A Novel Method to Calculate Nonlocal Symmetries from Local Symmetries

Abstract

We explore nonlocal symmetries in a class of Hamiltonian dynamical systems governed by second-order differential equations. Specifically, we establish an algorithm for deriving nonlocal symmetries by utilizing the Jacobi metric and the Eisenhart–Duval lift to geometrize the dynamical systems. The geometrized systems often exhibit additional local symmetries compared to the original systems, some of which correspond to nonlocal symmetries for the original formulation. This novel approach allows us to determine nonlocal symmetries in a systematic way. Within this geometric framework, we demonstrate that the second-order differential equation $\ddot{q}-F\left(q\right)=0$ admits an infinite number of nonlocal symmetries generated by the infinite-dimensional conformal algebra of a two-dimensional Riemannian manifold. Applications to higher-dimensional systems are also discussed.

1. Introduction

Symmetry methods are powerful tools for the analytic treatment of nonlinear differential equations; this approach has been applied for the study of nonlinear differential equations which describe the evolution laws in fluid dynamics [1,2], MHD [3,4] analytic mechanics [5,6], and other areas of applied mathematics [7]. Symmetry analysis provides a systematic approach for deriving invariant functions and solutions for nonlinear differential equations. The insights gained from determining symmetries are crucial for understanding the behavior of dynamical systems. Moreover, the existence of specific symmetries is essential in many areas of applied mathematics for modeling real-world phenomena.

The term symmetry commonly refers to the generator of local point transformations that leave a differential equation invariant. These symmetries are known as point symmetries. However, the concept of symmetry is more general and includes various other transformations both local and nonlocal under which a differential equation remains invariant. For instance, the Lewis invariant for the Ermakov–Pinney equation [8] and the Carter constant for the Kerr metric [9] are conservation laws related to local non-point symmetries, specifically the so-called contact vector fields. Nonlocal symmetries [10,11] are essential for the study of differential equations.

A notable example is the nonlinear Burgers equation, which admits a nonlocal symmetry, allowing it to be rewritten in terms of the linear heat equation. In gravitational physics, nonlocal symmetries have been employed to derive algebraic solutions in scalar field cosmology for arbitrary potential functions [12], as discussed in [13]. Additionally, an algorithm for employing nonlocal symmetries to derive conservation laws was recently proposed in [14]. Another method for the systematic construction of nonlocal symmetries is presented in [15], where nonlocal symmetries were applied to perform a reduction in differential equations. By using this approach, the integrability of nonlinear oscillators is investigated in [16]. Another application of nonlocal symmetries for the study of the integrability of nonlinar partial differential equations is presented in [17]. For further applications of nonlocal symmetries, see [18,19,20], and an interesting discussion can be found in [11].

However, calculating nonlocal transformations is not always straightforward and there is currently no universal approach. In this study, we focus on Hamiltonian systems of second-order differential equations and employ the Jacobi metric and the Eisenhart–Duval lift to geometrize the dynamical system. The equivalent geometrized system may possess additional point symmetries, which can lead to the identification of nonlocal symmetries in the original system. The structure of the paper is as follows.

In Section 2, we present the basic properties and definitions for the Eisenhart–Duval lift. Specifically, we consider a specific family of second-order differential equations and we show how the Eisenhart–Duval lift is employed to reduce the rewriting of the given dynamical system in an equivalent form of geodesic equations. We review previous results on the relation between Lie and Noether symmetries for geodesic equations in Riemannian manifolds. In Section 3, we present the basic elements for the symmetry analysis of differential equations. We focus on the Lie and Noether symmetries of geodesic equations and we review some results which relate the symmetries of the background space with the group properties of the background space.

Furthermore, in Section 4, Section 5, Section 6 and Section 7, we demonstrate how, from the symmetries of the equivalent geodesic system, we can derive local and nonlocal hidden symmetries for the original system. We perform this by considering various specific nonlinear non-maximally symmetric second-order differential equations. In Section 4, we consider the Ermakov–Pinney equation, where we discuss the origin of the $SL\left(2,R\right)$ Lie algebra. Furthermore, in Section 4, we discuss that any second-order order differential equation $\ddot{q}-F\left(q\right)=0$, admits infinity nonlocal symmetries generated by the infinity elements of the conformal algebra for the two-dimensional Riemannian manifold. In Section 6 and Section 7, we discuss the nonlocal symmetries of two-dimensional systems. Finally, in Section 8, we discuss possible extensions and applications of this geometric approach for the construction of nonlocal hidden symmetries.

2. Geometrization of Dynamical Systems

The geometrization of dynamical systems, which involves reformulating a dynamical system as a set of geodesic equations, has been extensively studied in the literature. Within this framework, non-affine parametrization serves as a linearization technique. This method reformulates the system to construct connection coefficients directly from the dynamical equations. However, these connections typically do not correspond to the Levi-Civita connection of any metric tensor, indicating that the resulting geometry is non-Riemannian.

In contrast, Riemannian geometry holds a central role in the physical description of dynamical systems. Therefore, geometrization approaches that preserve Riemannian characteristics are of particular interest.

Two such approaches that maintain the Riemannian properties of a dynamical system are the Eisenhart–Duval lift and the Jacobi metric. These methods are primarily applied to autonomous systems, where conservative forces are naturally incorporated into the geometric structure of the background manifold.

2.1. Jacobi Metric

In the Jacobi metric approach, the independent variable is redefined to encapsulate all dynamical terms within a modified kinetic metric [21,22,23]. This process involves a conformal transformation, enabling the dynamical system to be expressed as an equivalent conformally related system. In this framework, the original dynamical system is reformulated as a set of geodesic equations. Notably, the dynamical degrees of freedom remain unchanged. The method has been widely studied for the study of physical systems. Within a more general context of velocity-dependent terms, the Jacobi metric approach leads to Finsler–Randers geometry instead of Riemannian manifolds [24,25].

Consider the dynamical system described by the constraint Lagrangian function:

$$L\left(N,q^k,{\dot{q}}^k\right)={\displaystyle \frac{1}{2N}}{g}_{ij}\left(q^k\right){\dot{q}}^i{\dot{q}}^j-N\left(V\left(q^k\right)-h\right),$$

(1)

where $g_{ij}\left(q^k\right)$ is a second-rank tensor with inverse $g^{ij}\left(q^l\right)$ and describes the kinetic metric. Variables $N\left(t\right)$ and $q^i\left(t\right)$ are the $\left(1+n\right)$ degrees of freedom of the dynamical system; $V\left(q^k\right)$ is the potential function and h is the value for the energy. The singular variable $N\left(t\right)$ has been introduced such that the Hamiltonian is the constraint.

