Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

Abstract

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as the Kummer equation and the combined gravity-inertial-Rossbywave equation and certain classes of partial differential equations related to multidimensional wave equations.

1. Introduction

The method of equivalent Lagrangians is used to find the solutions of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic algebra of Noether point symmetries. The underlying idea of the method is to construct a regular point transformation which maps the Lagrangian of a “simpler” differential equation (with known solutions) to the Lagrangian of the differential equation in question. Once determined, this point transformation will then provide a way of mapping the solutions of the simpler differential equation to the solutions of the equation we seek to solve. This transformation can also be used to find conserved quantities for the equation in question, if the conserved quantities for the simpler differential equation are known. In the sections that follow, the method of equivalent Lagrangians is described for scalar second-order ordinary differential equations and for partial differential equations in two independent variables.

Some well-known ordinary differential equations in mathematical physics such as the Kummer equation and the combined gravity-inertial-rossby wave equation are analysed. Also, using the standard Lagrangian and previous knowledge of the (1+1) wave equation, we find some interesting properties of certain classes of partial differential equations like the canonical form of the wave equation, the wave equation with dissipation and the Klein-Gordon equation.

2. Equivalent Lagrangians

Consider an 𝑟th-order system of partial differential equations (DEs) of 𝑛-independent variables 𝑥=(𝑥¹,𝑥²,…,𝑥^𝑛) and 𝑚-dependent variables 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑚): 𝐺^𝜇𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑟)=0,𝜇=1,…,𝑚, (2.1) where 𝑢₍₁₎,𝑢₍₂₎,…,𝑢_(𝑟) denote the collections of all first-, second-, …, 𝑟th-order partial derivatives. The total differentiation operator with respect to 𝑥^𝑖 is given by 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+…,𝑖=1,…,𝑛. (2.2)

A current 𝑇=(𝑇¹,…,𝑇^𝑛) is conserved if it satisfies 𝐷_𝑖𝑇^𝑖=0 (2.3) along the solutions of (2.1).

The Euler-Lagrange (Euler) operator is defined by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼+_𝑠≥1(−1)^𝑠𝐷_𝑖1,…,𝐷_𝑖𝑠𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑠},𝛼=1,…,𝑚. (2.4)

Hence, the Euler-Lagrange (Euler) equations are of the form 𝜕𝐿𝜕𝑢^𝛼=0,𝛼=1,…,𝑚, (2.5) where 𝐿 is a Lagrangian of some order; the solutions of (2.5) are the optimizers of the functional 𝐿𝑥,𝑢,𝑢₍₁₎,…𝑑𝑥. (2.6)

A vector field 𝑋 of the form 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼,𝜉^𝑖,𝜂^𝛼∈𝒜, (2.7) which leaves (2.6) invariant is known as a Noether symmetry, where 𝒜 is the space of differential functions. Equivalently, 𝑋 is a Noether symmetry of 𝐿 if there is a vector 𝐵=(𝐵¹,…,𝐵^𝑛)∈𝒜 such that 𝑋(𝐿)+𝐿𝐷_𝑖𝜉^𝑖=𝐷_𝑖𝐵^𝑖, (2.8) where 𝑋 is prolonged to the degree of 𝐿 (see [1]). If the vector 𝐵 is identically zero, then 𝑋 is a strict Noether symmetry, see Ibragimov et al. [2].

It is well known that if the Noether symmetry algebras for two Lagrangians, 𝐿 and 𝐿, are isomorphic, the Lagrangians can be mapped from one to the other. In light of this, we define the notion of equivalent Lagrangians.

Definition 2.1. Two Lagrangians, 𝐿=𝐿(𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑟)) and 𝐿=𝐿(𝑋,𝑈,𝑈₍₁₎,…,𝑈_(𝑟)), are said to be equivalent if and only if there exists a transformation, 𝑋=𝑋(𝑥,𝑢) and 𝑈=𝑈(𝑥,𝑢), such that 𝐿𝑥,𝑢,𝑢₍₁₎,…=𝐿𝑋,𝑈,𝑈₍₁₎,…𝐽𝑥,𝑢,𝑢₍₁₎, (2.9) where 𝐽 is the determinant of the Jacobian matrix, see Kara [3].
For ordinary differential equations in which 𝑢=𝑢(𝑥), the definition of equivalence up to gauge is as follows.

Definition 2.2. Two Lagrangians, 𝐿 and 𝐿, are said to be equivalent up to gauge if and only if there exists a transformation, 𝑋=𝑋(𝑥,𝑢) and 𝑈=𝑈(𝑥,𝑢), such that 𝐿𝑥,𝑢,𝑢^′=𝐿𝑋,𝑈,𝑈^′𝑑𝑈𝑑𝑥+𝑓_𝑥+𝑢′𝑓_𝑢, (2.10) where the gauge function, 𝑓, is an arbitrary function of 𝑥 and 𝑢, see Kara and Mahomed [4].

