Lagrange points are the equilibrium points within a restricted three-body system, epitomized by the Trojan asteroids near the L_{4} and L_{5} points of the Sun–Jupiter system. They also play a crucial role in some space missions, including the James Webb Space Telescope which is located at the Sun–Earth L_{2} point. While the existence of five Lagrange points is a well-known feature of the restricted three-body problem, the equations describing the precise location of all five points are not extensively documented. This work presents a derivation of all Lagrange points using polar coordinates and a new normalization scheme that offers a simpler interpretation of solutions compared to prior analyses. A subtle issue concerning the treatment of angular momentum in the potential formulation of this problem is addressed and resolved. The supplementary material to this work contains additional mathematical details and discussion.

## I. INTRODUCTION

The equilibrium points of the restricted three-body system are known as Lagrange points.^{*} (The word “restricted” means that one mass is so much smaller than the other two that it can be ignored when calculating the dynamics of the two more massive bodies.) The near-Earth Lagrange points of the Sun–Earth system, located approximately $ 1.5 \xd7 10 6$ km from Earth, have been home to multiple satellites: SOHO, WMAP, Planck, and the James Webb Space Telescope (JWST).^{1} The first of these is located near the sun-facing L_{1} point for continuous solar observations. In their search for signals of cosmic origin, these other observatories have been parked near the L_{2} point to take advantage of the clear view of deep space.^{2} Dust clouds have been observed at the L_{4} and L_{5} points of the Earth–Moon system.^{3} Further afield, the Trojan asteroids cluster around Jupiter's L_{4} and L_{5} points. The L_{3} point has no known natural or man-made significance.

While the pattern of five Lagrange points for the Sun–Earth system is relatively well-known, the configuration of these points in systems with significantly larger mass ratios, like the Pluto–Charon system with a mass ratio of about 0.12 or an equal-mass binary star system, is likely less familiar to readers. A symmetry argument for the latter readily establishes that the configuration for such a system must look quite different from those for stellar–planetary systems. Figure 1 compares the locations of the five Lagrange points for two scenarios, one for a system with very disparate masses and the other with equal masses. Beyond the qualitative aspect of the relative location of the Lagrange points, one might want to know how exactly NASA mission engineers determined that the JWST should be located at a distance of $ 1.5 \xd7 10 6$ km from Earth?

The definitive text on the subject of Lagrange points is *Theory of Orbits: the Restricted Problem of Three Bodies*.^{4} Unfortunately, it and many other sources^{5–10} analyze the motion in cartesian coordinates, which does not reflect the natural geometry of the problem. Where this problem appears in undergraduate physics textbooks, it receives very abbreviated treatment.^{11–13} The number of articles aimed at the undergraduate and educator audiences are few and light on details.^{14,15} Of these sources, only Ref. 4 provides approximate solutions to higher than first-order.

This work presents a thorough analysis of all equilibrium points using physically motivated arguments and polar coordinates. Many mathematical curiosities will be encountered on this journey, including the implications of Galois theory of quintic polynomials, expansions in fractional powers of a small parameter, symmetries, symmetry breaking, and a challenge to the local/global dichotomy of conserved quantities. While this work does not calculate the stability of the equilibrium points, it should be noted that stability can be calculated by extending the formalism presented here to second derivatives of the potential and including the effect of the Coriolis force. Such calculations show that only the L_{4} and L_{5} points are truly stable.

Section II presents preliminary aspects, including a definition of the coordinate system and a discussion of the effective potential. Building on this foundation, Sec. III analyzes the solutions for three cases. A discussion in Sec. IV summarizes the findings and offers some reflections on a few of the subtler points of the analysis, which is followed by concluding remarks in Sec. V.

## II. PRELIMINARIES

The analytical goals of this work are formulas for the equilibrium points of a satellite that is gravitationally bound to two massive bodies. The two massive bodies, with masses *m*_{1} and *m*_{2}, will be called the primaries, and the satellite, with mass *m*_{3}, will be called the tertiary.^{†} Subsections II A–II G define the coordinate system, the rotation frequency of the system, the effective potential, discuss the curious nature of angular momentum in this problem, present the azimuthal and radial components of the force, and introduce the dimensionless form that will be used in the final analysis.

