\widetilde{{\varphi }^i})=\mathcal{L}(x^a,{\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i),\hfill \end{array}\right.$$

(15)

thus the Lagrangian variation can be written as

$$\delta \mathcal{L}={\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\varphi }^i}}\delta {\varphi }^i+{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\mathcal{\partial }}_a{\varphi }^i={\mathcal{\partial }}_a\left({\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i\right)=0,$$

(16)

where in the second equality we used the Euler–Lagrange equation. Under this assumption, both the Noether current and the corresponding Noether vector take on the simplified form:

$$\begin{array}{ccc}& & j^a={\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\varphi }^i,\hfill \\ & & X={\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\varphi }^i}}\delta {\varphi }^i+{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}\delta {\mathcal{\partial }}_a{\varphi }^i.\hfill \end{array}$$

(17)

Furthermore, when variations of the coordinates are set to zero, the symmetry condition given in Equation (14) simplifies to the requirement $X\mathcal{L}=0$. Although this represents a reduced form of the general theorem, it is particularly well suited for the systematic introduction of cyclic variables within the Lagrangian formalism. The objective is to determine a new variable, denoted by ${\psi }^1$, whose conjugate momentum is conserved, namely

$${\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\psi }^1\right)}}={\pi }_{{\psi }^1}^a=\mathrm{const}.$$

(18)

The first relation in Equation (17) shows that the Noether current can be identified with a constant only when the infinitesimal generator associated with ${\psi }^1$ is of order unity.

Let us therefore consider a general change of variables ${\varphi }^i\to {\psi }^i\left({\varphi }^j\right)$, with ${\psi }^1$ chosen to be a cyclic coordinate. In terms of the new variables ${\psi }^i$, the Noether vector X can be written by means of the interior derivative $i_X$. In particular, the infinitesimal generators corresponding to the original variables ${\varphi }^i$ are related to those of the transformed variables ${\psi }^i$ through the following relations:

$$\delta {\varphi }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\varphi }^i}}=\delta {\varphi }^i{\displaystyle \frac{\mathcal{\partial }{\psi }^j}{\mathcal{\partial }{\varphi }^i}}{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\psi }^j}}=i_{X}d{\psi }^j{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\psi }^j}}.$$

(19)

The relation above allows us to easily write the transformed Noether vector $X^{\prime }$ in the form

$$\begin{array}{c}\hfill X^{\prime }=\delta {\varphi }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\varphi }^i}}+{\mathcal{\partial }}_a\delta {\varphi }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}=\left(i_{X}d{\psi }^k\right){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\psi }^k}}+{\mathcal{\partial }}_a\left(i_{X}d{\psi }^k\right){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\psi }^k\right)}}.\end{array}$$

(20)

Consequently, the conserved quantity $j^a$ can be re-expressed in terms of the fields ${\psi }^i$ as follows:

$$j^a=\delta {\psi }^i{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\psi }^i\right)}}=i_{X}d{\psi }^i{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\psi }^i\right)}}.$$

(21)

In order for the conserved quantity to coincide with the momentum conjugate to ${\psi }^1$, the following conditions are required to hold:

$$i_{X}d{\psi }^1=\delta {\varphi }^j{\displaystyle \frac{\mathcal{\partial }{\psi }^1}{\mathcal{\partial }{\varphi }^j}}=1,i_{X}d{\psi }^i=\delta {\varphi }^j{\displaystyle \frac{\mathcal{\partial }{\psi }^i}{\mathcal{\partial }{\varphi }^j}}=0,i\ne 1,$$

(22)

so that, from Equation (21), one obtains

$$j^a=\delta {\varphi }^i{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}={\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\psi }^1\right)}}\to {\pi }_{{\psi }^1}^a=\mathrm{constant}.$$

(23)

As a consequence, the momentum conjugate to the cyclic variable ${\psi }^1$, denoted by ${\pi }_{{\psi }^1}^a$, is exactly identified with the Noether current, in full agreement with the construction. It is worth stressing that the condition $X^{\prime }{\mathcal{L}}^{\prime }=X\mathcal{L}=0$ holds independently of the choice of variables, thereby guaranteeing that the Noether symmetry is preserved under changes of field variables. Moreover, the requirement $X\mathcal{L}=0$ is equivalent to demanding that the Lie derivative of the Lagrangian along the vector field X vanishes, namely $X\mathcal{L}=L_X\mathcal{L}=0$, which signals the presence of internal symmetries in the system.

Finally, it should be emphasized that the variable transformation defined in Equation (22) is by no means unique, since infinitely many field redefinitions are in principle allowed. As a result, the choice of new variables aimed at simplifying the dynamics must be carried out with care.

3. From the Noether Theorem to the Noether Symmetry Approach

In this section, we focus on the central theme of the present work, namely the selection of theories that admit symmetries from a wide class of admissible Lagrangians. In the preceding discussion, we showed how conserved quantities and symmetry generators can be determined once the Lagrangian of the system is specified. In the standard approach, the symmetries of a theory are inferred from the explicit form of the Lagrangian, which in turn allows one to identify the transformations that leave the equations of motion invariant. Within this framework, the associated Noether current (11) arises naturally.

Here, we emphasize that Noether’s theorem can also be exploited as a constructive tool for identifying physically meaningful models that are amenable to exact treatment. Rather than fixing the Lagrangian from the outset, we begin with broad families of Lagrangians and restrict their functional form by demanding the existence of symmetries. In practice, this amounts to imposing the condition (14) to reduce the complexity of the dynamics and to single out those Lagrangians that satisfy the required symmetry properties. Although the explicit form of the symmetry transformation is not known beforehand, we assume the existence of at least one transformation that leaves the Euler–Lagrange equations invariant. This strategy not only leads to dynamical systems that can be integrated exactly, but also provides a selection principle for viable theories. We note, however, that no general prescription exists for determining in advance whether a given Lagrangian will satisfy the Noether symmetry approach, even though the resulting models are typically physically consistent.