Specifically, variation with respect to the singular variable $N\left(t\right)$ leads to the equation of motion

$${\displaystyle \frac{1}{2N^2}}{g}_{ij}{\dot{q}}^i{\dot{q}}^j+V\left(q^k\right)-h=0.$$

(2)

or

$$H\left(q^k,p^k\right)\equiv {\displaystyle \frac{1}{2}}{g}^{ij}{p}_i{p}_j+V\left(q^k\right)=h,$$

(3)

in which $p_i={\displaystyle \frac{1}{N}}{g}_{ij}{\dot{q}}^j$ is the momentum.

Variation with respect to the variable $q^i\left(t\right)$ of the Lagrangian function (1) leads to the equations of motion:

$${\ddot{q}}^i+{\Gamma }_{jk}^i{\dot{q}}^j{\dot{q}}^k+V^{,i}-2{(lnN)}^{,i}\left({\displaystyle \frac{1}{2}}{g}_{ij}{\dot{q}}^i{\dot{q}}^j+V\left(q^k\right)-h\right)=0,$$

(4)

where ${\Gamma }_{jk}^i$ is the Levi-Civita connection for the metric tensor $g_{ij}$. Thus, with the use of the constraint Equation (2) the latter dynamical system is reduced

$${\ddot{q}}^i+{\Gamma }_{jk}^i{\dot{q}}^j{\dot{q}}^k+V^{,i}=0.$$

(5)

Hence, the equations of motion are invariant under time-reparametrization. Thus, if we consider the function

$$N\left(t\right)={\displaystyle \frac{1}{\overline{N}\left(V\left(q^k\right)-h\right)}},$$

where $\overline{N}\left(t\right)$ is another function, then in this case, Lagrangian (1) reads

$$L\left(\overline{N},q^k,{\dot{q}}^k\right)={\displaystyle \frac{1}{2\overline{N}}}{\overline{g}}_{ij}{\dot{q}}^i{\dot{q}}^j-\overline{N},$$

(6)

where ${\overline{g}}_{ij}=\left(V\left(q^k\right)-h\right)g_{ij}$ is the Jacobi metric which defines the modified kinetic term. The tensor field  ${\overline{g}}_{ij}$ is conformally related to the tensor field $g_{ij}$.

The Euler–Lagrange equations are

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2{\overline{N}}^2}}{\overline{g}}_{ij}{\dot{q}}^i{\dot{q}}^j+V\left(q^k\right)-h& =& 0,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\ddot{q}}^i+{\overline{\Gamma }}_{jk}^i{\dot{q}}^j{\dot{q}}^k& =& 0,\hfill \end{array}$$

(8)

where now ${\overline{\Gamma }}_{jk}^i$ is the Levi-Civita connection for the Jacobi metric ${\overline{g}}_{ij}$. The latter system describes a set of geodesic equations within the Riemannian manifold with metric tensor $g_{ij}$.

2.2. Eisenhart–Duval Lift

On the other hand, in the Eisenhart–Duval lift, geometrization is achieved by introducing additional degrees of freedom [26]. The resulting geometric dynamical system is expressed as a set of geodesic equations on a higher-dimensional Riemannian manifold. This new Riemannian manifold possesses a sufficient number of isometries/Killing symmetries, which are used to derive conservation laws for the geodesic equations. When these conservation laws are applied, the geodesic equations reduce to the original dynamical system. The method was introduced by Eisenhart in [26] but it was rediscovered later by Duval [27] in order to reformulate the equations of motions for particles in the classical and quantum regimes in a covariant form. The application of Eisenhart–Duval lift covers many areas in the study of dynamical systems [28,29,30,31]. Although the Eisenhart–Duval lift is mainly applied for the second-order dynamical systems, the extension of the method to dynamical system with higher-order derivatives was presented in [32]. It is important to mention that within the relativistic limit, the Eisenhart–Duval lift is described by the Kaluza–Klein theory [33,34].

Consider the regular dynamical system described by the Lagrangian function

$$\tilde{L}\left(N,q^k,{\dot{q}}^k\right)={\displaystyle \frac{1}{2}}{g}_{ij}{\dot{q}}^i{\dot{q}}^j-V\left(q^k\right),$$

(9)

and Hamiltonian function

$$H\left(q^k,p^k\right)\equiv {\displaystyle \frac{1}{2}}{g}^{ij}{p}_i{p}_j+V\left(q^k\right)=h.$$

(10)

with $p_i=g_{ij}{\dot{q}}^j$.

Within the Hamiltonian formalism, the equations of motion are

$$\begin{array}{ccc}\hfill {\dot{q}}^i& =& g^{ij}{\dot{p}}_j,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\dot{p}}_k& =& -{\displaystyle \frac{1}{2}}{g}_{,k}^{ij}{p}_i{p}_j-V_{,k}.\hfill \end{array}$$

(12)

Assume now the new higher-dimensional Hamiltonian function

$$H_{n+1}={\displaystyle \frac{1}{2}}{g}^{ij}{p}_i{p}_j+{\displaystyle \frac{1}{2}}V\left(q^k\right)p_z^2=h_{n+1},$$

(13)

with equations of motion

$$\begin{array}{ccc}\hfill {\dot{q}}^i& =& g^{ij}{\dot{p}}_j,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \dot{z}& =& {\displaystyle \frac{1}{V\left(q^k\right)}}{p}_z,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\dot{p}}_k& =& -{\displaystyle \frac{1}{2}}{g}_{,k}^{ij}{p}_i{p}_j-{\displaystyle \frac{1}{2}}{V}_{,k}{p}_z^2,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\dot{p}}_z& =& 0.\hfill \end{array}$$

(17)

The Hamiltonian function (13) describes the geodesic equations for the line element

$$d{s}_{n+1}^2=g_{ij}\left(q^k\right)d{q}^{i}d{q}^j+{\displaystyle \frac{1}{V\left(q^k\right)}}d{z}^2.$$

(18)

This extended metric is known as the Eisenhart–Duval metric. It possesses the Killing symmetry ${\partial }_z$, with the conservation law for the momentum $p_z$, whereby the latter follows easily from the equation of motion (17).

Thus, by replacing the conservation law $p_z=p_z^0$ in the dynamical system (13)–(17), the original dynamical systems (11) and (12) are recovered, where the free parameters $h_{1+1}$ and $p_z^0$ are constrained as

$$h_{1+1}=h\mathrm{and}{\left(p_z^0\right)}^2=2.$$

(19)

The extended metric (18) is known as the Riemannian lift [28]. However, this is not the unique approach to write an equivalent geodesic system. A second approach is the Lorentzian lift. In this consideration the dynamical system is written in the equivalent form of a set of geodesic equations for a metric spacetime expressed in Brinkmann coordinates, similar to the pp-wave spaces.