Remark 2.3. The definitions imply that given a variational differential equation with corresponding Lagrangian 𝐿, we can find a regular point transformation 𝑋=𝑋(𝑥,𝑢) and 𝑈=𝑈(𝑥,𝑢) which maps 𝐿 to another (equivalent) Lagrangian 𝐿. This regular point transformation also maps the solutions of the differential equation associated with 𝐿 to the solutions of the original differential equation.
Also, once we have found the regular point transformation 𝑋=𝑋(𝑥,𝑢) and 𝑈=𝑈(𝑥,𝑢) mentioned above, it is possible to use this transformation to map the (known) conserved quantities of the differential equation associated with 𝐿 to the conserved quantities of the equation in question.
As an illustration, consider the well-known harmonic oscillator ordinary differential equation 𝑦^′′+𝑦=0, (2.11) with Lagrangian 1𝐿=2𝑦^′2−12𝑦². (2.12)
Using the method of equivalent Lagrangians detailed in the following sections, one can find the regular point transformation 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦) that maps the Lagrangian 1𝐿=2𝑌^′2, (2.13) associated with the free particle differential equation 𝑌^′′=0, (2.14) to the Lagrangian (2.12) associated with the differential equation (2.11). The transformation in question is given by the equations 𝑋=tan𝑥 and 𝑌=𝑦sec𝑥. This transformation in turn maps the solutions of (2.14) to the solutions of (2.11). Furthermore, we can use it to find the conserved quantities of (2.11).
Consider, for example, the known conserved quantity 𝐼=𝑋𝑌′−𝑌 (2.15) of (2.14). Using transformations 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦) above, it follows that a conserved quantity for (2.11) is 𝐼=𝑋𝑑𝑌𝑑𝑋−𝑌=𝑦^′sin𝑥+𝑦sin𝑥tan𝑥−𝑦sec𝑥. (2.16) This is verified by 𝑑𝐼/𝑑𝑥=𝑦^′′sin𝑥+𝑦sin𝑥=sin𝑥(𝑦^′′+𝑦). 𝐼 is the well-known integral 𝑦^′sin𝑥−𝑦cos𝑥.

3. Applications to ODEs

Second-order ordinary differential equations (ODEs) can be divided into equivalence classes based on their Lie symmetries [5]. Two equations belong to the same equivalence class if there exists a diffeomorphism that transforms one of the equations to the other [5]. If a second-order ordinary differential equation admits eight Lie symmetries (the maximum number of Lie symmetries of a scalar second-order ordinary differential equation, by Lie’s “Counting Theorem,” it belongs to the equivalence class of the equation 𝑌^′′=0 [5]. Hence, it can be mapped to this equation by means of a regular point transformation.

Mahomed et al. [5] prove that the maximum dimension of the Noether symmetry algebra for a scalar second-order ordinary differential equation is five and that (2.14) with standard Lagrangian (2.13) attains this maximum. This five-dimensional Noether algebra is unique (see [5]), and so for any scalar second-order ordinary differential equation with Lagrangian, 𝐿, generating a five-dimensional Noether algebra, 𝐿 can be mapped to 𝐿 by means of a regular point transformation 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦) (this transformation evidently also transforms the corresponding Euler-Lagrange equations, for 𝐿 and 𝐿, respectively, from one to the other [5]).

We use the method of equivalent Lagrangians detailed above to find solutions and conserved quantities for two scalar second-order ordinary differential equations, namely, the Kummer equation and the combined gravity-inertial Rossby wave equation.

3.1. The Kummer Equation

The Kummer equation, also called the confluent hypergeometric function, has several applications in theoretical physics. It models the velocity distribution of electrons in a high-frequency gas discharge. Using the solutions of this equation, together with kinetic theory, it is thus possible to predict the high-frequency breakdown electric field for gases (see [6]). The differential equation is given by 𝑥𝑦^′′+(2𝑘−𝑥)𝑦^′−𝑘𝑦=0, (3.1) where 𝑘 is an arbitrary constant. By rearranging this equation and multiplying by an integrating factor 𝑘𝑥𝑒−𝑥, we discover that a Lagrangian for this equation is 1𝐿=2𝑥²𝑘𝑒^−𝑥𝑘𝑦′+𝑥𝑦². (3.2)

Equation (3.1) has 8 Lie symmetries. Therefore, it can be mapped, via a point transformation 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦) to equation (2.14), with Lagrangian (2.13), which is known to have five Noether symmetries. It can be shown that the Lagrangian for the Kummer equation (3.1), given by (3.2), also has five Noether symmetries. Therefore, Lagrangians (3.2) and (2.13) are equivalent. Invoking Definition 2.2 and substituting 𝐿 and 𝐿 into (2.10), we can find the point transformations 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦) that map (3.2) to (2.13), and hence (3.1) to (2.14).

Equation (2.10) gives us 12𝑥^2𝑘𝑒^−𝑥𝑦^′2+𝑘𝑥𝑦²=12𝑦^′2𝑑𝑋𝑑𝑥+𝑓_𝑥+𝑦′𝑓_𝑦=12𝑌_𝑥+2𝑌_𝑥𝑌_𝑦𝑦′+𝑦^′2𝑌²_𝑦𝑋_𝑥+𝑦′𝑋_𝑦+𝑓_𝑥+𝑦′𝑓_𝑦. (3.3) In order to simplify the above equation, we assume that 𝑋 is a function of 𝑥 only and is of the form 𝑋=∫𝑎²𝑑𝑥, where 𝑎 is a function of 𝑥. In fact, this assumption is not essential. It turns out that the coefficient of the cubic term in the subsequent separation leads to 𝑋_𝑦=0. The above equation becomes 12𝑥^2𝑘𝑒^−𝑥𝑦^′2+𝑘𝑥𝑦²=12𝑌²_𝑥+2𝑌_𝑥𝑌_𝑦𝑦′+𝑦^′2𝑌²_𝑦𝑎²+𝑓_𝑥+𝑦′𝑓_𝑦. (3.4)

Now, since the variables 𝑥, 𝑦, and 𝑦′ are all linearly independent, we can separate (3.4) by powers of 𝑦′, after which we obtain a system of three equations: 12𝑥^2𝑘𝑒^−𝑥=12𝑌²_𝑦𝑎²,10=22𝑌_𝑥𝑌_𝑦𝑎²+𝑓_𝑦1,2𝑥^2𝑘𝑒^−𝑥𝑦²=12𝑌²_𝑥𝑎²+𝑓_𝑥. (3.5)(3.6)(3.7) From (3.5), we get that 𝑌_𝑦=𝑎𝑥^𝑘𝑒^−(1/2)𝑥. (3.8) Integrating with respect to 𝑦 results in the expression 𝑌=𝑎𝑥^𝑘𝑒^−(1/2)𝑥𝑦+𝑏(𝑥), (3.9) for which we assume that 𝑏(𝑥)=0. We can differentiate (3.9) partially with respect to 𝑥, and substitute expressions for 𝑌_𝑥 and 𝑌_𝑦 (given above) into (3.6), in order to obtain the expression 𝑓_𝑦=−𝑥^2𝑘−1𝑒^−𝑥𝑦𝑎̇𝑎1𝑥+𝑘−2𝑥, (3.10) from which we get 1𝑓=−2𝑥^2𝑘−1𝑒^−𝑥𝑦²𝑎̇𝑎1𝑥+𝑘−2𝑥+𝑐(𝑥), (3.11) where we again assume that 𝑐(𝑥)=0.