In the analysis of symmetries, we will make use of both the Noether vector X and its first prolongation $X^{\left[1\right]}$, clarifying their respective roles on a case-by-case basis. The infinitesimal generators $\delta x$, $\delta \varphi$, and $\delta \left({\mathcal{\partial }}_a\varphi \right)$ are not specified a priori; instead, they emerge naturally as solutions of the Noether system. Consequently, these generators are functions of the configuration-space variables and must be determined through the application of Noether’s theorem. For notational convenience, we introduce the following definitions:

$$\begin{array}{c}\delta x^a\equiv {\xi }^a(x^a,{\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i),\delta {\varphi }^i={\eta }^i(x^a,{\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i),\\ \delta {\mathcal{\partial }}_a{\varphi }^i={\mathcal{\partial }}_a{\eta }^i(x^a,{\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i),\end{array}$$

(24)

such that the first prolongation of the Noether vector defined in Equation (13) takes the form:

$$X^{\left[1\right]}={\xi }^a{\mathcal{\partial }}_a+{\eta }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\varphi }^i}}+\left({\mathcal{\partial }}_a{\eta }^i-{\mathcal{\partial }}_a{\varphi }^i{\mathcal{\partial }}_b{\xi }^b\right){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}.$$

(25)

The ordinary Noether vector X is readily recovered by imposing ${\xi }^a=0$. Furthermore, because the transformation generated by X is independent of the spacetime coordinates, the corresponding non-extended Noether vector can be written in the form:

$$X={\eta }^i({\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\varphi }^i}}+{\mathcal{\partial }}_a{\eta }^i({\varphi }^i,{\mathcal{\partial }}_a{\varphi }^i){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}}$$

(26)

and, thus, the expressions for internal symmetries become, respectively,

$$\left\{\begin{array}{c}\left(X^{\left[1\right]}+{\mathcal{\partial }}_a{\xi }^a\right)\mathcal{L}={\mathcal{\partial }}_a{g}^a\hfill \\ L_X\mathcal{L}=X\mathcal{L}=0,\hfill \end{array}\right.$$

(27)

with corresponding conserved quantities of the form

$$\left\{\begin{array}{c}{\displaystyle j^{a\left[1\right]}=\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}{\eta }^i-\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}{\mathcal{\partial }}_b{\varphi }^i{\xi }^b+\mathcal{L}{\xi }^a-g^a}\hfill \\ {\displaystyle j^a=\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\left({\mathcal{\partial }}_a{\varphi }^i\right)}{\eta }^i.}\hfill \end{array}\right.$$

(28)

In both cases, one obtains a homogeneous polynomial of second order in the velocities, supplemented by an inhomogeneous term depending on the fields ${\varphi }^i$. Requiring this polynomial to vanish identically forces each of its coefficients to be zero independently. As a result, the corresponding symmetry condition is overdetermined: whenever a solution exists, it is characterized by integration constants rather than by boundary conditions.

Application to Canonical Lagrangians

In what follows, we summarize the main steps leading to the general Noether identity (14) and to the condition $X\mathcal{L}=0$, which corresponds to the vanishing of the Lie derivative, focusing in particular on Lagrangians with explicit time dependence. We then formulate a set of guidelines for systematically constructing a suitable change of variables. The examples discussed in this section provide useful and illustrative settings that clarify how the method can be applied to cosmological Lagrangians.

Let us therefore consider a Lagrangian of the form $\mathcal{L}(q^i,{\dot{q}}^i)$, depending on generalized coordinates and their associated velocities. Introducing the quantity ${\eta }^{i\left[1\right]}={\dot{\eta }}^i-{\dot{q}}^i\dot{\xi }$, the symmetry transformations given in Equation (29) can be rewritten as

$$\left\{\begin{array}{c}\tilde{t}=t+ϵ\xi (t,q^i)+O\left({ϵ}^2\right)\hfill \\ {\tilde{q}}^i=q^i+ϵ{\eta }^i(t,q^i)+O\left({ϵ}^2\right)\hfill \\ {\dot{\tilde{q}}}^i={\dot{q}}^i+ϵ{\eta }^{i\left[1\right]}\hfill \end{array}\right.$$

(29)

yielding a first prolongation of Noether’s vector of the form:

$$X^{\left[1\right]}=\xi {\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }t}}+{\eta }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{q}^i}}+{\eta }^{i\left[1\right]}{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\dot{q}}^i}}=\xi {\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }t}}+{\eta }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{q}^i}}+({\dot{\eta }}^i-{\dot{q}}^i\dot{\xi }){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\dot{q}}^i}}.$$

(30)

We emphasize that the Lagrangian is assumed to depend only on the generalized coordinates and their first derivatives, so transformations involving higher-order terms such as ${\ddot{q}}^i$ do not need to be considered. As demonstrated in the previous section, if the coordinate transformation (29) leaves the equations of motion invariant, then the system necessarily fulfills the Noether identity

$$X^{\left[1\right]}\mathcal{L}+\dot{\xi }\mathcal{L}=\dot{g}(t,q^i),$$

(31)

from which the integral of motion

$${\displaystyle I(t,q^i,{\dot{q}}^i)=\xi \left({\dot{q}}^i\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\dot{q}}^i}-\mathcal{L}\right)-{\eta }^i\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\dot{q}}^i}+g(t,q^i)}$$

(32)

automatically follows.

As discussed above, imposing the condition $L_X\mathcal{L}=0$ restricts the analysis to internal symmetries, for which the infinitesimal coordinate variation $\delta x^a={\xi }^a$ does not explicitly contribute to the transformation. Despite this, the function ${\xi }^a(x^a,{\varphi }^i)$ often plays an important role in the identification of symmetry properties. Our goal is therefore to determine whether a general condition on the infinitesimal generator ${\xi }^a$ can be formulated that holds for any canonical Lagrangian. To derive such a constraint, we apply the Noether symmetry approach to the most general canonical Lagrangian, namely

$$\mathcal{L}\left(q^i,{\dot{q}}^i\right)=a_{ij}\left(q^i\right){\dot{q}}^i{\dot{q}}^j-V\left(q^i\right)-A_i\left(q^i\right){\dot{q}}^i,$$

(33)

where $a_{ij}$ denotes a matrix-valued function of the generalized coordinates $q^i$, $V\left(q\right)$ represents the potential term, and $A_i$ is a vector function depending on $q^i$. In the case of a two-particle system, with the index i taking values from 1 to 2, the first prolongation of the Noether vector takes the form:

$$X^{\left[1\right]}=\xi {\mathcal{\partial }}_t+\alpha {\mathcal{\partial }}_{q_1}+\beta {\mathcal{\partial }}_{q_2}+(\dot{\alpha }-{\dot{q}}_1\dot{\xi }){\mathcal{\partial }}_{{\dot{q}}_1}+(\dot{\beta }-{\dot{q}}_2\dot{\xi }){\mathcal{\partial }}_{{\dot{q}}_2},$$