In the Lorentzian lift, we consider the equivalent Hamiltonian function [28]

$$H_{n+2}\equiv {\displaystyle \frac{1}{2}}{g}^{ij}{p}_i{p}_j+V\left(q^k\right)p_u^2+p_u{p}_v=h_{n+2},$$

(20)

where $p_u$ and $p_v$ are the two new dependent variables. From the geodesic Hamiltonian $H_{n+2}$, we derive the following geodesic equations:

$$\begin{array}{ccc}\hfill {\dot{q}}^i& =& g^{ij}{\dot{p}}_j,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill \dot{u}& =& 2V\left(q^k\right)p_u+p_v,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill \dot{v}& =& p_u,\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\dot{p}}_k& =& -{\displaystyle \frac{1}{2}}{g}_{,k}^{ij}{p}_i{p}_j-V_{,k}{p}_u^2,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\dot{p}}_u& =& 0,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\dot{p}}_v& =& 0.\hfill \end{array}$$

(26)

Hence, the extended metric which defines the kinetic energy reads as

$$d{s}_{\left(n+2\right)}^2=g_{ij}\left(q^k\right)d{q}^{i}d{q}^j+2dudv-2V\left(q^k\right)d{u}^2.$$

(27)

For arbitrary metric $g_{ij}$ and potential V, the latter line element possesses two Killing symmetries, the translations ${\partial }_u$ and ${\partial }_v$, with corresponding conservation laws $p_v=p_v^0$ and $p_u=p_u^0$, respectively. Thus, by replacing in the geodesic equations, we recover the original dynamical system when the free parameters $h_{n+2}$, $p_u^0$, and $p_v^0$ are constrained as follows:

$$p_u^0{p}_v^0-h_{n+2}=h,{\left(p_u^0\right)}^2=1.$$

(28)

New lift methods which generalize the Riemannian and the Lorentzian lift were introduced recently in [35]. These lifts are based on the introduction of three new dependent variables with the corresponding conservation laws. In the following lines, we review these two new lifts.

Let us assume the Hamiltonian function [35]

$$H_{n+3}\equiv {\displaystyle \frac{1}{2}}{g}^{ij}{p}_i{p}_j+{\displaystyle \frac{1}{2}}{F}_1\left(q^k\right)p_z^2+F_2\left(q^k\right)p_u^2+p_u{p}_v=h_{n+3},$$

(29)

which describes the geodesic equations for the metric tensor with line element

$$d{s}_{\left(n+3\right)}^2=g_{ij}\left(q^k\right)d{q}^{i}d{q}^{j}d{x}^2+{\displaystyle \frac{1}{F_1\left(q^k\right)}}d{z}^2+2dudv-2F_2\left(q^k\right)d{u}^2.$$

(30)

The latter line element always admits the isometries ${\partial }_z,{\partial }_u$, and ${\partial }_v$. The corresponding conservation laws are $p_z=p_z^0$, $p_v=p_v^0$, and $p_u=p_u^0$. Moreover, the geodesic equations are

$$\begin{array}{ccc}\hfill {\dot{q}}^i& =& g^{ij}{\dot{p}}_j,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \dot{z}& =& F_1\left(q^k\right)p_z\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill \dot{u}& =& 2F_2\left(q^k\right)p_u+p_v,\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill \dot{v}& =& p_u,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill {\dot{p}}_k& =& -{\displaystyle \frac{1}{2}}{g}_{,k}^{ij}{p}_i{p}_j-\left({\displaystyle \frac{1}{2}}{F}_{1,k}{p}_z^2+F_{2,k}{p}_u^2\right),\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\dot{p}}_z& =& 0,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\dot{p}}_u& =& 0,\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill {\dot{p}}_v& =& 0.\hfill \end{array}$$

(38)

Therefore, by replacing the conservation laws within the geodesic equations, we recover the original system with constraints

$$V\left(q^k\right)={\displaystyle \frac{1}{2}}{F}_1\left(q^k\right){\left(p_z^0\right)}^2+F_2\left(q^k\right){\left(p_u^0\right)}^2+p_u^0{p}_v^0,$$

(39)

and

$$p_u^0{p}_v^0-h_{n+3}=h.$$

(40)

Furthermore, the Hamiltonian function

$${\widehat{H}}_{n+3}={\displaystyle \frac{1}{2}}{p}_x^2+{\displaystyle \frac{1}{2}}{F}_1\left(q^k\right)p_z^2+V_1\left(q^k\right)p_u^2+V_2\left(q^k\right)p_u{p}_v={\widehat{h}}_{n+3},$$

(41)

describes the geodesic equations for the Riemannian space with the line element

$$d{s}_{\left(n+3\right)}^2=d{x}^2+{\displaystyle \frac{1}{F_1\left(q^k\right)}}d{z}^2+{\displaystyle \frac{2}{V_2\left(q^k\right)}}dudv-2{\displaystyle \frac{V_1\left(q^k\right)}{{\left(V_2\left(q^k\right)\right)}^2}}d{u}^2$$

(42)

with three Killing symmetries: the translations ${\partial }_z,{\partial }_u$, and ${\partial }_v$. The corresponding geodesic equations are thus calculated:

$$\begin{array}{ccc}\hfill {\dot{q}}^i& =& g^{ij}{\dot{p}}_j,\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill \dot{z}& =& F_1\left(q^k\right)p_z\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill \dot{u}& =& 2V_1\left(q^k\right)p_u+V_2\left(q^k\right)p_v,\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill \dot{v}& =& V_2\left(q^k\right)p_u,\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill {\dot{p}}_k& =& -{\displaystyle \frac{1}{2}}{g}_{,k}^{ij}{p}_i{p}_j-\left({\displaystyle \frac{1}{2}}{F}_{1,k}{p}_z^2+V_{1,k}{p}_u^2+V_{2,k}{p}_u{p}_v\right),\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill {\dot{p}}_z& =& 0,\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill {\dot{p}}_u& =& 0,\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill {\dot{p}}_v& =& 0.\hfill \end{array}$$

(50)

Thus, the original system is recovered when

$$V\left(q^k\right)={\displaystyle \frac{1}{2}}{F}_1\left(q^k\right){\left(p_z^0\right)}^2+V_1\left(q^k\right){\left(p_u^0\right)}^2+V_2\left(q^k\right)p_u^0{p}_v^0,$$

(51)

and

$${\widehat{h}}_{n+3}=h.$$

(52)

3. Symmetries of Differential Equations

In the following, we present the fundamental properties and definitions relevant to the symmetry analysis of differential equations, with a particular focus on the point symmetries of second-order ordinary differential equations. Additionally, we discuss variational symmetries, emphasizing Noether’s theorems for point symmetries. It is noted that the Einstein summation convention is applied in this work.

3.1. Symmetries of Differential Equations

Geometrically, a differential equation (DE) may be considered as a function $H=H(t,q^k,{\dot{q}}^k,{\ddot{q}}^k)$ in the jet space $B=B\left(t,q^k,{\dot{q}}^k,{\ddot{q}}^k\right)$, where t is the independent variable and $q^i=q^i\left(t\right)$ are the dependent variables, where the dot denotes total derivative, that is, ${\dot{q}}^i={\displaystyle \frac{d{q}^i}{dt}}$.