For (3.7), we can substitute our expression for 𝑌_𝑥 to obtain 𝑎̇𝑥𝑘=𝑎𝑥²𝑎̇+2𝑎𝑥1𝑘−2𝑥+1𝑘−2𝑥²+𝑓_𝑥. (3.12)

Making the substitution 𝑎̇𝐴=𝑎 (3.13) simplifies the above equation to 𝑥𝑘=𝐴𝑥²1+2𝐴𝑥𝑘−2𝑥+1𝑘−2𝑥²+𝑓_𝑥. (3.14)

We then differentiate (3.11) partially with respect to 𝑥 and obtain an expression for 𝑓_𝑥, which we can substitute into the above equation. This simplifies to 0=𝐴²−̇𝑘𝐴−𝑥𝑘𝑥−1−14. (3.15)

Integrating the equation 𝐴=𝑎/̇𝑎 gives us the expression 𝑎=𝑒^{∫𝐴𝑑𝑥}, (3.16) where 𝐴 satisfies (3.15). Hence, we have that 𝑋=𝑒^{2∫𝐴𝑑𝑥}𝑑𝑥,𝑌=𝑒^{∫𝐴𝑑𝑥}𝑥^𝑘𝑒^−(1/2)𝑥𝑦. (3.17)

Equation (3.17) defines our regular point transformations 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦), which transform (3.1) to (2.14).

We know that the solution to (2.14) is given by 𝑌=𝛼𝑋+𝛽, where 𝛼 and 𝛽 are arbitrary constants. Therefore, we can substitute expressions (3.17), for 𝑋 and 𝑌, respectively, to obtain an expression for 𝑦 which is the solution to the Kummer equation (3.1). As before, the point transformations found above can also be used to find the conserved quantities of the Kummer equation.

3.2. The Combined Gravity-Inertial-Rossby Wave Equation

The combined gravity-inertial-Rossby wave equation is given by 𝑦^′′+𝑔(𝑥)𝑦=0, (3.18) where 𝑔(𝑥) is an arbitrary function of 𝑥. The derivation of this equation is outlined in MCkenzie [7]. Very briefly, the governing equations for the combined gravity-inertial-Rossby waves on a 𝛽-plane reduce to a partial differential equation, which, with Fourier plane wave analysis, becomes a second-order ordinary differential equation describing the latitudinal structure of the perturbations. In (3.18), 𝑥 and 𝑦 are local Cartesian coordinates and 𝑔(𝑥) is the wave number, see MCkenzie [7]. By inspection, we find that 1𝐿=2𝑦′²−12𝑔(𝑥)𝑦² (3.19) is a Lagrangian for (3.18). As for the Kummer equation and its corresponding Lagrangian, it can be shown that (3.18) with Lagrangian (3.19) has an eight-dimensional Lie symmetry algebra and a five-dimensional Noether algebra. Therefore, this equation can be mapped to (2.14) using the method of equivalent Lagrangians. We follow the same procedure as for the Kummer equation in the previous section, with our aim being to find the regular point transformations 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦) that map (3.18) to (2.14).

As before, we begin by substituting expressions for 𝐿 and 𝐿 (given in (3.19) and (2.13), resp.) into (2.10). This gives the equation 12𝑦′²−12𝑔(𝑥)𝑦²=12𝑌_𝑥+2𝑌_𝑥𝑌_𝑦𝑦′+𝑦′²𝑌²_𝑦𝑋_𝑥+𝑦′𝑋_𝑦+𝑓_𝑥+𝑦′𝑓_𝑦. (3.20) Again we assume that 𝑋 is of the form 𝑋=∫𝑎²𝑑𝑥, where 𝑎=𝑎(𝑥), which simplifies the above equation to 12𝑦^′2−12𝑔(𝑥)𝑦²=12𝑌²_𝑥+2𝑌_𝑥𝑌_𝑦𝑦^′+𝑦^′2𝑌²_𝑦𝑎²+𝑓_𝑥+𝑦^′𝑓_𝑦. (3.21) Separating by powers of 𝑦′, we obtain the following system of three equations: 12=12𝑌²_𝑦𝑎²,𝑌0=^𝑥𝑌_𝑦𝑎²+𝑓_𝑦,−12𝑔(𝑥)𝑦²=12𝑌²_𝑥𝑎²+𝑓_𝑥. (3.22)(3.23)(3.24)

From (3.22), we deduce that 𝑌=𝑎𝑦+𝑏(𝑥), (3.25) where we can assume that 𝑏(𝑥)=0. Substituting expressions for 𝑌_𝑦 and 𝑌_𝑥 into (3.23), and then integrating with respect to 𝑦, we have that 1𝑓=−2𝑎̇𝑎𝑦²+𝑐(𝑥), (3.26) for which we again assume that 𝑐(𝑥)=0. Finally, after substituting expressions for 𝑌_𝑥 and 𝑓_𝑥 into (3.24), and making the substitution 𝐴=𝑎/̇𝑎, we obtain the equation −12𝑔(𝑥)𝑦²=12𝐴²𝑦²−12𝐴̇𝑦², (3.27) which simplifies to 0=𝐴̇−𝐴²−𝑔(𝑥). (3.28)