(34)

with ${\eta }^i\equiv (\alpha ,\beta )$ and $q^i\equiv (q_1\left(t\right),q_2\left(t\right))$. We note that Equation (34) corresponds to the two-dimensional specialization of the vector introduced in Equation (13). Enforcing the identity given in Equation (14) then leads to a system consisting of 14 differential equations:

$$\left\{\begin{array}{c}{\displaystyle \alpha {\mathcal{\partial }}_{q_1}V+\beta {\mathcal{\partial }}_{q_2}V+\left({\mathcal{\partial }}_t\alpha \right)A_1+\left({\mathcal{\partial }}_t\beta \right)A_2+\left({\mathcal{\partial }}_t\xi \right)V-{\mathcal{\partial }}_{t}g=0}\hfill \\ {\displaystyle -\alpha {\mathcal{\partial }}_{q_1}{A}_1-\beta {\mathcal{\partial }}_{q_2}{A}_1-A_1{\mathcal{\partial }}_{q_1}\alpha +2\left({\mathcal{\partial }}_t\alpha \right)a_{12}-A_2{\mathcal{\partial }}_{q_1}\beta +\left({\mathcal{\partial }}_t\beta \right)a_{12}+\left({\mathcal{\partial }}_t\beta \right)a_{21}-{\mathcal{\partial }}_{q_1}g=0}\hfill \\ {\displaystyle -\alpha {\mathcal{\partial }}_{q_1}{A}_2-\beta {\mathcal{\partial }}_{q_2}{A}_2-A_1{\mathcal{\partial }}_{q_2}\alpha +\left({\mathcal{\partial }}_t\alpha \right)a_{12}+\left({\mathcal{\partial }}_t\alpha \right)a_{21}-A_2{\mathcal{\partial }}_{q_2}\beta +2\left({\mathcal{\partial }}_t\beta \right)a_{22}-{\mathcal{\partial }}_{q_2}g=0}\hfill \\ {\displaystyle \alpha {\mathcal{\partial }}_{q_1}{a}_{11}+\beta {\mathcal{\partial }}_{q_2}{a}_{11}+2\left({\mathcal{\partial }}_{q_1}\alpha \right)a_{11}-\left({\mathcal{\partial }}_t\xi \right)a_{11}+\left({\mathcal{\partial }}_{q_1}\beta \right)a_{12}+\left({\mathcal{\partial }}_{q_1}\beta \right)a_{21}=0}\hfill \\ {\displaystyle \alpha {\mathcal{\partial }}_{q_1}{a}_{22}+\beta {\mathcal{\partial }}_{q_2}{a}_{22}+\left({\mathcal{\partial }}_{q_2}\alpha \right)a_{12}+\left({\mathcal{\partial }}_{q_2}\alpha \right)a_{21}+2\left({\mathcal{\partial }}_{q_2}\beta \right)a_{22}-\left({\mathcal{\partial }}_q\xi \right)a_{22}=0}\hfill \\ {\displaystyle \alpha {\mathcal{\partial }}_{q_1}{a}_{12}+\alpha {\mathcal{\partial }}_{q_1}{a}_{21}+\beta {\mathcal{\partial }}_{q_2}{a}_{12}+\beta {\mathcal{\partial }}_{q_2}{a}_{21}+2\left({\mathcal{\partial }}_{q_2}\alpha \right)a_{11}}\hfill \\ {\displaystyle +\left({\mathcal{\partial }}_{q_1}\alpha \right)a_{12}+\left({\mathcal{\partial }}_{q_1}\alpha \right)a_{21}-\left({\mathcal{\partial }}_t\xi \right)a_{12}-\left({\mathcal{\partial }}_t\xi \right)a_{21}+\left({\mathcal{\partial }}_{q_2}\beta \right)a_{12}+\left({\mathcal{\partial }}_{q_2}\beta \right)a_{21}+2\left({\mathcal{\partial }}_{q_1}\beta \right)a_{22}=0}\hfill \\ {\displaystyle -\left({\mathcal{\partial }}_{q_1}\xi \right)a_{12}-\left({\mathcal{\partial }}_{q_1}\xi \right)a_{21}-\left({\mathcal{\partial }}_{q_2}\xi \right)a_{11}=0}\hfill \\ {\displaystyle -\left({\mathcal{\partial }}_{q_2}\xi \right)a_{12}-\left({\mathcal{\partial }}_{q_2}\xi \right)a_{21}-\left({\mathcal{\partial }}_{q_1}\xi \right)a_{22}=0}\hfill \\ {\displaystyle -2\left({\mathcal{\partial }}_{q_1}\xi \right)a_{11}+\left({\mathcal{\partial }}_{q_1}\xi \right)a_{11}=0}\hfill \\ {\displaystyle -2\left({\mathcal{\partial }}_{q_2}\xi \right)a_{22}+\left({\mathcal{\partial }}_{q_2}\xi \right)a_{22}=0.}\hfill \end{array}\right.$$

(35)

The large number of unknown functions makes it impossible to obtain a unique solution for the full system, rendering it overdetermined. Nevertheless, only the final two equations are required to derive a general constraint on the function $\xi$. In particular, assuming that $a_{11}\ne 0$ and $a_{22}\ne 0$, these two equations are satisfied if and only if ${\mathcal{\partial }}_{q_1}\xi ={\mathcal{\partial }}_{q_2}\xi =0$. This condition implies that $\xi$ can depend exclusively on time. Such a general result may be employed a priori in subsequent applications involving canonical Lagrangians, as will be illustrated in Section 4.1.

4. The Noether Symmetry Approach for Theories of Gravity

As discussed above, Noether’s theorem may be regarded as a powerful tool for identifying theories that are invariant under given transformations and for extracting the corresponding symmetry generators. This perspective is particularly effective in the context of electromagnetic, weak, and strong interactions, which are successfully described within the Yang–Mills framework. In the gravitational sector, however, such an approach becomes even more significant, since no single theory is currently capable of providing a complete and consistent description of gravity at all energy scales.

Modified theories of gravity are often characterized by a certain degree of arbitrariness, as their actions typically involve unknown functions of curvature or torsion invariants. By analyzing the symmetry properties of gravitational Lagrangians, one can impose meaningful constraints on the functional form of the underlying action, following the strategy described in Section 3.

It should be stressed that the application of the Noether symmetry approach to gravitational systems is not free of subtleties. For instance, one might be tempted to apply Noether’s theorem directly to the fully covariant action of a modified gravity theory without fixing a background metric. However, since Noether’s theorem was originally formulated in the context of flat spacetime, a more appropriate procedure is to work with the corresponding point-like Lagrangian rather than the full Lagrangian density. Within this viewpoint, the geometric degrees of freedom appearing in the action are treated as dynamical fields evolving on a flat background.