Consider the infinitesimal point transformation in the base manifold $\left\{t,x\right\}$

$$\begin{array}{cc}\hfill \overline{t}& =t+\epsilon \xi (t,q^k),\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\overline{q}}^i& =q^i+\epsilon {\eta }^i(t,q^k),\hfill \end{array}$$

(54)

with the vector field as the generator

$$\mathbf{X}=\xi (t,q^k){\partial }_t+{\eta }^i(t,q^k){\partial }_i.$$

(55)

The extension of the generator in the jet space is

$${\mathbf{X}}^{\left[2\right]}=\mathbf{X}+{\eta }^{i\left[1\right]}{\partial }_{{\dot{q}}^i}+{\eta }^{i\left[2\right]}{\partial }_{{\ddot{q}}^i},$$

(56)

in which ${\eta }^{i\left[1\right]}$ and ${\eta }^{i\left[2\right]}$ are defined as [36,37]

$$\begin{array}{ccc}\hfill {\eta }^{i\left[1\right]}& =& {\displaystyle \frac{d\overline{q}}{d\overline{t}}},\hfill \end{array}$$

(57)

$$\begin{array}{ccc}\hfill {\eta }^{i\left[2\right]}& =& {\displaystyle \frac{d}{d\overline{t}}}\left({\displaystyle \frac{d\overline{q}}{d\overline{t}}}\right),\hfill \end{array}$$

(58)

that is,

$$\begin{array}{ccc}\hfill {\eta }^{i\left[1\right]}& =& {\dot{\eta }}^i-{\dot{q}}^i\dot{\xi },\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill {\eta }^{i\left[2\right]}& =& {\dot{\eta }}^{i\left[2\right]}-{\ddot{q}}^i\dot{\xi }.\hfill \end{array}$$

(60)

The vector field $\mathbf{X}$, is a Lie point symmetry of the differential equation $H=H(t,q^k,{\dot{q}}^k,{\ddot{q}}^k)$ if and only if there exists a function $\kappa$ such that [38,39]

$${\mathbf{X}}^{\left[2\right]}\left(H\right)=\kappa H,\mathrm{mod}H=0,$$

(61)

which is equivalent to

$${\mathbf{X}}^{\left[2\right]}\left(H\right)=0.$$

(62)

The latter condition yields an algebraic equation involving the derivatives of the dependent variable. From the definition of the monomials, we construct the set of conditions known as the determining system. This system is a linear set of partial differential equations with respect to the variables $\xi$ and ${\eta }^i$. Solving this system provides the general form of the point transformations (53) and (54) that leave the differential equation invariant.

Within the concept, a more general transformation is obtained when coefficients $\xi ,{\eta }^i$ can be functions of the derivatives or integrals of the dependent variables. The symmetry condition (61) remains true, but the solution process is different, which leads to specific forms of the coefficients $\xi$ and ${\eta }^i$. These kinds of symmetries are known as generalized, constant, hidden, nonlocal etc. [40,41,42,43].

Lie symmetries can be utilized in various ways to study differential equations. For any given symmetry vector, a corresponding set of invariant functions can be defined, which facilitates the reduction in the order of differential equations. For the Lie point symmetry $\mathbf{X}$, we define the Lagrange system

$${\displaystyle \frac{dt}{\xi }}={\displaystyle \frac{d{q}^i}{{\eta }^i}}={\displaystyle \frac{d{\dot{q}}^i}{{\eta }^{i\left[1\right]}}},$$

(63)

whose solution leads to the invariant functions $W^{i\left[0\right]}=W^{i\left[0\right]}\left(t,q^k\right)$, $W^{i\left[1\right]}=W^{i\left[1\right]}\left(t,q^k\right)$. For some applications, we refer the reader to [38,39,44,45,46]. At this point, it is important to mention that for partial differential equations, the application of invariant functions results in the reduction in the number of independent variables [39]. Moreover, Lie symmetries can be employed to construct conservation laws. Various approaches for constructing conservation laws exist, and they are reviewed in [38].

The application of invariant functions leads to new differential equations with altered algebraic properties. If the original equation admits at least two point symmetries, i.e., $X_1$ and $X_2$, with a commutator $[X_1,X_2]=c{X}_2$, where c may be zero, reduction by the invariants of $X_1$ results in $X_2$ being a nonlocal symmetry for the reduced differential equation, while reduction by $X_2$ results in $X_1$ being inherited as a Lie point symmetry of the reduced equation [47]. In this context, we will construct nonlocal symmetries and nonlocal transformations for a specific family of second-order differential equations.

A key result of Lie symmetry analysis of differential equations is the linearization criterion. Linearization is important as it enables the expression of analytic solutions using simple functions, thereby offering a better understanding of the dynamics of the dynamical system. For second-order ordinary differential equations $H=H(t,q,\dot{q},\ddot{q})$, the S. Lie theorem states that the equation admits eight Lie point symmetries, forming the $sl\left(3,R\right)$ Lie algebra; then, there exists a change in variables $\left\{t,q\right\}\to \left\{T,Q\right\}$ such that the equation to be expressed as the free particle ${\displaystyle \frac{d^{2}Q}{d{T}^2}}=0$. There are various generalizations of this result for higher-order differential equations and for higher-dimensional systems; see, for instance, [48,49,50,51,52].

3.2. Variational Symmetries

One of the most systematic ways for the construction of conservation laws within the symmetries is by applying Noether’s theorems [53]. They provide a one-to-one relation between the variational symmetries of the dynamical system and the admitted conservation laws. Noether’s first theorem provides an algebraic relation that can be used to constrain the Lie symmetries, which keep the variational principle invariant. Furthermore, Noether’s second theorem gives an algebraic formula which is applied to construct the conservation law from the symmetry vector.

Let us assume that the differential equation $H=H(t,q^k,{\dot{q}}^k,{\ddot{q}}^k)$ follows from the variation in the action integral:

$$S=\int L\left(t,q^k,{\dot{q}}^k\right)dt,$$

(64)

where $L\left(t,q^k,{\dot{q}}^k\right)$ is the Lagrangian function.

Noether’s first theorem states that if $\mathbf{X}$ is a Lie symmetry for the differential equation, it is also a variational symmetry if and only if there exists a function g such that

$${\mathbf{X}}^{\left[1\right]}L+L\dot{\xi }=\dot{g}.$$

(65)

Function g is a boundary term introduced to account for the infinitesimal changes in the value of the action integral resulting from infinitesimal modifications to the domain boundary caused by the transformation of variables in the action integral.

Condition (65) is also known as the Noether condition. The admitted Noether symmetries for a given dynamical system form a group.