### A. Coordinate system

*r*

_{1},

*r*

_{2}, and

*r*) are measured from the center of mass of the system, with

*r*

_{1}and

*r*

_{2}related through $ m 1 r 1 = m 2 r 2$, since

*m*

_{3}is considered negligible in comparison to

*m*

_{1}and

*m*

_{2}. This analysis can be extended to elliptical orbits, but the approximation of circular orbits is quite good for many planetary bodies: Earth's orbital eccentricity is less than 0.02 and Jupiter's is approximately 0.05. The angular position of the tertiary in the rotating frame of the

*m*

_{1}-

*m*

_{2}system, denoted by

*θ*, is measured from the line connecting the center of mass and

*m*

_{2}. A tertiary located at an equilibrium point will co-rotate with the

*m*

_{1}-

*m*

_{2}system, meaning that both

*r*and

*θ*will be constant. The distances between the tertiary and the primaries are described by $ r 1 \u2032$ and $ r 2 \u2032$, which can be expressed in single form as

*j*is either 1 (evaluated with the upper sign) or 2 (evaluated with the lower sign). The net force on the tertiary will involve the derivatives of Eq. (1) with respect to

*θ*and

*r*. They are

### B. Rotation frequency of the *m*_{1}-*m*_{2} system

*m*

_{1}-

*m*

_{2}system is determined only by the masses of the primaries and the distance between them. Newton's second law for the primaries can be expressed in a single equation as

*ω*

_{0}is the angular rotation frequency. An expression for

*ω*

_{0}that is symmetric in the parameters describing the primaries will be useful. Adding $ m 1 r 2$ to both sides of the center of mass condition, $ m 1 r 1 = m 2 r 2$, and rearranging yields $ m 1 / r 2 = ( m 1 + m 2 ) / ( r 1 + r 2 )$. Evaluating Eq. (4) for

*m*

_{2}and using the last equality gives

### C. The effective potential

*Analytical Mechanics*by Fowles and Cassiday,

^{11}

^{,}

*Classical Mechanics*by Goldstein

*et al.*,

^{12}

^{,}

*An Introduction to Mechanics*by Kleppner and Kolenkow,

^{13}and

*Orbital Mechanics for Engineering Students*by Curtis.

^{9}The latter analyzes the problem in cartesian coordinates. The text by Kleppner and Kolenkow presents a very limited treatment of the L

_{4}and L

_{5}points, which assumes without proof that the geometry is described by an equilateral triangle. The

*Classical Mechanics*text introduces the problem through the Lagrangian, but stops short of any further analysis and motivates the solutions through a qualitative analysis by reference to an illustration of equipotential contours. A brief discussion of the Lagrangian approach can be found in Appendix A. These and other sources often provide a visual representation of the solution space through contour plots of the effective potential (see, for example, Fig. 7.4.4 on page 294 of Ref. 11 and Fig. 3.30 on page 124 of Ref. 12). The

*Analytical Mechanics*text by Fowles and Cassiday presents the effective potential that was used to generate such figures, which is of the form

Curiously, the sign of the inertial term in Eq. (6) is negative. One might be inclined to chalk this up to a typographical error, except that it yields correct solutions to the Lagrange point problem. The effective potential of Eq. (6) is also known as Jacobi's constant, a lesser-known conserved quantity that is found by integrating the radial force multiplied by $ r \u0307$ under the constraint of constant angular velocity.^{16} Further commentary on the nature of Eq. (6) follows in Sec. II F.

*θ*and

*r*through Eq. (1) and L refers to the angular momentum of the tertiary. As simple as it is, Eq. (7) contains a problem. This would be a fine equation to use if the angular momentum of the tertiary were a conserved quantity. However, it is only the total angular momentum that is conserved in a three-body system and it cannot be assumed that the angular momentum of any single body is itself conserved.

^{‡}It is, therefore, unclear how the angular momentum should be treated in this potential. The resolution to this conundrum is presented subsequently.