For these reasons, the analysis carried out in this section is performed by adopting a spatially flat cosmological line element of the form:

$$d{s}^2=d{t}^2-a{\left(t\right)}^{2}d{\left|\mathbf{x}\right|}^2,$$

(36)

where $\mathbf{x}$ denotes the spatial coordinates and $a\left(t\right)$ is the scale factor, which depends on the cosmic time t. The line element given in Equation (36) will be employed to derive the point-like Lagrangian from the fully covariant $f\left(R\right)$ gravitational action. Since this reduced Lagrangian depends solely on the cosmic time, the scale factor $a\left(t\right)$, and the Ricci scalar $R\left(t\right)$, the Noether identity (14) simplifies to

$$X^{\left[1\right]}\mathcal{L}+\dot{\xi }\mathcal{L}=\dot{g}(t,q^i),$$

(37)

with the first prolongation of the Noether vector and the associated conserved quantity given, respectively, by

$$X^{\left[1\right]}=\xi {\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }t}}+{\eta }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{q}^i}}+{\eta }^{i\left[1\right]}{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\dot{q}}^i}}=\xi {\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }t}}+{\eta }^i{\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{q}^i}}+({\dot{\eta }}^i-{\dot{q}}^i\dot{\xi }){\displaystyle \frac{\mathcal{\partial }}{\mathcal{\partial }{\dot{q}}^i}},$$

(38)

$${\displaystyle j(t,q^i,{\dot{q}}^i)=\xi \left({\dot{q}}^i\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\dot{q}}^i}-\mathcal{L}\right)-{\eta }^i\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\dot{q}}^i}+g(t,q^i).}$$

(39)

Also notice that $q^i=\{a\left(t\right),R\left(t\right)\}$, $\xi =\xi (t,a,R)$, ${\eta }^i=\{\alpha (t,a,R),\beta (t,a,R),\gamma (t,a,R)\}$.

Furthermore, in the spacetime described by Equation (36), the metric does not explicitly depend on the spatial coordinates $\mathbf{x}$. As a result, the Lagrangian density can be integrated over the spatial hypersurface, yielding

$$S=\int \mathcal{L}{d}^{D}x=\int \mathcal{L}{d}^{D-1}xdt=\int Ldt,$$

(40)

where we have defined $L\equiv \int \sqrt{-g}\mathcal{L}{d}^{D-1}x$, with D being the number of space-time dimensions.

4.1. The Case of $f\left(R\right)$ Gravity

We begin by formulating the Lagrangian framework for $f\left(R\right)$ gravity, starting from the action given by

$$S=\int \sqrt{-g}f\left(R\right)d^{4}x,$$

(41)

where g denotes the determinant of the metric tensor $g_{\mu \nu }$, which, in the scenario considered here, is a diagonal rank-two tensor given by $g_{\mu \nu }=\mathrm{diag}\left(1,-a^2\left(t\right),-a^2\left(t\right),-a^2\left(t\right)\right)$. Note that we adopt the metric signature $(+,-,-,-)$.

To obtain the corresponding point-like Lagrangian, we employ the method of Lagrange multipliers in a cosmological setting. Specifically, for a spatially flat Friedmann–Lemaître–Robertson–Walker spacetime described by the line element (36), the Ricci scalar R takes the cosmological form

$$R=-6\left({\displaystyle \frac{\ddot{a}}{a}}+{\displaystyle \frac{{\dot{a}}^2}{a^2}}\right).$$

(42)

By introducing a Lagrange multiplier $\lambda$ and taking into account that the minisuperspace variables depend only on time, the action can be rewritten as

$$S=\int \left[a^{3}f\left(R\right)-\lambda \left(R+6{\displaystyle \frac{\ddot{a}}{a}}+6{\displaystyle \frac{{\dot{a}}^2}{a^2}}\right)\right]dt.$$

(43)

Because all variables are independent of the spatial coordinates, the integration over the spatial hypersurface can be performed explicitly, reducing the general integral in Equation (41) to an integral over time only. Varying the action with respect to the scalar curvature then allows one to determine the Lagrange multiplier $\lambda$ as

$${\displaystyle \frac{\delta S}{\delta R}}=a^3{f}_R\left(R\right)-\lambda =0,\lambda =a^3{f}_R\left(R\right).$$

(44)

In order to arrive at a canonical Lagrangian, the terms containing second-order derivatives must be integrated by parts, after which the divergence theorem is applied. As a result, one obtains the following point-like Lagrangian [39].

$$\mathcal{L}=a^3\left[f\left(R\right)-R{f}_R\left(R\right)\right]+6a{\dot{a}}^2{f}_R\left(R\right)+6a^2\dot{a}\dot{R}{f}_{RR}\left(R\right),$$

(45)

The Euler–Lagrange equations provide the equations of motion for a(t) and R(t). In addition, minisuperspace gravitational Lagrangians satisfy the energy condition (Hamiltonian constraint)

$$E_L\equiv \dot{a}{\displaystyle \frac{\mathcal{\partial }L}{\mathcal{\partial }\dot{a}}}+\dot{R}{\displaystyle \frac{\mathcal{\partial }L}{\mathcal{\partial }\dot{R}}}-L=0,$$

(46)

which is equivalent to the (0, 0) component of the field equations. For Equation (45), this gives

$$6a^2\dot{a}\dot{R}{f}_{RR}\left(R\right)+6a{\dot{a}}^2{f}_R\left(R\right)-a^3\left[f\left(R\right)-R{f}_R\left(R\right)\right]=0,$$

i.e., the first equation in system (46).

From this Lagrangian, two Euler–Lagrange equations can be obtained by varying with respect to the scale factor and the scalar curvature, respectively. The resulting equations of motion must be supplemented by an additional relation, commonly referred to as the energy condition. In fact, minisuperspace gravitational Lagrangians satisfy the Hamiltonian constraint $E_{\mathcal{L}}\equiv {\dot{q}}^i{\displaystyle \frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }{\dot{q}}^i}}=0$, which in the present case of $f\left(R\right)$ gravity with a three-dimensional minisuperspace can be recast as ${\displaystyle \dot{a}\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\dot{a}}+\dot{R}\frac{\mathcal{\partial }\mathcal{L}}{\mathcal{\partial }\dot{R}}-\mathcal{L}=0}$.

This condition corresponds to the vanishing of the gravitational energy and is associated with the $(0,0)$ component of the field equations derived from varying the action with respect to the metric tensor. Its inclusion is essential to ensure the equivalence between the Lagrangian formulation and the standard variational approach. Alternatively, one can also consider a lapse function $N\left(t\right)$ as a $(0,0)$ component of the metric and get the corresponding point-like Lagrangian. The related Euler–Lagrange equation with respect to the lapse function will be equivalent to the energy condition, after setting $N\left(t\right)=1$ (see [18] for further details). The complete set of equations of motion is therefore given by:

$$\left\{\begin{array}{c}{\displaystyle 6a^2\dot{a}\dot{R}{f}_{RR}\left(R\right)+6a{\dot{a}}^2{f}_R\left(R\right)-a^3[f\left(R\right)-R{f}_R\left(R\right)]=0}\hfill \\ {\displaystyle {\dot{R}}^2{f}_{RRR}\left(R\right)+\ddot{R}{f}_{RR}\left(R\right)+\frac{{\dot{a}}^2}{a^2}{f}_R\left(R\right)+2\frac{\ddot{a}}{a}{f}_R\left(R\right)-\frac{1}{2}[f\left(R\right)-R{f}_R\left(R\right)]+2\frac{\dot{a}}{a}\dot{R}{f}_{RR}\left(R\right)=0}\hfill \\ {\displaystyle R=-6\left(\frac{\ddot{a}}{a}+\frac{{\dot{a}}^2}{a^2}\right).}\hfill \end{array}\right.$$

(47)

It is worth noting that the third equation, namely the equation of motion obtained by varying with respect to R, reproduces the cosmological expression for the scalar curvature that was introduced in Equation (42) when constructing the original action. This equation is directly associated with the Lagrange multiplier and, as such, its appearance is expected by design, providing a useful consistency check of the procedure.