For each Noether symmetry $\mathbf{X}$, a conservation law for the dynamical system can be derived using Noether’s second theorem. Specifically, the conserved quantity is given by

$$I\left(t,\mathbf{x},\dot{\mathbf{x}}\right)=\xi \mathcal{H}-{\displaystyle \frac{\partial L}{\partial \dot{\mathbf{x}}}}\dot{\mathbf{x}}+g,$$

(66)

in which

$$\mathcal{H}={\displaystyle \frac{\partial L}{\partial \dot{\mathbf{x}}}}\dot{\mathbf{x}}-L,$$

(67)

is the Hamiltonian function.

Even though Noether’s theorems are typically associated with point symmetries, they are not limited to this case and remain valid for other types of transformations as well [54].

3.3. Symmetries for the Geodesic Lagrangian

We focus on the geodesic Lagrangian

$$L\left(t,q^k,{\dot{q}}^k\right)={\displaystyle \frac{1}{2}}{g}_{ij}\left(q^k\right){\dot{q}}^i{\dot{q}}^j,$$

(68)

with Euler–Lagrange equations

$${\ddot{q}}^i+{\Gamma }_{jk}^i{\dot{q}}^j{\dot{q}}^k=0,$$

(69)

The Lie and the Noether point symmetries for this geometric dynamical system have been widely studied in the literature. It has been found that the collineations of the underlying Riemannian manifold are the generators for the symmetry vectors of the differential equations [55].

In particular, it was found that the Lie point symmetries for the geodesic Equation (69) are derived by the elements of the special projective collineations of the connection coefficients ${\Gamma }_{jk}^i$. The admitted Lie symmetries form the projective algebra for the decomposable spacetime with the line element [56]:

$$d{s}^2=d{t}^2+g_{ij}d{q}^{i}d{q}^j.$$

On the other hand, the Noether symmetries for the geodesic Lagrangian (68) are determined by the elements of the homothetic algebra of the metric tensor $g_{ij}$. Detailed relationships are presented in [55]. However, the key point is that studying the algebraic properties of the Riemannian manifold allows us to address the symmetry classification problem.

As a result, the task of determining the symmetries of the geodesic equations is effectively reduced to analyzing the properties of the metric $g_{ij}$.

3.4. Symmetries of Null Geodesics

In the case of null geodesics, that is, the kinetic energy has zero value, i.e.,

$$\mathcal{H}={\displaystyle \frac{1}{2}}{g}_{ij}\left(q^k\right){\dot{q}}^i{\dot{q}}^j=0,$$

(70)

the geodesic equations are invariant under conformal transformations.

Thus, we introduce the constraint Lagrangian function

$$L\left(N,q^k,{\dot{q}}^k\right)={\displaystyle \frac{1}{2N}}{g}_{ij}{\dot{q}}^i{\dot{q}}^j,$$

which describes the null geodesics, and N plays the role of the conformal factor. Due to the existence of the constraint equation, the Lie and the Noether symmetries of the null geodesic equations are different from those discussed before. Specifically, the Lie and the Noether symmetries are derived by the Conformall Killing vectors for the metric tensor $g_{ij}$. Recall that the conformal symmetries are independent of the conformal factor, while this is not true for the projective collineations. For more detail, we refer the reader to [57].

4. Nonlocal Symmetries for the Ermakov–Pinney Equation

Consider the Ermakov–Pinney equation

$$\ddot{\rho }+{\rho }^{-3}=0.$$

(71)

Its group properties have been widely studied in the literature [58]. The equation plays an important role in many aspects of physical problems; see the discussion in [59,60,61] and references therein.

The Ermakov–Pinney equation possesses the three Lie point symmetries, the vector fields

$$\begin{array}{ccc}\hfill X^1& =& {\partial }_t,\hfill \end{array}$$

(72)

$$\begin{array}{ccc}\hfill X^2& =& 2t{\partial }_t+\rho {\partial }_{\rho },\hfill \end{array}$$

(73)

$$\begin{array}{ccc}\hfill X^3& =& t^2{\partial }_t+t\rho {\partial }_{\rho },\hfill \end{array}$$

(74)

Which form the $SL\left(2,R\right)$ Lie algebra.

The analytic solution of the Ermakov–Pinney equation is expressed in terms of the two solutions of the free particle, that is, the solution is

$$\rho =\sqrt{x_1^2+x_2^2},$$

(75)

in which ${\ddot{x}}_1=0$ and ${\ddot{x}}_2=0$.

Moreover, the variation in the Lagrangian function

$$L\left(\rho ,\dot{\rho }\right)={\displaystyle \frac{1}{2}}\left({\dot{\rho }}^2-{\rho }^{-2}\right),$$

(76)

gives rise to the Ermakov–Pinney equation. The three vector fields are also variational symmetries for the latter Lagrangian, with corresponding conservation laws:

$$\begin{array}{ccc}\hfill I_1\left(X^1\right)& =& {\displaystyle \frac{1}{2}}\left({\dot{\rho }}^2+{\rho }^{-2}\right)\equiv \mathcal{H},\hfill \end{array}$$

(77)

$$\begin{array}{ccc}\hfill I_2\left(X^2\right)& =& 2t\mathcal{H}-\rho \dot{\rho },\hfill \end{array}$$

(78)

$$\begin{array}{ccc}\hfill I_3\left(X^3\right)& =& t^2\mathcal{H}-t\rho \dot{\rho }+{\displaystyle \frac{1}{2}}{\rho }^2.\hfill \end{array}$$

(79)

Recently, in [62], the origin of the $SL\left(2,R\right)$ algebra was discussed in detail. An equivalent dynamical system, which shares the same solution with the the Ermakov–Pinney equation, is the free particle in the two-dimensional plane, with the Lagrangian function:

$$L\left(x,\dot{x},y,\dot{y}\right)={\displaystyle \frac{1}{2}}\left({\dot{x}}^2+{\dot{y}}^2\right).$$

(80)

Lagrangian (80) is maximally symmetric, that is, it admits the Noether point symmetries

$$\begin{array}{ccc}\hfill Y^1& =& {\partial }_t,Y^2={\partial }_x,Y^3={\partial }_y,Y^4=y{\partial }_x-x{\partial }_y,\hfill \\ \hfill Y^5& =& t{\partial }_x,Y^6=t{\partial }_y,Y^7=2t{\partial }_t+x{\partial }_x+y{\partial }_y,\hfill \\ \hfill Y^8& =& t^2{\partial }_t+tx{\partial }_x+ty{\partial }_y.\hfill \end{array}$$

The corresponding conservation laws are

$$\begin{array}{ccc}\hfill I\left(Y^1\right)& =& {\displaystyle \frac{1}{2}}\left({\dot{x}}^2+{\dot{y}}^2\right)\equiv {\mathcal{H}}_1,\hfill \end{array}$$