### D. Conservation of angular momentum

The only possible resolution to the aforementioned puzzle is that the angular momentum must be conserved in a small region around each equilibrium point but is not conserved on larger scales. This sense of local but not global conservation is rather opposite to the conventional understanding of conservation laws where local conservation (e.g., of charge) necessarily implies global conservation. The phrase “regionally conserved quantity” is introduced here to describe the peculiar nature of angular momentum conservation over a small region. This concept is similar to quasi-local conservation in general relativity where a locally flat but globally curved spacetime allows energy and momentum to be treated as both conserved and not-conserved depending on scale.^{17–20}

In the vicinity of a Lagrange point, the azimuthal force is very weak given a small displacement from equilibrium. Because this force can be approximated as a linear function of the angular deviation over these scales, the first correction to the angular momentum will average to zero over an epicycle due to the fact that this force is an odd symmetric function. It follows that any net change to the angular momentum over longer time scales must enter as a second-order or higher effect and will, therefore, have zero derivative at the equilibrium point. It is in this sense that we can take the angular momentum to be conserved in a local region around each equilibrium point.

### E. The angular derivative

*L*as a constant of the motion. Applying the $ \u2212 r \u2212 1 ( \u2202 / \u2202 \theta )$ operator to

*V*gives the azimuthal force,

_{eff}Equilibrium points of the system require $ F \theta = 0$, which admits three cases: (A) *θ* = 0, (B) $ \theta = \pi $, and (C) $ r 1 \u2032 = r 2 \u2032$. Case A yields Lagrange points L_{1} and L_{2}, and case B yields Lagrange point L_{3}. The third case indicates that the position of the equilibrium point must lie along the bisector of the axis between the primaries, which describes Lagrange points L_{4} and L_{5}.

### F. The radial derivative

*after*the radial derivative is calculated. The radial force, calculated from $ \u2212 ( \u2202 / \u2202 r )$ applied to Eq. (7), is

*L*can now be replaced with $ m r 2 \omega 0$, which results in

*ω*

_{0}as

*r*varies. One may wonder why Eq. (6) is used at all if it has questionable physical meaning? One answer is that is a convenient short-hand that skirts the complications of the angular momentum. Another is that the physically meaningful form of the effective potential described by Eq. (7) does not lend itself to visual analysis of the solution space since it has no extrema at the Lagrange points, whereas Eq. (6) does. This point is revisited in Sec. IV and comparative illustrations of Eqs. (6) and (7) are presented in the supplementary material.

^{21}

*r*given in Eq. (3) and the definition of $ \omega 0 2$ from Eq. (5) to Eq. (11) yields the final form that will be used in the subsequent analysis

### G. Dimensionless forms

*R*, and the masses by normalization mass,

*M*. Ideal dimensionless forms of Eq. (12) have $ ( r 1 + r 2 ) / R = ( m 1 + m 2 ) / M$ because this results in a greatly simplified centrifugal term. The method used here takes

*R*=

*r*

_{2}and

*M*=

*m*

_{1}, which satisfies the prior condition since $ r 1 / r 2 = m 2 / m 1$ is equivalent to the center of mass condition. Under this transformation, the dimensionless form of

*m*

_{1}becomes 1 and the dimensionless form of

*m*

_{2}is defined as $ m 2 / m 1 = \mu $, which is known as the mass ratio. It follows that the dimensionless forms of

*r*

_{1}and

*r*

_{2}are

*μ*and 1, respectively. Since both the mass and radial scales are described by a single parameter,

*μ*, cases with $ \mu \u2032 > 1$ need not be considered since they are described by $ \mu = 1 / \mu \u2032$ with the locations of

*m*

_{1}and

*m*

_{2}swapped. Therefore,

*μ*is limited to the range $ 0 < \mu \u2264 1$ in this analysis. Under this normalization scheme, the radial force is

*x*located at a distance of $ 1 + \mu $ beyond the tertiary. While this interpretation does not have any physical meaning, viewing it this way may help to understand the importance of the Coriolis force as an essential stabilizing effect.