The system outlined above can be solved only once a specific form of the otherwise arbitrary function $f\left(R\right)$ is chosen through the imposition of Noether symmetries. To this end, we note that the first prolongation of the Noether vector acts on the minisuperspace of variables $\mathcal{S}\equiv a,R$ and therefore assumes the form:

$$\begin{array}{ccc}\hfill X^{\left[1\right]}& =& \xi (a,R,t){\mathcal{\partial }}_t+\alpha (a,R,t){\mathcal{\partial }}_a+\beta (a,R,t){\mathcal{\partial }}_R+\left(\dot{\alpha }(a,R,t)-\dot{\xi }(a,R,t)\dot{a}\right){\mathcal{\partial }}_{\dot{a}}\hfill \\ & +& \left(\dot{\beta }(a,R,t)-\dot{\xi }(a,R,t)\dot{R}\right){\mathcal{\partial }}_{\dot{R}}.\hfill \end{array}$$

(48)

By writing the time derivatives of the infinitesimal generators in terms of the canonical variables and applying the Noether identity (37) to the point-like Lagrangian (45), we get a system of nine differential equations [39,40]:

$$\left\{\begin{array}{c}{\displaystyle 3\alpha f\left(R\right)-3\alpha R{f}_R\left(R\right)-\beta a{f}_{RR}\left(R\right)-\frac{{\mathcal{\partial }}_{t}g}{a^2}+{\mathcal{\partial }}_t\xi a[f\left(R\right)-R{f}_R\left(R\right)]=0}\hfill \\ {\displaystyle {\mathcal{\partial }}_R\alpha f_{RR}\left(R\right)=0}\hfill \\ {\displaystyle {\mathcal{\partial }}_R\xi f_{RR}\left(R\right)=0}\hfill \\ {\displaystyle 6{\mathcal{\partial }}_t\alpha f_{RR}\left(R\right)+a[f\left(R\right)-R{f}_R\left(R\right)]{\mathcal{\partial }}_R\xi -\frac{{\mathcal{\partial }}_{R}g}{a^2}=0}\hfill \\ {\displaystyle 2\alpha f_{RR}\left(R\right)+\beta a{f}_{RRR}-{\mathcal{\partial }}_t\xi a{f}_{RR}\left(R\right)+2{\mathcal{\partial }}_R\alpha f_R\left(R\right)+{\mathcal{\partial }}_R\beta a{f}_{RR}\left(R\right)+{\mathcal{\partial }}_a\alpha a{f}_{RR}\left(R\right)=0}\hfill \\ {\displaystyle f_R{\mathcal{\partial }}_a\xi =0}\hfill \\ {\displaystyle 12{\mathcal{\partial }}_t\alpha f_R\left(R\right)+6a{\mathcal{\partial }}_t\beta f_{RR}\left(R\right)+a^2[f\left(R\right)-R{f}_R\left(R\right)]{\mathcal{\partial }}_a\xi -\frac{{\mathcal{\partial }}_{a}g}{a}=0}\hfill \\ {\displaystyle \alpha f_R\left(R\right)+\beta a{f}_{RR}\left(R\right)-a{\mathcal{\partial }}_t\xi f_R\left(R\right)+2a{\mathcal{\partial }}_a\alpha f_R\left(R\right)+a^2{\mathcal{\partial }}_a\beta f_{RR}\left(R\right)=0}\hfill \\ {\displaystyle {\mathcal{\partial }}_R\xi f_R\left(R\right)+a{\mathcal{\partial }}_a\xi f_{RR}\left(R\right)=0,}\hfill \end{array}\right.$$

(49)

where $\alpha$ and $\beta$ denote the components of the function ${\eta }^i$ introduced in Equation (25), that is ${\eta }^i\equiv (\alpha ,\beta )$, with the index i taking values from 1 to 2. The system above is obtained by enforcing the condition (37) and by separately setting to zero all terms with the same order in time derivatives. A particularly simple solution arises by imposing $f_{RR}\left(R\right)=0$, which leads to

$$f\left(R\right)=f_{0}R+\Lambda ,\alpha =\pm \sqrt{{\alpha }_{0}a+{\alpha }_1},\xi ={\xi }_{0}t+{\xi }_1,\beta =\beta (a,R,t),$$

(50)

with ${\alpha }_0$, ${\alpha }_1$, ${\xi }_0$, ${\xi }_1$ and $f_0$ being integration constants. The conserved quantity associated with this symmetry vanishes identically, independently of the explicit time dependence of the minisuperspace variables. Consequently, this case corresponds to a trivial solution lacking physical relevance. In contrast, when $f_{RR}\left(R\right)\ne 0$, the system can be consistently reduced to three differential equations of the form:

$$\left\{\begin{array}{c}{\displaystyle 3\alpha [f\left(R\right)-R{f}_R\left(R\right)]-\beta aR{f}_{RR}\left(R\right)+a[f\left(R\right)-R{f}_R\left(R\right)]{\mathcal{\partial }}_t\xi =0}\hfill \\ {\displaystyle 2\alpha f_{RR}\left(R\right)+\beta a{f}_{RRR}\left(R\right)-{\mathcal{\partial }}_t\xi a{f}_{RR}\left(R\right)+{\mathcal{\partial }}_R\beta a{f}_{RR}\left(R\right)+a{\mathcal{\partial }}_a\alpha f_{RR}\left(R\right)=0}\hfill \\ {\displaystyle \alpha f_R\left(R\right)+\beta a{f}_{RR}\left(R\right)-{\mathcal{\partial }}_t\xi a{f}_R\left(R\right)+2a{\mathcal{\partial }}_a\alpha f_R\left(R\right)+a^2{\mathcal{\partial }}_a\beta f_{RR}\left(R\right)=0}\hfill \\ {\displaystyle \xi \equiv \xi \left(t\right),\alpha \equiv \alpha \left(a\right),\beta \equiv \beta (a,R).}\hfill \end{array}\right.$$

(51)