(81)

$$\begin{array}{ccc}\hfill I\left(Y^2\right)& =& \dot{x},\hfill \end{array}$$

(82)

$$\begin{array}{ccc}\hfill I\left(Y^3\right)& =& \dot{y},\hfill \end{array}$$

(83)

$$\begin{array}{ccc}\hfill I\left(Y^4\right)& =& y\dot{x}-x\dot{y},\hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill I\left(Y^5\right)& =& t\dot{x}-x,\hfill \end{array}$$

(85)

$$\begin{array}{ccc}\hfill I\left(Y^6\right)& =& t\dot{y}-y,\hfill \end{array}$$

(86)

$$\begin{array}{ccc}\hfill I\left(Y^7\right)& =& 2t{\mathcal{H}}_1-x\dot{x}-y\dot{y},\hfill \end{array}$$

(87)

$$\begin{array}{ccc}\hfill I\left(Y^8\right)& =& t^2{\mathcal{H}}_1-t\left(x\dot{x}+y\dot{y}\right)+{\displaystyle \frac{1}{2}}\left(x^2+y^2\right).\hfill \end{array}$$

(88)

In terms of polar coordinates, i.e., $x=\rho cos\theta ,y=\rho sin\theta$, the Lagrangian function reads

$$L\left(\rho ,\dot{\rho },\theta ,\dot{\theta }\right)={\displaystyle \frac{1}{2}}\left({\dot{\rho }}^2+{\rho }^2{\dot{\theta }}^2\right).$$

(89)

We define the momentum $p_{\rho }=\dot{\rho },p_{\theta }={\rho }^2\dot{\theta }$. Therefore, the Hamiltonian function reads

$${\mathcal{H}}_1={\displaystyle \frac{1}{2}}\left(p_{\rho }^2+{\displaystyle \frac{p_{\theta }^2}{{\rho }^2}}\right).$$

(90)

Conservation law $I\left(Y^4\right)$ reads $I\left(Y^4\right)=p_{\theta }$, which is that of the angular momentum. Hence, by replacing $p_{\theta }=p_{\theta }^0$ in the Hamiltonian (90), we end with the Hamiltonian (77) for the Ermakov–Pinney equation. The transformation from the Cartesian to polar coordinates within the two-dimensional system is essential in order to perform the global geometric linearization of the Ermakov–Pinney equation [62].

From the symmetry vectors, only the vector fields $Y^1,Y^7$ and $Y^8$ have zero commutator with the rotation $Y^4$, thus, these are the only symmetries which survive as point symmetries. Indeed, the application of the angular momentum gives $\theta \left(t\right)=\int {\displaystyle \frac{I\left(Y^4\right)}{{\rho }^2}}dt$. Consequently, the rest of the symmetry vectors become nonlocal symmetries.

Indeed, the vector field $Y^2$ in polar coordinates is expressed as $Y^2=cos\theta {\partial }_{\rho }-{\displaystyle \frac{sin\theta }{\rho }}{\partial }_{\theta }$. Therefore, we write the nonlocal symmetry for the Ermakov–Pinney equation:

$$Y^2=cos\left(\int {\displaystyle \frac{I\left(Y^4\right)}{{\rho }^2}}dt\right){\partial }_{\rho },$$

(91)

while from $Y^3$, it follows that

$$Y^3=sin\left(\int {\displaystyle \frac{I\left(Y^4\right)}{{\rho }^2}}dt\right){\partial }_{\rho }.$$

(92)

In order to demonstrate the novelty of the derivation of these nonlocal symmetries, consider the equation for the free particle $\ddot{x}=0.$ Under the nonlocal transformation

$$x=\rho cos\left(\int {\displaystyle \frac{I\left(Y^4\right)}{{\rho }^2}}dt\right)$$

(93)

or

$$x=\rho sin\left(\int {\displaystyle \frac{I\left(Y^4\right)}{{\rho }^2}}dt\right),$$

(94)

we end with Equation (71).

At this point, it is important to mention that for equation

$${\displaystyle \frac{d^{2}x}{d{s}^2}}+\omega {\left(s\right)}^{2}x+{\displaystyle \frac{1}{x^3}}=0,$$

(95)

the change in variables

$$\begin{array}{ccc}\hfill s& =& \int {\displaystyle \frac{1}{u\left(t\right)}}dt,\hfill \\ \hfill x\left(t\right)& =& \rho \left(t\right)u\left(t\right)cossect,\hfill \end{array}$$

(96)

where $u\left(t\right)$ is a solution of the linear equation $\ddot{u}+{\omega }^{2}u=0$, eliminates the oscillator term, that is, Equation (95) takes the form of Equation (71) [10]. Equation (95) is also known as the Ermakov–Pinney equation with solution $x=\sqrt{u_1^2+u_2^2}$, where $u_{1,2}$ is the solution of the linear equation. Consequently, the above method can be applied to construct nonlocal symmetries.

5. Infinity Nonlocal Symmetries for $\ddot{\mathit{q}}+\mathit{F}\left(\mathit{q}\right)=\mathbf{0}$

Consider the second-order differential equation

$$\ddot{q}-\Phi \left(q\right)=0,$$

(97)

which follows from the Hamiltonian

$$\mathcal{H}\equiv {\displaystyle \frac{1}{2}}{p}^2+V\left(q\right)-h=0,F\left(q\right)=-V_{,q}.$$

(98)

Without loss of generality, we can define the potential function $U\left(q\right)=V\left(q\right)-h$ so that we can absorb the constant of integration.

The extended Eisenhart–Duval Hamiltonian within the Riemannian lift reads

$$d{s}^2=d{q}^2+{\displaystyle \frac{1}{U\left(q\right)}}d{z}^2.$$

(99)

with the vector field ${\partial }_z$ as the isometry.

We employ the change in variables $dq=F\left(x\right)dx$, such that

$$d{s}^2=F^2\left(x\right)d{x}^2+{\displaystyle \frac{1}{U\left(\int F\left(x\right)dx\right)}}d{z}^2,$$

(100)

Moreover, we select function $F\left(x\right)$ such that $F^2\left(x\right)={\displaystyle \frac{1}{U\left(\int F\left(x\right)dx\right)}}$, and the latter line element reads

$$d{s}^2={\displaystyle \frac{1}{U\left(\int F\left(x\right)dx\right)}}\left(d{x}^2+d{z}^2\right).$$

(101)

Thus, the extended geodesic Hamiltonian becomes

$$H_{1+1}\equiv U\left(\int F\left(x\right)dx\right)\left(p_x^2+p_z^2\right)=0.$$

(102)

The symmetries of the null geodesic equations are the Conformal Killing Symmetries of the two-dimensional space. We remark that all two-dimensional spaces are conformally flat and they admit an infinite number of Conformal Killing Symmetries [63]. In the diagonal coordinates (101), the generic form of the infinity Conformal Killing Symmetries is

$$X=\left[{\xi }^1\left(x+iz\right)+{\xi }^2\left(x-iz\right)\right]{\partial }_x+\left[{\xi }^1\left(x+iz\right)-{\xi }^2\left(x-iz\right)\right]{\partial }_z.$$