Thus, as before, 𝑎=𝑒^{∫𝐴𝑑𝑥}, where 𝐴 satisfies (3.28). Hence, we have that our regular point transformations 𝑋=𝑋(𝑥,𝑦) and 𝑌=𝑌(𝑥,𝑦), which transform (3.18) to (2.14), are given by 𝑋=𝑒^{2∫𝐴𝑑𝑥}𝑑𝑥,𝑌=𝑒^{∫𝐴𝑑𝑥}𝑦. (3.29)

4. Applications to PDEs

We now study the application of the method to some classes of partial differential equations (PDEs) in two independent variables. We first demonstrate that given a Lagrangian, 𝐿, and a known transformation, one can construct an equivalent Lagrangian 𝐿. Following this, we turn our attention to the construction of a standard form for the Lagrangian equivalent to the usual Lagrangian of the standard wave equation. This will enable us to apply the method to partial differential equations whose Lagrangians are known to be equivalent to that of the standard wave equation. In this latter situation, the aim of the method is to construct a transformation that maps one Lagrangian, 𝐿, to its equivalent 𝐿.

4.1. Illustrative Example 1

In the first example, we use a given Lagrangian 𝐿 and a given transformation, 𝑋=𝑋(𝑥,𝑡,𝑢), 𝑇=𝑇(𝑥,𝑡,𝑢), and 𝑈=𝑈(𝑥,𝑡,𝑢), in order to construct an equivalent Lagrangian 𝐿.

Consider the (1+1) wave equation with unit wave speed, 𝑈_𝑇𝑇−𝑈_𝑋𝑋=0. (4.1)

Equation (4.1) is known to have the Lagrangian 1𝐿=2𝑈²_𝑇−𝑈²_𝑋. (4.2)

Suppose we are given the transformation 𝑋=𝑡+𝑥,𝑇=𝑡−𝑥,𝑈=𝑢, (4.3) which is the standard transformation to canonical form, see Kara [3]. By making the correct substitutions into (2.9), we can calculate 𝐿.

Firstly, the determinant of the Jacobean matrix, 𝐽, is given by 𝐽=𝑑𝑋𝑑𝑥𝑑𝑋𝑑𝑡𝑑𝑇𝑑𝑥𝑑𝑇𝑑𝑡, (4.4) for two independent variables 𝑥 and 𝑡, see Kara [3].

The Lagrangian 𝐿 is a function of the variables 𝑥, 𝑡 and 𝑢, where 𝑢=𝑢(𝑥,𝑡). Hence using our canonical transformation above, we have that 𝐽=2.

It follows that 1𝐿=2𝑈²_𝑇−𝑈²_𝑋⋅2. (4.5)

In order to find 𝑈_𝑇 and 𝑈_𝑋, we note 𝑑𝑈=𝑑𝑡𝜕𝑇𝜕𝑡𝜕𝑈+𝜕𝑇𝜕𝑋𝜕𝑡𝜕𝑈,𝜕𝑋𝑑𝑈=𝑑𝑥𝜕𝑇𝜕𝑥𝜕𝑈+𝜕𝑇𝜕𝑋𝜕𝑥𝜕𝑈.𝜕𝑋 (4.6)

Using our canonical transformation, 𝑋=𝑋(𝑥,𝑡,𝑢), 𝑇=𝑇(𝑥,𝑡,𝑢), and 𝑈=𝑈(𝑥,𝑡,𝑢), we have the equations 𝑢_𝑡=𝑈_𝑇+𝑈_𝑋 and 𝑢_𝑥=−𝑈_𝑇+𝑈_𝑋. Solving these simultaneously, we get that 𝑈_𝑇=12𝑢_𝑡−𝑢_𝑥,𝑈_𝑋=12𝑢_𝑡+𝑢_𝑥. (4.7)

These, expressions into (4.5) yield 1𝐿=4𝑢²_𝑡−2𝑢_𝑥𝑢_𝑡+𝑢²_𝑥−14𝑢²_𝑡+2𝑢_𝑥𝑢_𝑡+𝑢²_𝑥=−𝑢_𝑥𝑢_𝑡. (4.8)

Hence, 𝐿 is equivalent to 𝐿 in the sense of Definition 2.1. The Euler-Lagrange equation associated with 𝐿 is 𝑢_𝑥𝑡=0, (4.9) which is the canonical form of the wave equation given in (4.1).

4.2. Illustrative Example 2

In the previous example, we made use of a canonical transformation in order to find a Lagrangian equivalent to 𝐿. In this example, however, transformed variables are concluded as a consequence of the underlying symmetry structure from which an equivalent Lagrangian is constructed.

It can be verified that 𝐺()=𝜕𝑋,𝑇,𝑈𝜕𝑇 (4.10) is a Noether point symmetry generator for the Lagrangian 𝐿 given by (4.2). Suppose we wish to map 𝐺 to the dilation symmetry generator 𝐺()𝜕𝑥,𝑡,𝑢=𝑥+1𝜕𝑥2𝑢𝜕.𝜕𝑢 (4.11)

Once this mapping is found, it can be used in formula (2.9) to determine 𝐿. The formula for change of variables is given by 𝐺()(𝑋)𝜕𝑋,𝑇,𝑈=𝐺(𝑇)𝜕𝜕𝑋+𝐺(𝑈)𝜕𝜕𝑇+𝐺.𝜕𝑈 (4.12)

Substituting the relevant values into (4.12), we obtain the three equations: 𝐺(𝑋)=0,𝐺(𝑇)=1,𝐺(𝑈)=0. (4.13)

We solve these equations using the method of invariants, for which we get that 𝑢𝑋=ℱ𝑡,²𝑥𝑢,𝑇=ln𝑥+𝒢𝑡,²𝑥𝑢,𝑈=ℋ𝑡,²𝑥, (4.14) where ℱ, 𝒢, and ℋ are arbitrary functions. As an illustration, we choose 𝑢𝑋=𝑡,𝑇=ln𝑥,𝑈=²𝑥. (4.15)