Since the Lagrangian (45) is written in canonical form, the function $\xi$ is constrained to depend solely on the cosmic time, as already discussed in Section 3. Imposing the conditions $f_{RR}\ne 0$ and $\xi \left(t\right)\ne 0$, the system admits three further nontrivial solutions, which can be summarized as follows:

$$\left\{\begin{array}{c}{\displaystyle I:\alpha =\frac{7{\xi }_0}{36}a,\beta =-\frac{7{\xi }_0}{6}R,\xi ={\xi }_{0}t+{\xi }_1,f\left(R\right)=f_0{R}^{\frac{19}{14}}+\Lambda }\hfill \\ {\displaystyle II:\xi ={\xi }_0,\alpha =\beta =0,f\left(R\right)=f_0{R}^{\frac{19}{14}}+\Lambda }\hfill \\ {\displaystyle III:\xi ={\xi }_{0}t+{\xi }_1,\alpha ={\alpha }_{0}a,\beta =-2{\xi }_{0}R,f\left(R\right)=f_0{R}^k,k=1+\frac{3{\alpha }_0}{2{\xi }_0},}\hfill \end{array}\right.$$

(52)

hence the corresponding generators and conserved quantities are

$$\left\{\begin{array}{c}{\displaystyle X_1=({\xi }_{0}t+{\xi }_1){\mathcal{\partial }}_t+\frac{7{\xi }_0}{36}a{\mathcal{\partial }}_a-\frac{7{\xi }_0}{6}R{\mathcal{\partial }}_R}\hfill \\ {\displaystyle j_1=a^{3}t[f\left(R\right)-R{f}_R\left(R\right)]+6t[a{\dot{a}}^2{f}_R\left(R\right)+a^2\dot{a}\dot{R}{f}_{RR}\left(R\right)]+\Lambda t{a}^3}\hfill \\ {\displaystyle +7a^2\left[-\frac{1}{3}\dot{a}{f}_R\left(R\right)-\frac{1}{12}a\dot{R}{f}_{RR}\left(R\right)+\dot{a}R{f}_{RR}\left(R\right)\right]}\hfill \\ \\ {\displaystyle X_2={\xi }_0{\mathcal{\partial }}_t}\hfill \\ {\displaystyle j_2=a^2[f\left(R\right)-R{f}_R\left(R\right)]+6[a{\dot{a}}^2{f}_R\left(R\right)+a^2\dot{a}\dot{R}{f}_{RR}\left(R\right)]+\Lambda a^3}\hfill \\ \\ {\displaystyle X_3=({\xi }_{0}t+{\xi }_1){\mathcal{\partial }}_t+{\alpha }_{0}a{\mathcal{\partial }}_a-2{\xi }_{0}R{\mathcal{\partial }}_R}\hfill \\ {\displaystyle j_3=2{\xi }_{0}aR\left(6a\dot{a}{f}_{RRR}\dot{R}+6{\dot{a}}^2{f}_{RR}-a^{2}R{f}_{RR}\right)}\hfill \\ {\displaystyle -{\alpha }_{0}a\left[6a\dot{a}\dot{R}{f}_{RR}+6{\dot{a}}^2{f}_R+a\left(6\dot{a}\dot{R}{f}_{RR}+2a\left(f-R{f}_R\right)\right)+a^2\left(f-R{f}_R\right)\right]}\hfill \\ {\displaystyle +{\xi }_{0}t\left[6a^2\dot{a}\dot{R}{f}_{RR}+a\dot{a}\left(12\dot{a}{f}_R+6a\dot{R}{f}_{RR}\right)\right.}\hfill \\ \left.{\displaystyle -a\left(6a\dot{a}\dot{R}{f}_{RR}+6{\dot{a}}^2{f}_R+a^2\left(f-R{f}_R\right)\right)}\right],\hfill \end{array}\right.$$

(53)

It is worth noting that the third solution involves a free parameter k, which can be constrained through observational data, as discussed in Refs. [24,41,42,43]. In addition, by setting $\xi \left(t\right)=0$, one obtains four further symmetry solutions, as shown in Refs. [44,45]:

$$\left\{\begin{array}{c}{\displaystyle X_5=\frac{{\alpha }_0}{a^2}{\mathcal{\partial }}_a-3{\alpha }_0\frac{R}{a^3}{\mathcal{\partial }}_R}\hfill \\ {\displaystyle j_5=24{\alpha }_0{f}_0\frac{\dot{a}}{a},f\left(R\right)=f_{0}R}\hfill \\ {\displaystyle X_6={\alpha }_0\sqrt{a}{\mathcal{\partial }}_a-\frac{{\alpha }_0}{2}\frac{R}{\sqrt{a^3}}{\mathcal{\partial }}_R}\hfill \\ {\displaystyle j_6=24{\alpha }_0{f}_0{a}^{\frac{3}{2}}\dot{a},f\left(R\right)=f_{0}R}\hfill \\ {\displaystyle X_7=\frac{{\alpha }_0}{a}{\mathcal{\partial }}_a-2{\alpha }_0\frac{R}{a^2}{\mathcal{\partial }}_R}\hfill \\ {\displaystyle j_7=9{\alpha }_0{f}_0(2R\dot{a}+a\dot{R})R^{-\frac{1}{2}},f\left(R\right)=f_0{R}^{\frac{3}{2}}}\hfill \\ {\displaystyle X_8=\frac{{\beta }_0}{a}{\mathcal{\partial }}_R}\hfill \\ {\displaystyle j_8=24{\beta }_0{f}_{1}a\dot{a},f\left(R\right)=f_{0}R+f_1{R}^2.}\hfill \end{array}\right.$$

(54)

Since the solutions in Equation (54) correspond to internal symmetries, they can equivalently be obtained by requiring the Lie derivative of the Lagrangian (45) to vanish. The complete collection of solutions arising from the system (49) is summarized in

Table 1. Noether solutions in $f\left(R\right)$ cosmology.

$\mathit{\xi }\left(\mathit{t}\right)$	$\mathit{\alpha }\left(\mathit{a}\right)$	$\mathit{\beta }(\mathit{a},\mathit{R})$	$\mathit{f}\left(\mathit{R}\right)$
${\xi }_{0}t+{\xi }_1$	${\displaystyle \pm \sqrt{{\alpha }_{0}a+{\alpha }_1}}$	$\beta (a,R,t)$	$f_{0}R+\Lambda$
${\xi }_{0}t+{\xi }_1$	${\displaystyle \frac{7{\xi }_0}{36}a}$	${\displaystyle -\frac{7{\xi }_0}{6}R}$	${\displaystyle f_0{R}^{\frac{19}{14}}+\Lambda }$
${\xi }_{0}t+{\xi }_1$	${\alpha }_{0}a$	$-2{\xi }_{0}R$	$f_0{R}^{1+{\displaystyle \frac{3{\alpha }_0}{2{\xi }_0}}}$
${\xi }_0$	0	0	${\displaystyle f_0{R}^{\frac{19}{14}}+\Lambda }$
0	${\displaystyle \frac{{\alpha }_0}{a^2}}$	${\displaystyle -3{\alpha }_0\frac{R}{a^3}}$	$f_{0}R$
0	${\displaystyle {\alpha }_0\sqrt{a}}$	${\displaystyle -\frac{{\alpha }_0}{2}\frac{R}{\sqrt{a^3}}}$	$f_{0}R$
0	${\displaystyle \frac{{\alpha }_0}{a}}$	${\displaystyle -2{\alpha }_0\frac{R}{a^2}}$	${\displaystyle f_0{R}^{\frac{3}{2}}}$
0	0	${\displaystyle \frac{{\beta }_0}{a}}$	${\displaystyle f_{0}R+f_1{R}^2}$

.