(103)

The conservation law related to the isometry ${\partial }_z$, which leads to the original system, that is,

$$z\left(t\right)=\int {\displaystyle \frac{p_z^0}{U\left(\int F\left(x\right)dx\right)}}dt,$$

(104)

thus, under this change in variables, the infinite Conformal Killing Symmetries become nonlocal symmetries:

$$X=\left[{\xi }^1\left(x+i\int {\displaystyle \frac{p_z^0}{U\left(\int F\left(x\right)dx\right)}}dt\right)+{\xi }^2\left(x-i\int {\displaystyle \frac{p_z^0}{U\left(\int F\left(x\right)dx\right)}}dt\right)\right]{\partial }_x.$$

(105)

Exponential Potential

Consider the exponential potential $U\left(q\right)=e^q$, where, for simplicity, we consider the value of the energy to be zero, i.e., $h=0$. Under the change in variables $q=2ln\left({\displaystyle \frac{r}{2}}\right)$, the extended Hamiltonian (102) becomes

$$H_{1+1}=r^2\left(p_r^2+p_z^2\right)=0.$$

(106)

The equations of motion are

$$\begin{array}{ccc}\hfill \dot{r}& =& 2r^2{p}_r,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \dot{z}& =& 2r^2{p}_z,\hfill \end{array}$$

(107)

$$\begin{array}{ccc}\hfill {\dot{p}}_r& =& 2r\left(p_r^2+p_z^2\right)=0,\hfill \end{array}$$

(108)

$$\begin{array}{ccc}\hfill {\dot{p}}_z& =& 2r\left(p_r^2+p_z^2\right)=0.\hfill \end{array}$$

(109)

This set of null geodesic equations admits infinity symmetries, the infinite Conformal Killing Vectors of the two-dimensional Riemannian manifold, that is,

$$\overline{X}=\left({\xi }^1\left(r+iz\right)+{\xi }^2\left(r-iz\right)\right){\partial }_r+\left({\xi }^1\left(r+iz\right)-{\xi }^2\left(r-iz\right)\right){\partial }_z.$$

(110)

After the application of the conservation law $p_z=p_z^0$, and $z=2p_z^0\int r^{2}dt$, the local conservation laws become nonlocal, that is,

$$\overline{X}=\left({\xi }^1\left(r+2i{p}_z^0\int r^{2}dt\right)+{\xi }^2\left(r-2i{p}_z^0\int r^{2}dt\right)\right){\partial }_x.$$

(111)

Consider the symmetry vector $X_1=r{\partial }_r+iz{\partial }_z$, and the conservation law $I\left(X_1\right)=r{p}_r+iz{p}_z$, that is, $I\left(X_1\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{r}}\dot{r}+i{\displaystyle \frac{z}{r^2}}\dot{z}\right)$. Therefore, after the application of the transformation $p_z=p_z^0$, and $z=2p_z^0\int r^{2}dt$, we end with the nonlocal conservation law:

$$I\left(X_1\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{r}}\dot{r}+2{\left(p_z^0\right)}^{2}i\int r^{2}dt\right)$$

(112)

Consequently, the derivative of the latter gives

$$\ddot{q}+4{\left(p_z^0\right)}^{2}i{e}^q=0.$$

(113)

which is the original equation for $4{\left(p_z^0\right)}^{2}i=1$.

6. Nonlocal Symmetries for the Szekeres System

The two-dimensional dynamical system

$$\begin{array}{ccc}\hfill \ddot{u}+u^{-2}& =& 0,\hfill \end{array}$$

(114)

$$\begin{array}{ccc}\hfill \ddot{v}-2v{u}^{-3}& =& 0,\hfill \end{array}$$

(115)

is known as the Szekeres system [64] and describes inhomogeneous and anisotropic cosmological universes [65].

The Hamiltonian function for the Szekeres system is

$$\mathcal{H}\equiv p_u{p}_v+{\displaystyle \frac{v}{u^2}}=h,$$

(116)

where $p_u=\dot{v},p_v=\dot{u}$

We employ the Einsenhart–Duval lift and we write the equivalent Hamiltonian:

$${\mathcal{H}}_{2+1}\equiv p_u{p}_v+{\displaystyle \frac{1}{2}}{\displaystyle \frac{v}{u^2}}{p}_z^2=h.$$

(117)

The extended metric tensor has the line element

$$d{s}^2=2dudv+{\displaystyle \frac{u^2}{v}}d{z}^2.$$

(118)

This line element is conformally flat. The geodesic equations for $h\ne 0$ admit the variational symmetries:

$$\begin{array}{ccc}\hfill X^1& =& {\partial }_t,\hfill \\ \hfill X^2& =& u{\partial }_u-v{\partial }_v-{\displaystyle \frac{3}{2}}z{\partial }_z\hfill \\ \hfill X^3& =& 2t{\partial }_t+2u{\partial }_u-z{\partial }_z,\hfill \\ \hfill X^4& =& {\partial }_z.\hfill \end{array}$$

The vector field ${\partial }_z$, is the one which generates the conservation law $p_z=p_z^0$, which is used to recover the original system.

However, for $h=0$, that is, when ${\mathcal{H}}_{2+1}$ describes the null geodesics for the three-dimensional space, the admitted variational symmetries are the ten Conformal Killing Symmetries with a generator:

$$\begin{array}{ccc}\hfill Y& =& \left(2{\alpha }_3{u}^2{z}^2-2{\alpha }_{4}uz+{\alpha }_9{u}^{2}z+{\alpha }_8{u}^2+{\alpha }_6+{\alpha }_{7}u\right){\partial }_u\hfill \\ & & +\left({\alpha }_6{\displaystyle \frac{z^2}{2v}}+{\alpha }_{4}vz+{\alpha }_5{\displaystyle \frac{z}{v}}+{\alpha }_{1}v+{\alpha }_2{\displaystyle \frac{1}{v}}+{\alpha }_3{v}^3\right){\partial }_v\hfill \\ & & +\left(-{\alpha }_4{\displaystyle \frac{v^2}{u}}-{\alpha }_6{\displaystyle \frac{z}{u}}-{\alpha }_5{\displaystyle \frac{1}{u}}+2{\alpha }_3{v}^{2}z+{\alpha }_9{\displaystyle \frac{v^2}{2}}+{\alpha }_4{z}^2-{\alpha }_7{\displaystyle \frac{z}{2}}+{\alpha }_{1}z+{\alpha }_{10}\right){\partial }_z.\hfill \end{array}$$

(119)