This gives us our transformation. From (4.4), 𝐽=−1/𝑥. Thus, 𝑈_𝑋=2𝑢𝑢_𝑡/𝑥 and 𝑈_𝑇=2𝑢𝑢_𝑥−(𝑢²/𝑥) so that, by (2.9), we get 1𝐿=2𝑈²_𝑇−𝑈²_𝑋.−1𝑥=−12𝑥³𝑢²4𝑢²_𝑥𝑥²−4𝑢𝑢_𝑥𝑥+𝑢²−4𝑢²_𝑡. (4.16)

4.3. Equivalent Lagrangian for the Wave Equation in (1 + 1) Dimension

We now find an expression for the form of a Lagrangian, 𝐿, which is equivalent to the usual Lagrangian of the wave equation, 𝐿. Once we have this form, given any 𝐿 equivalent to 𝐿, we can find the transformation that maps 𝐿 to 𝐿, and hence the solutions and conserved quantities of the differential equation associated with 𝐿 to those of the standard wave equation.

Since 𝑈_𝑇=(𝑑𝑈/𝑑𝑡)𝑋_𝑥−𝑋_𝑡(𝑑𝑈/𝑑𝑥)𝑇_𝑡𝑋_𝑥−𝑋_𝑡𝑇_𝑥,𝑈_𝑋=𝑇_𝑡(𝑑𝑈/𝑑𝑥)−𝑇_𝑥(𝑑𝑈/𝑑𝑡)𝑇_𝑡𝑋_𝑥−𝑋_𝑡𝑇_𝑥, (4.17) we get 12𝑈²_𝑇−𝑈²_𝑋=12𝑑𝑈²/𝑑𝑡𝑋²_𝑥−𝑇²_𝑥+𝑑𝑈²/𝑑𝑥𝑋²_𝑡−𝑇²_𝑡+2(𝑑𝑈/𝑑𝑥)(𝑑𝑈/𝑑𝑡)𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥𝑇²_𝑡𝑋²_𝑥−2𝑇_𝑡𝑇_𝑥𝑋_𝑡𝑋_𝑥+𝑋²_𝑡𝑇²_𝑥. (4.18) Here, 𝐽=𝑋_𝑥(𝑇_𝑡+𝑢_𝑡𝑇_𝑢)−𝑋_𝑡(𝑇_𝑥+𝑢_𝑥𝑇_𝑢)+𝑋_𝑢(𝑢_𝑥𝑇_𝑡−𝑢_𝑡𝑇_𝑥) so that 1𝐿=2𝑈²_𝑇−𝑈²_𝑋𝐽𝑋_𝑥𝑇_𝑡+𝑢_𝑡𝑇_𝑢−𝑋_𝑡𝑇_𝑥+𝑢_𝑥𝑇_𝑢+𝑋_𝑢𝑢_𝑥𝑇_𝑡−𝑢_𝑡𝑇_𝑥. (4.19)

It can be shown that 𝑋_𝑢=0 and 𝑇_𝑢=0 (i.e., 𝑋=𝑋(𝑥,𝑡) and 𝑇=𝑇(𝑥,𝑡)). Then, the above expression for the Lagrangian reduces to 1𝐿=2𝑑𝑈²/𝑑𝑡𝑋²_𝑥−𝑇²_𝑥+𝑑𝑈²/𝑑𝑥𝑋²_𝑡−𝑇²_𝑡+2(𝑑𝑈/𝑑𝑥)(𝑑𝑈/𝑑𝑡)𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥²×𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥=12𝑑𝑈²/𝑑𝑡𝑋²_𝑥−𝑇²_𝑥+𝑑𝑈²/𝑑𝑥𝑋²_𝑡−𝑇²_𝑡+2(𝑑𝑈/𝑑𝑥)(𝑑𝑈/𝑑𝑡)𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥. (4.20) With 𝑑𝑈/𝑑𝑡=𝑈_𝑡+𝑢_𝑡𝑈_𝑢 and 𝑑𝑈/𝑑𝑥=𝑈_𝑥+𝑢_𝑥𝑈_𝑢, 1𝐿=2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥𝑈_𝑡+𝑢_𝑡𝑈_𝑢²𝑋²_𝑥−𝑇²_𝑥+𝑈_𝑥+𝑢_𝑥𝑈_𝑢²𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑥+𝑢_𝑥𝑈_𝑢𝑈_𝑡+𝑢_𝑡𝑈_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥=12𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥𝑢_𝑥𝑢_𝑡2𝑈²_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥+𝑢²_𝑥𝑈²_𝑢𝑋²_𝑡−𝑇²_𝑡+𝑢²_𝑡𝑈²_𝑢𝑋²_𝑥−𝑇²_𝑥+𝑢_𝑥2𝑈_𝑢𝑈_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥+𝑢_𝑡2𝑈_𝑡𝑈_𝑢𝑋²_𝑥−𝑇²_𝑥+2𝑈_𝑢𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥+𝑈²_𝑡𝑋²_𝑥−𝑇²_𝑥+𝑈²_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥. (4.21)

This is the general form for a Lagrangian equivalent to the Lagrangian of the wave equation, where 𝑈=𝑈(𝑥,𝑡,𝑢), 𝑋=𝑋(𝑥,𝑡), and 𝑇=𝑇(𝑥,𝑡). Once we have a Lagrangian which we know to be equivalent to the Lagrangian 𝐿 given by (4.2), we can reverse the process of the examples above and use the form of the Lagrangian in the previous equation in order to find the transformations that map the solutions of the standard wave equation (4.1) to the solutions of the equivalent Euler differential equation.