It is worth emphasizing that the functions reported in Table 1 continue to satisfy Noether’s system even after the inclusion of a matter contribution in the form of the Lagrangian ${\mathcal{L}}_m={\rho }_0{a}^s$, with s a constant, within Equation (45). In this case, although the presence of matter modifies the infinitesimal symmetry generators, the functional dependence of $f\left(R\right)$ itself remains unchanged. This outcome follows from the fact that the addition of the matter sector introduces an extra equation into the system (49), which does not yield additional constraints on the function $f\left(R\right)$. Consequently, while the matter Lagrangian can influence the structure of the symmetry generators and, in turn, the conserved quantities, it does not further restrict the original gravitational action.

In the following, the forms of $f\left(R\right)$ selected through Noether’s approach are inserted into the equations of motion (47) in order to investigate the resulting cosmological solutions for the different models under consideration.

The choice $f\left(R\right)=f_{0}R+\Lambda$ reproduces the standard Einstein–de Sitter vacuum expansion and will therefore not be analyzed further. By contrast, the power-law model $f\left(R\right)=f_0{R}^k$, which reduces to the second case of the system (52) in the limit $\Lambda \to 0$, leads to the Lagrangian

$$\mathcal{L}=f_{0}a{R}^{k-2}\left(6k(k-1)a\dot{a}\dot{R}+6kR{\dot{a}}^2-(k-1)a^2{R}^2\right).$$

(55)

and the Euler–Lagrange equations lead to the solution

$$a\left(t\right)=a_0{t}^{{\displaystyle \frac{(2k-1)(k-1)}{2-k}}},R\left(t\right)={\displaystyle \frac{6k(5-4k)(2k-1)(k-1)}{{(k-2)}^2{t}^2}},$$

(56)

which holds only for $k\ne 2$. When $k=2$, the scalar curvature is constant and leads to exponential solutions of the form

$$a\left(t\right)=a_0{e}^{\ell t},R\left(t\right)=-12{\ell }^2,$$

(57)

with ℓ being a real constant. By choosing $k=19/4$ in Equation (56), one finds that the corresponding scale factor coincides with that associated with the second model in the system (52). Moreover, for suitable values of the exponent k, the geometric sector alone is able to effectively reproduce the behavior of radiation, stiff matter and dust-dominated fluids:

• ${\displaystyle k=\frac{2}{3}\pm \frac{\sqrt{10}}{6}\to }$ Stiff Matter: $a\left(t\right)=a_0{t}^{{\displaystyle \frac{1}{3}}}$
• ${\displaystyle k=\frac{5}{4}\to }$ Radiation: $a\left(t\right)=a_0{t}^{{\displaystyle \frac{1}{2}}}$
• ${\displaystyle k=\frac{7}{12}\pm \frac{\sqrt{73}}{12}\to }$ Dust Matter: $a\left(t\right)=a_0{t}^{{\displaystyle \frac{2}{3}}}$

The above correspondence between symmetry-selected $f\left(R\right)$ models and effective perfect-fluid behaviors can be naturally extended to imperfect-fluid descriptions. In particular, bulk-viscous cosmology introduces the effective pressure $p_{eff}=p-3H\zeta$, so that viscosity contributes a negative, generally time-dependent pressure able to trigger acceleration. This mechanism is explicitly exploited in the recent literature, e.g., in [37], where different bulk-viscosity prescriptions are constrained with BAO, Pantheon and $H\left(z\right)$ data and the phase-space analysis exhibits a late-time de Sitter attractor. In our case, a de Sitter accelerated phase emerges already at the level of the vacuum geometric sector for the symmetry-selected model $f\left(R\right)\sim R^2$. Therefore, the qualitative de Sitter evolution often discussed in viscous cosmology can also be realized within a purely geometric framework determined by Noether symmetries, providing a complementary, exactly integrable route to cosmic acceleration.

A further class of solutions is obtained by choosing the functional form $f\left(R\right)=f_0{R}^{3/2}$, which leads to the Lagrangian:

$$\mathcal{L}=f_{0}a{R}^{-{\displaystyle \frac{1}{2}}}\left(-a^2{R}^2+18R{\dot{a}}^2+9a\dot{a}\dot{R}\right),$$

(58)

which yields the following equations of motion:

$$\left\{\begin{array}{c}{\displaystyle 36R^2{\dot{a}}^2+36aR(\dot{a}\dot{R}+2R\ddot{a})+3a^2(2R^3-3{\dot{R}}^2+6R\ddot{R})=0}\hfill \\ {\displaystyle R=-6\left(\frac{\ddot{a}}{a}+\frac{{\dot{a}}^2}{a^2}\right)}\hfill \\ {\displaystyle a^2{R}^2+18R{\dot{a}}^2+9a\dot{a}\dot{R}=0}\hfill \end{array}\right.$$

(59)

The above system admits the solution

$$a\left(t\right)=a_0{\left[c_4{t}^4+c_3{t}^3+c_2{t}^2+c_{1}t+c_0\right]}^{1/2},$$

(60)

where the $c_i$ are integration constants. It is worth noting that, by setting $c_0=c_1=c_2=c_3=0$ and choosing $k=3/2$, Equations (56) and (60) coincide. Nevertheless, with respect to Equation (56), the case $k=3/2$ introduces four additional contributions that cannot be obtained analytically for a generic value of k. The constants $c_i$ thus remain free parameters, which may be fixed by comparison with observational data at both astrophysical and cosmological scales.

Within this framework, a detailed study of the solution (60) could be used to constrain the values of the $c_i$ that best reproduce current observations at low and high energies. For example, a nonvanishing $c_4$ term is associated with power-law inflation, whereas a cosmological evolution dominated by the linear contribution proportional to $c_1$ corresponds to a radiation-dominated era. Further discussion on these aspects can be found in Ref. [46]. As pointed out in Ref. [21], the special case $k=3/2$ is related to Liouville field theory and represents a remarkable example in which a fourth-order Lagrangian can be recast into a form involving elementary functions through a conformal transformation to the Einstein frame.