Consequently, after the nonlocal transformation $z=p_z^0\int {\displaystyle \frac{v}{u^2}}dt$, we determine the generic nonlocal symmetry for the Szekeres system:

$$\begin{array}{ccc}\hfill Y_{nonlocal}& =& \left(2{\alpha }_3{u}^2{\left(p_z^0\int {\displaystyle \frac{v}{u^2}}dt\right)}^2-2{\alpha }_{4}u{p}_z^0\int {\displaystyle \frac{v}{u^2}}dt+{\alpha }_9{u}^2{p}_z^0\int {\displaystyle \frac{v}{u^2}}dt\right){\partial }_u\hfill \\ & & +\left({\alpha }_6{\displaystyle \frac{1}{2v}}{\left(p_z^0\int {\displaystyle \frac{v}{u^2}}dt\right)}^2+{\alpha }_{4}v{p}_z^0\int {\displaystyle \frac{v}{u^2}}dt+{\alpha }_5{\displaystyle \frac{1}{v}}{p}_z^0\int {\displaystyle \frac{v}{u^2}}dt\right){\partial }_v.\hfill \end{array}$$

(120)

7. Nonlocal Symmetries for the Exponential Interaction

We proceed our discussion with the consideration of the two-dimensional system with the Lagrangian function

$$L\left(q_1,{\dot{q}}_1,q_2,{\dot{q}}_2\right)={\displaystyle \frac{1}{2}}\left({\dot{q}}_1^2+{\dot{q}}_2^2\right)-V_0{e}^{\left(q_1-q_2\right)},$$

(121)

which describes the equations of motion of two particles with an exponential interaction similar to the Toda lattice. For simplicity, we introduce the non-diagonal variables $q_1=i{\displaystyle \frac{u+v}{2}},q_2={\displaystyle \frac{u-v}{2}}$, where the Lagrangian function reads

$$L\left(u,\dot{u},v,\dot{v}\right)=\dot{u}\dot{v}-V_0{e}^v.$$

(122)

We introduce the extended Hamiltonian function

$${\mathcal{H}}_{2+1}\equiv p_u{p}_v-e^v{p}_z^2=h,$$

where $p_u=\dot{v},p_v=\dot{u}$. This Hamiltonian describes the geodesic equations for the conformally flat metric with the line element:

$$d{s}_{2+1}=2dudv-e^{-v}d{z}^2.$$

(123)

For $h\ne 0$, the geodesic equations admit the variational symmetries:

$$\begin{array}{ccc}\hfill X^1& =& {\partial }_t,\hfill \\ \hfill X^2& =& {\partial }_u,\hfill \\ \hfill X^3& =& z{\partial }_u-e^v{\partial }_z\hfill \\ \hfill X_4& =& {\partial }_v+{\displaystyle \frac{z}{2}}{\partial }_z\hfill \\ \hfill X^5& =& 2t{\partial }_t+2u{\partial }_u+z{\partial }_z,\hfill \\ \hfill X^6& =& {\partial }_z.\hfill \end{array}$$

Application of the conservation law $z=p_z^0\int e^{v}dt$ leads to the determination of the nonlocal symmetry:

$$X_{nonlocal}^3=\int e^{v}dt{\partial }_u.$$

However, for $h=0$, the null geodesic equations admit as symmetries the ten Conformal Killing Vectors of the three-dimensional manifold with a generic vector field:

$$\begin{array}{ccc}\hfill Y& =& \left({\displaystyle \frac{\left(2{\alpha }_9+2{\alpha }_{4}u\right)}{2}}z-{\alpha }_3{\displaystyle \frac{z^2}{2}}+{\alpha }_{7}u+{\alpha }_8-{\alpha }_6{u}^2\right){\partial }_u\hfill \\ & & +\left(\left({\displaystyle \frac{{\alpha }_6}{2}}{z}^2+{\alpha }_{5}z+{\alpha }_2\right)e^{-v}+{\alpha }_{4}z+{\alpha }_1+{\alpha }_3{e}^v\right){\partial }_v\hfill \\ & & +\left(\left({\alpha }_{3}z-{\alpha }_9-{\alpha }_{4}u\right)e^v+{\displaystyle \frac{{\alpha }_4}{2}}{z}^2+{\displaystyle \frac{\left({\alpha }_7+{\alpha }_1-2{\alpha }_{6}u\right)}{2}}z+{\alpha }_{10}-{\alpha }_{5}z\right){\partial }_z.\hfill \end{array}$$

(124)

Consequently, for the null geodesics, the application of the conservation law $z=p_z^0\int e^{v}dt$ leads to the derivation of the nonlocal symmetries:

$$\begin{array}{ccc}\hfill Y& =& \left({\displaystyle \frac{\left(2{\alpha }_9+2{\alpha }_{4}u\right)}{2}}{p}_z^0\int e^{v}dt-{\displaystyle \frac{{\alpha }_3}{2}}{\left(p_z^0\int e^{v}dt\right)}^2\right){\partial }_u\hfill \\ & & +\left(\left({\displaystyle \frac{{\alpha }_6}{2}}{\left(p_z^0\int e^{v}dt\right)}^2+{\alpha }_5{p}_z^0\int e^{v}dt\right)e^{-v}+{\alpha }_4{p}_z^0\int e^{v}dt\right){\partial }_v.\hfill \end{array}$$

(125)

8. Conclusions

In this work, we presented a novel approach for calculating nonlocal symmetries by leveraging the local symmetries of higher-dimensional equivalent systems. Specifically, we utilized the Jacobi metric and the Eisenhart–Duval lift to geometrize a class of Hamiltonian systems governed by second-order ordinary differential equations. Through this process, we defined equivalent dynamical systems that describe geodesic equations in Riemannian manifolds.

The geodesic equations exhibit more symmetries than the original system. However, among these symmetries, one vector field generates a conservation law necessary for reducing the system back to its original form. Consequently, applying the reduction process to the local symmetries of the geodesic equations enables the construction of nontrivial nonlocal symmetries for the original dynamical system.

We demonstrated the algorithm by deriving nonlocal symmetries for a variety of problems. These results can be extended to higher-dimensional models and higher-order systems, which can be reduced to second-order systems using Lagrange multipliers. The nonlocal symmetries obtained in this manner can be applied to geometrically linearize dynamical systems, as previously demonstrated. Additionally, they provide a foundation for constructing alternative Lagrangians or conservation laws.

In physical science, Noetherian conservation laws are associated with observables. In Newtonian physics, for instance, the conservation of momentum is directly linked to the local transformation induced by translation symmetry. Similarly, the conservation of angular momentum corresponds to the rotational symmetry of Euclidean space. Meanwhile, the Noetherian conservation law arising from autonomous symmetry is identified as energy. This raises the following question: what is the physical significance of the nonlocal symmetries derived in this study? Their physical interpretation remains unknown. However, as previously discussed, these conservation laws are valuable mathematical tools that aid in the analysis and derivation of solutions for nonlinear differential equations.

In future work, we plan to further explore and expand upon this direction of research.