4.3.1. Finding Transformations: Example 1

Consider (4.9) with its Lagrangian 𝐿=−𝑢_𝑥𝑢_𝑡, (4.22) which we found to be equivalent to (4.2). We use the form of the equivalent Lagrangian given above in order to find the transformations that map this Lagrangian to the Lagrangian (4.2). Substituting (4.22) for 𝐿 and separating by monomials, we arrive at the equations: 𝑢_𝑡𝑢_𝑥∶−1=2𝑈²_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢²_𝑥𝑈∶0=^2𝑢𝑋²_𝑡−𝑇²_𝑡2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢²_𝑡𝑈∶0=^2𝑢𝑋²_𝑥−𝑇²_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢_𝑥∶0=2𝑈_𝑢𝑈_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢_𝑡∶0=2𝑈_𝑡𝑈_𝑢𝑋²_𝑥−𝑇²_𝑥+2𝑈_𝑢𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑈1∶0=^2𝑡𝑋²_𝑥−𝑇²_𝑥+𝑈²_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥. (4.23)

Using software to solve this overdetermined system of equations, we get the transformation 𝑋=𝑓(𝑥)+𝑔(𝑡),𝑇=𝑓(𝑥)−𝑔(𝑡),𝑈=𝑢, (4.24) for 𝑓 and 𝑔 arbitrary functions of 𝑥 and 𝑡, respectively. The well-known transformation 𝑋=𝑡+𝑥, 𝑇=𝑡−𝑥, 𝑈=𝑢, used in (4.1), is in fact a special case.

4.3.2. Finding Transformations: Example 2

Utilising the Lagrangian 𝐿=−12𝑥³𝑢²4𝑢²_𝑥𝑥²−4𝑢𝑢_𝑥𝑥+𝑢²−4𝑢²_𝑡, (4.25) which was constructed to be equivalent to the Lagrangian 𝐿; this was done in Subsection 4.2 above, the procedure yields 𝑋=𝑓(𝑡+ln𝑥)−𝑔(𝑡−ln𝑥), 𝑇=𝑓(𝑡+ln𝑥)+𝑔(𝑡−ln𝑥), and 𝑈=±𝑢²/𝑥, where 𝑓 and 𝑔 are arbitrary functions and 𝑥 and 𝑡 as well. If we choose 𝑓 such that 𝑓(𝑡+ln𝑥)=(1/2)(𝑡+ln𝑥) and 𝑔 such that 𝑔(𝑡−ln𝑥)=−(1/2)(𝑡−ln𝑥), we have the transformation as the one used in Example 2 of the previous section that resulted in the Lagrangian (4.25).

4.4. The Equivalence of the Dissipative Wave and Klein-Gordon Equations

The equation 𝑈_𝑇𝑇+𝑈_𝑇−𝑈_𝑋𝑋=0, (4.26) is the one-dimensional case of the linear wave equation with dissipation (see [8]). This equation has the well-known Lagrangian, after multiplication by a variational factor, 1𝐿=2𝑒^𝑇𝑈²_𝑇−𝑈²_𝑋. (4.27)

We map 𝐿 to the Lagrangian 1𝐿=2𝑢²_𝑡−12𝑢²_𝑥−12𝑢𝑢_𝑡+18𝑢², (4.28) giving rise to the Euler equation: 𝑢_𝑡𝑡−𝑢_𝑥𝑥−14𝑢=0, (4.29) which we note to be a Klein-Gordon equation. The Noether symmetries 𝑋=𝜉𝜕_𝑥+𝜏𝜕_𝑡+𝜙𝜕_𝑢 with gauge (𝑓,𝑔) satisfy 0=𝑢𝜙4−𝜙𝑢_𝑡2+12𝑢𝑢_𝑥𝜉_𝑡−𝑢_𝑡𝑢_𝑥𝜉_𝑡+12𝑢𝑢_𝑡𝑢_𝑥𝜉_𝑢−𝑢²_𝑡𝑢_𝑥𝜉_𝑢+𝑢³_𝑥𝜉_𝑢+𝑢²_𝑥𝜉_𝑥+12𝑢𝑢_𝑡𝜏_𝑡−𝑢²_𝑡𝜏_𝑡+12𝑢𝑢²_𝑡𝜏_𝑢−𝑢³_𝑡𝜏_𝑢+𝑢_𝑡𝑢²_𝑥𝜏_𝑢+𝑢_𝑡𝑢_𝑥𝜏_𝑥−𝑢𝜙_𝑡2+𝑢_𝑡𝜙_𝑡−12𝑢𝑢_𝑡𝜙_𝑢+𝑢²_𝑡𝜙_𝑢−𝑢²_𝑥𝜙_𝑢−𝑢_𝑥𝜙_𝑥+12𝑢²_𝑡−12𝑢²_𝑥−12𝑢𝑢_𝑡+18𝑢²𝜏_𝑡+𝑢_𝑡𝜏_𝑢+𝜉_𝑥+𝑢_𝑥𝜉_𝑢−𝑓_𝑡+𝑢_𝑡𝑓_𝑢+𝑔_𝑥+𝑢_𝑥𝑔_𝑢, (4.30) which separates into an overdetermined system of partial differential equations whose solution is 𝑋₁=𝜕_𝑡𝑋,𝑓=0,𝑔=0,₂=𝜕_𝑥𝑋,𝑓=0,𝑔=0,₃=𝑡𝜕_𝑥+𝑥𝜕_𝑡1,𝑓=0,𝑔=4𝑢²,𝑋_∞=𝐹(𝑥,𝑡)𝜕_𝑢1,𝑓=−2𝑢𝐹+𝑢𝐹_𝑡,𝑔=−𝑢𝐹_𝑥, (4.31) where 𝐹 satisfies (1/4)𝐹+𝐹_𝑥𝑥−𝐹_𝑡𝑡=0. The Lie algebra is isomorphic to the Noether algebra corresponding to the Lagrangian 𝐿 (see [9]). Hence, 𝐿 and 𝐿 are equivalent Lagrangians.

We can therefore use (2.9) in order to find the transformations 𝑋=𝑋(𝑥,𝑡,𝑢), 𝑇=𝑇(𝑥,𝑡,𝑢) and 𝑈=𝑈(𝑥,𝑡,𝑢) that map 𝐿 to 𝐿.