Finally, the last functional form admitting Noether symmetries is given by $f\left(R\right)=f_{0}R+f_1{R}^2$. This choice corresponds to the well-known Starobinsky model [47], which, as discussed earlier, plays a central role in inflationary cosmology. The associated Lagrangian reads:

$$\mathcal{L}=2a[-f_1{a}^2{R}^2+6(f_0+2f_{1}R){\dot{a}}^2+12f_{1}a\dot{a}\dot{R}],$$

(61)

In this case, the equations of motion admit closed-form solutions only when either $f_0$ or $f_1$ vanishes. If $f_1=0$, the model reduces to standard GR, whereas when $f_0=0$ the corresponding solutions take the form:

$$a\left(t\right)=a_0\sqrt{t},f\left(R\right)=f_1{R}^2,\mathrm{and}a\left(t\right)=a_0{e}^{\ell t},f\left(R\right)=f_1{R}^2.$$

(62)

This result shows that cosmological behaviors typical of radiation-dominated epochs and de Sitter–like expansions are readily recovered. It is worth noting that, when the scale factor evolves as $a\left(t\right)$∼$\sqrt{t}$, the corresponding Ricci scalar satisfies $R\left(t\right)=0$. In this situation, the field equations are identically fulfilled, independently of the particular functional form of $f\left(R\right)$. For power-law models of the type $f\left(R\right)=f_0{R}^k$, the specific choice $k=2$ leads to an exponential behavior of the scale factor. This feature is closely related to the scale invariance of $f\left(R\right)=f_0{R}^2$ gravity, as discussed in Ref. [48]. Such properties can be qualitatively understood by expanding the action around a Gaussian fixed point and analyzing the scaling dimensions of the coupling constants.

Remarkably, conformal invariance emerges naturally among the solutions of the Noether symmetry system, indicating a close connection between the Noether symmetry approach and conformal symmetry itself [49].

The application of the Noether symmetry method to $f\left(R\right)$ gravity in spherically symmetric settings has also been investigated in several studies, including Refs. [42,50,51]. In particular, the analysis presented in Ref. [51] employs a symmetry-selected function to reproduce the mass–radius relation of neutron stars. Another illustrative example is provided in Ref. [52], where Noether symmetries within $f\left(R\right)$ gravity are used to constrain the Fundamental Plane of galaxies.

4.2. The Minisuperspace Reduction

We now illustrate how the change of variables introduced in Equation (22) can be used to reduce the dimensionality of the minisuperspace by introducing a cyclic variable into the Lagrangian. As a concrete example, we focus on the model defined by the function ${\displaystyle f\left(R\right)=f_0{R}^{3/2}}$, which belongs to the class of symmetry-admitting functions reported in Table 1. In particular, by applying Equation (22), we obtain a system of differential equations that maps the original minisuperspace $\mathcal{S}\equiv \{a,R\}$ onto a new set of variables ${\mathcal{S}}^{\prime }\equiv \{z,w\}$:

$$\left\{\begin{array}{c}\alpha {\mathcal{\partial }}_{a}z(a,R)+\beta {\mathcal{\partial }}_{R}z(a,R)=1\hfill \\ \alpha {\mathcal{\partial }}_{a}w(a,R)+\beta {\mathcal{\partial }}_{R}w(a,R)=0,\hfill \end{array}\right.$$

(63)

with z being the new cyclic variable. A possible solution is

$$\left\{\begin{array}{c}{\displaystyle w=w_0{\left(a\sqrt{R}\right)}^{\ell },z=\frac{a^2}{2{\alpha }_0}}\hfill \\ {\displaystyle a=\sqrt{2{\alpha }_{0}z},R=\frac{1}{2{\alpha }_0}z{\left(\frac{w}{w_0}\right)}^{\frac{2}{\ell }},}\hfill \end{array}\right.$$

(64)

where ℓ and $w_0$ are integration constants. Substituting Equation (64) into the Lagrangian (45), one finds that it takes the form:

$$\mathcal{L}={\displaystyle \frac{f_0}{\ell w}}{\left({\displaystyle \frac{w}{w_0}}\right)}^{{\displaystyle \frac{1}{\ell }}}\left[18{\alpha }_0\dot{w}\dot{z}-w\ell {\left({\displaystyle \frac{w}{w_0}}\right)}^{{\displaystyle \frac{2}{\ell }}}\right],$$

(65)

which is cyclic in the variable z. Although the discussion has focused on the specific case $f\left(R\right)=f_0{R}^{3/2}$, the same procedure can be straightforwardly applied to the other functional forms listed in Equations (54). This strategy proves particularly advantageous in the contexts of quantum cosmology and the Hamiltonian formulation, since the reduced Lagrangian greatly simplifies the construction of a suitable Hamiltonian and enables exact solutions of the Wheeler–DeWitt equation [53,54]. In particular, the latter work illustrates how the reduced dynamics arising from Noether symmetries can be exploited to derive a class of tachyonic potentials.

5. Conclusions

In this work, we reviewed the essential features of the Noether Symmetry Approach, presenting several applications and illustrating how Noether’s theorem can be employed as a constructive tool for selecting theories that admit symmetries. Starting from canonical Lagrangians, we introduced a systematic procedure aimed at constraining the symmetry generator. The strength of this method lies in its capability to single out physically meaningful models and, in many cases, to render their dynamics exactly solvable, thereby providing new perspectives for the investigation of gravitational phenomena.

We have shown that the first prolongation of the Noether vector allows for a broader class of symmetries, which reduce to those obtained from the vanishing Lie derivative when $g=\mathrm{const}$ and $\xi =0$. Furthermore, a key outcome of applying the Noether symmetry approach to canonical Lagrangians is that the infinitesimal generator $\xi$, associated with spacetime translations, is constrained to depend only on time. This property makes it possible to further streamline the procedure by reducing the associated system of differential equations. It should be emphasized, however, that this simplification does not generally extend to higher-order, non-canonical Lagrangians, such as those considered in Refs. [31,55,56,57].

The article further emphasizes the application of the Noether symmetry approach to $f\left(R\right)$ gravity models in a cosmological setting. As discussed in detail, the existence of a symmetry naturally induces a reduction of the minisuperspace through the introduction of a cyclic variable in the Lagrangian. This reduction simplifies the dynamical system and, in many cases, enables the derivation of exact solutions to the equations of motion. The method is particularly relevant in modified theories of gravity and in quantum cosmology, where the presence of a conserved quantity allows one to restrict the full superspace of variables to a reduced minisuperspace capable of describing physically meaningful universes [58,59,60].

More broadly, this framework provides a novel viewpoint for investigating the space of admissible physical theories within extended and alternative theories of gravity, where the explicit form of the action is often unknown a priori. Rather than starting from a prescribed action and subsequently identifying its symmetries, the Noether symmetry approach reverses this logic by postulating the existence of symmetries and using them as guiding principles to reconstruct the functional form of the action itself.