Assuming that 𝑋_𝑢=0 and 𝑇_𝑢=0 as before, we get 𝑒𝐿=^𝑇2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥𝑢_𝑥𝑢_𝑡2𝑈²_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥+𝑢²_𝑥𝑈²_𝑢𝑋²_𝑡−𝑇²_𝑡+𝑢²_𝑡𝑈²_𝑢𝑋²_𝑥−𝑇²_𝑥+𝑢_𝑥2𝑈_𝑢𝑈_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥+𝑢_𝑡2𝑈_𝑡𝑈_𝑢𝑋²_𝑥−𝑇²_𝑥+2𝑈_𝑢𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥+𝑈²_𝑡𝑋²_𝑥−𝑇²_𝑥+𝑈²_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥. (4.32)

Substituting for 𝐿 and then separating by derivative terms, we arrive at the system 𝑢_𝑡𝑢_𝑥∶0=2𝑈²_𝑢𝑒^𝑇𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢²_𝑥1∶−2=𝑈²_𝑢𝑒^𝑇𝑋²_𝑡−𝑇²_𝑡2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢²_𝑡∶12=𝑈²_𝑢𝑒^𝑇𝑋²_𝑥−𝑇²_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢_𝑥𝑒∶0=^𝑇2𝑈_𝑢𝑈_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑢𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,𝑢_𝑡1∶−2𝑒𝑢=^𝑇2𝑈_𝑡𝑈_𝑢𝑋²_𝑥−𝑇²_𝑥+2𝑈_𝑢𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥,11∶8𝑢²=𝑒^𝑇𝑈²_𝑡𝑋²_𝑥−𝑇²_𝑥+𝑈²_𝑥𝑋²_𝑡−𝑇²_𝑡+2𝑈_𝑡𝑈_𝑥𝑇_𝑡𝑇_𝑥−𝑋_𝑡𝑋_𝑥2𝑋_𝑥𝑇_𝑡−𝑋_𝑡𝑇_𝑥. (4.33)

For this overdetermined system of equations, the software yields the result 𝑋=𝑓₁(𝑡+𝑥)−𝑓₂(𝑡−𝑥),𝑇=𝑓₁(𝑡+𝑥)+𝑓₂(𝑡−𝑥),𝑈=−𝑒^−(1/2)𝑡𝑢. (4.34)

Notes
(1)the special case 𝑓₁(𝑡+𝑥)=(1/2)(𝑡+𝑥), 𝑓₂(𝑡−𝑥)=(1/2)(𝑡−𝑥), from which we get that the transformation 𝑋=𝑥,𝑇=𝑡,𝑈=−𝑒^−(1/2)𝑡𝑢 (4.35) is a known transformation mapping (4.26) to (4.29), see Ibragimov [8]. The transformations also transform the Noether symmetries of 𝐿 to those of 𝐿;(2)this result is true for 𝑛-dimensional case.

4.5. Equivalent Lagrangians and Wave Equations on Spacetime Manifolds

The three-dimensional linear wave equation is the well-known second-order PDE: 𝑈_𝑇𝑇−𝑈_𝑋𝑋−𝑈_𝑌𝑌−𝑈_𝑍𝑍=0, (4.36) see [8]. Here, 𝑇, 𝑋, 𝑌, and 𝑍 are the independent variables, and 𝑈=𝑈(𝑇,𝑋,𝑌,𝑍) is the dependent variable. The usual Lagrangian for this equation is 1𝐿=2𝑈²_𝑇−12𝑈²_𝑋−12𝑈²_𝑌−12𝑈²_𝑍, (4.37) [8]. It is well known that the Lagrangian for the three-dimensional wave equation in spherical co-ordinates is given by 1𝐿=2𝑥²sin𝑦𝑢²_𝑡−12sin𝑦𝑥²𝑢²_𝑥−12sin𝑦𝑢²_𝑦−1𝑢2sin𝑦^2𝑧. (4.38)𝐿 and 𝐿 are naturally equivalent because they are Lagrangians of the same equation. However, it can also be verified that they generate isomorphic Noether algebras of point symmetries.

In this section, we apply the method of equivalent Lagrangians to 𝐿 and 𝐿, in an attempt to recover the transformation that maps 𝐿 to 𝐿. We shall demonstrate that (2.9) of Definition 2.1 is satisfied by the transformation from Cartesian to polar coordinates, which is to be expected from our choice of Lagrangians.

As before, we make use of Definition 2.1, which relates two equivalent Lagrangians to each other by means of the transformation that maps one to the other. Substituting 𝐿 and 𝐿 into (2.9) gives us 12𝑥²sin𝑦𝑢²_𝑡−12sin𝑦𝑥²𝑢²_𝑥−12sin𝑦𝑢²_𝑦−1𝑢2sin𝑦^2𝑧=12𝑈²_𝑇−12𝑈²_𝑋−12𝑈²_𝑌−12𝑈²_𝑍𝐽. (4.39)

We first calculate the Jacobian 𝐽. We assume that 𝑇_𝑥=𝑇_𝑦=𝑇_𝑧=𝑋_𝑡=𝑌_𝑡=𝑍_𝑡=0. In other words, 𝑇=𝑇(𝑡), 𝑋=𝑋(𝑥,𝑦,𝑧), 𝑌=𝑌(𝑥,𝑦,𝑧), and 𝑍=𝑍(𝑥,𝑦,𝑧). Furthermore, we assume that 𝑇=𝑡. Hence 𝑇_𝑡=1. It follows that 𝐽=𝑋_𝑥𝑌_𝑦𝑍_𝑧−𝑋_𝑥𝑌_𝑧𝑍_𝑦−𝑋_𝑦𝑌_𝑥𝑍_𝑧+𝑋_𝑦𝑌_𝑧𝑍_𝑥+𝑋_𝑧𝑌_𝑥𝑍_𝑦−𝑋_𝑧𝑌_𝑦𝑍_𝑥. Then, 𝑈_𝑇𝑈_𝑋𝑈_𝑌𝑈_𝑍=1𝐽𝑢_𝑡𝑋_𝑥𝑌_𝑦𝑍_𝑧−𝑋_𝑥𝑌_𝑧𝑍_𝑦−𝑋_𝑦𝑌_𝑥𝑍_𝑧+𝑋_𝑦𝑌_𝑧𝑍_𝑥+𝑋_𝑧𝑌_𝑥𝑍_𝑦−𝑋_𝑧𝑌_𝑦𝑍_𝑥𝑢_𝑧𝑌_𝑥𝑍_𝑦−𝑌_𝑦𝑍_𝑥−𝑢_𝑦𝑌_𝑥𝑍_𝑧−𝑌_𝑧𝑍_𝑥+𝑢_𝑥𝑌_𝑦𝑍_𝑧−𝑌_𝑧𝑍_𝑦𝑢_𝑦𝑋_𝑥𝑍_𝑧−𝑋_𝑧𝑍_𝑥−𝑢_𝑧𝑋_𝑥𝑍_𝑦−𝑋_𝑦𝑍_𝑥−𝑢_𝑥𝑋_𝑦𝑍_𝑧−𝑋_𝑧𝑍_𝑦𝑢_𝑧𝑋_𝑥𝑌_𝑦−𝑋_𝑦𝑌_𝑥−𝑢_𝑦𝑋_𝑥𝑌_𝑧−𝑋_𝑧𝑌_𝑥+𝑢_𝑥𝑋_𝑦𝑌_𝑧−𝑋_𝑧𝑌_𝑦, (4.40) from which we can read the expressions for 𝑈_𝑇, 𝑈_𝑋, 𝑈_𝑌, and 𝑈_𝑍 in terms of 𝑢_𝑡, 𝑢_𝑥, 𝑢_𝑦, and 𝑢_𝑧. Substituting these into (4.39), along with our expression for 𝐽, we have the equation 12𝑥²sin𝑦𝑢²_𝑡−12sin𝑦𝑥²𝑢²_𝑥−12sin𝑦𝑢²_𝑦−1𝑢2sin𝑦^2𝑧=12−𝑋_𝑧𝑌_𝑦𝑍_𝑥+𝑋_𝑦𝑌_𝑧𝑍_𝑥+𝑋_𝑧𝑌_𝑥𝑍_𝑦−𝑋_𝑥𝑌_𝑧𝑍_𝑦−𝑋_𝑦𝑌_𝑥𝑍_𝑧+𝑋_𝑥𝑌_𝑦𝑍_𝑧=×𝑢²_𝑡−𝑢_𝑧𝑋_𝑦𝑌_𝑥−𝑢_𝑦𝑋_𝑧𝑌_𝑥−𝑢_𝑧𝑋_𝑥𝑌_𝑦+𝑢_𝑥𝑋_𝑧𝑌_𝑦+𝑢_𝑦𝑋_𝑥𝑌_𝑧−𝑢_𝑥𝑋_𝑦𝑌_𝑧²𝑋_𝑧𝑌_𝑦𝑍_𝑥−𝑋_𝑦𝑌_𝑧𝑍_𝑥−𝑋_𝑧𝑌_𝑥𝑍_𝑦+𝑋_𝑥𝑌_𝑧𝑍_𝑦+𝑋_𝑦𝑌_𝑥𝑍_𝑧−𝑋_𝑥𝑌_𝑦𝑍_𝑧²𝑢=−^𝑧𝑌_𝑦𝑍_𝑥−𝑢_𝑦𝑌_𝑧𝑍_𝑥−𝑢_𝑧𝑌_𝑥𝑍_𝑦+𝑢_𝑥𝑌_𝑧𝑍_𝑦+𝑢_𝑦𝑌_𝑥𝑍_𝑧−𝑢_𝑥𝑌_𝑦𝑍_𝑧²𝑋_𝑧𝑌_𝑦𝑍_𝑥−𝑋_𝑦𝑌_𝑧𝑍_𝑥−𝑋_𝑧𝑌_𝑥𝑍_𝑦+𝑋_𝑥𝑌_𝑧𝑍_𝑦+𝑋_𝑦𝑌_𝑥𝑍_𝑧−𝑋_𝑥𝑌_𝑦𝑍_𝑧²−𝑢_𝑧𝑋_𝑦𝑍_𝑥−𝑢_𝑦𝑋_𝑧𝑍_𝑥−𝑢_𝑧𝑋_𝑥𝑍_𝑦+𝑢_𝑥𝑋_𝑧𝑍_𝑦+𝑢_𝑦𝑋_𝑥𝑍_𝑧−𝑢_𝑥𝑋_𝑦𝑍_𝑧²−𝑋_𝑧𝑌_𝑦𝑍_𝑥+𝑋_𝑦𝑌_𝑧𝑍_𝑥+𝑋_𝑧𝑌_𝑥𝑍_𝑦−𝑋_𝑥𝑌_𝑧𝑍_𝑦−𝑋_𝑦𝑌_𝑥𝑍_𝑧+𝑋_𝑥𝑌_𝑦𝑍_𝑧². (4.41)

It can be shown, amongst others, a solution is given by 𝑇=𝑡, 𝑋=𝑥sin𝑦cos𝑧, 𝑌=𝑥sin𝑦sin𝑧, and  𝑍=𝑥cos𝑦.

This procedure is particularly useful in mapping variational equations, like the wave equation, between equivalent “curved manifolds.”

5. Conclusion

In this paper, we have applied the notion of equivalent Lagrangians to determine transformations that map differential equations one to another in order to generate solutions, conservation laws, inter alia. An additional consequence of the procedure is recovering some well-known transformations like the mapping from the standard wave equation to the canonical form. The procedure, although cumbersome, holds for partial differential equations of any number of independent variables. Finally, transformation maps between variational equations (like the wave equation) on equivalent manifolds can be determined by the procedure.