Szebehely,^{4} Cornish,^{5} and Widnall *et al.*^{6} take an alternate approach and use $ R = r 1 + r 2$ and $ M = m 1 + m 2$, which also satisfies $ ( r 1 + r 2 ) / R = ( m 1 + m 2 ) / M$, where $ \mu * = m 2 / ( m 1 + m 2 )$ is the dimensionless maas ratio (note: while those texts also use the symbol *μ*, the “ $*$” is added here to differentiate this definition from the definition of *μ* used in this work). These two definitions of the mass ratio are related through $ \mu = \mu * / ( 1 \u2212 \mu * )$. While the use of *μ* instead of $ \mu *$ is a departure from the standard of the literature, it is argued here that it has several advantages. First, there is the aesthetic quality that *μ* spans the range $ 0 < \mu \u2264 1$ whereas $ \mu *$ spans the range $ 0 < \mu * \u2264 1 / 2$. Second, the first-order approximations of the radial force equation for the L_{1} and L_{2} points are much simpler when described in terms of *μ*. Third, Ref. 4 presents series solutions for the radial locations of L_{1} and L_{2} in terms of a quantity $ \mu * / ( 1 \u2212 \mu * )$, which is equal to *μ* and suggests that *μ* is in fact an optimal dimensionless form. Finally, and most importantly, the radial location of the L_{2} point is non-monotonic when described in terms of $ \mu *$ and the L_{4} and L_{5} points decrease with increasing $ \mu *$, both of which are rather counterintuitive and require significant care to correctly interpret. In contrast, the formulas for the L_{2} point and the L_{4} and L_{5} points are increasing monotonic functions when described in terms of *μ*. Figure 3 depicts the pattern of solutions with increasing *μ* for all five Lagrange points. A detailed comparison of these normalization schemes, their associated solutions for the Lagrange points, and the transformations between them can be found in the supplementary material.^{21}

## III. SOLUTIONS TO THE LAGRANGE POINT PROBLEM

The analysis is now reduced to a purely mathematical problem of finding the radial and angular coordinates that allow both components of the force to vanish simultaneously. Solutions are discovered analyzing the dimensionless part of Eq. (13) subject to the three cases defined in Sec. II E. For multiple reasons, it is helpful to begin by exploring first-order approximations to Eq. (13) before delving into the search for exact and higher-order solutions. The results of these analyses are summarized in Fig. 4, which shows how the radial coordinates of the Lagrange points vary with *μ*.

### A. First-order approximations to the radial force equation

_{4}and L

_{5}exist, so no approximations for these cases are needed. This section focuses on first-order approximate solutions for the other three Lagrange points. The analysis of the L

_{3}point is the easiest to understand and provides a good starting point for this discussion. The equation for the L

_{3}point is found by setting $ \theta = \pi $ in the terms in square brackets of Eq. (13). Equilibrium at L

_{3}requires

*m*

_{3}to

*m*

_{1}, the second represents the attraction of

*m*

_{3}to

*m*

_{2}, and the last is the inertial term. Because

*m*

_{1}is far more massive than

*m*

_{2}in the limit of small

*μ*and the latter is located further from the tertiary, the motion of

*m*

_{3}is dominated by its interaction with

*m*

_{1}. Therefore, the orbital radius of the tertiary should be similar to that of

*m*

_{2}. That is, the analysis is best done when considering $ r = r 2 + \Delta r$ in dimensional variables, which is equivalent to $ x = 1 + \Delta $ with $ \Delta \u226a 1$ in dimensionless variables. Two effects increase the gravitational force at

*x*= 1 for finite

*μ*compared to the

*μ*= 0 case:

*m*

_{1}is shifted slightly toward

*m*

_{3}, and the gravitational force of

*m*

_{2}on

*m*

_{3}adds to that of

*m*

_{1}on

*m*

_{3}. These effects require an outward shift of the tertiary (positive Δ) to maintain synchronous orbit. Changing variables from

*x*to Δ and conducting a first-order Taylor series expansion for small parameters (

*μ*and Δ) gives

*μ*in Eq. (16) differs from that found in other texts on the subject, such as Refs. 5 and 6, which present the coefficient of Δ as having the value 5/12. This difference is a result of the different normalization schemes used between the analyses, and the above is readily shown to be equivalent to the previously established results.

^{§}

_{1}and L

_{2}points that exist in the neighborhood of

*m*

_{2}, the gravitational force from

*m*

_{2}enters much more strongly as $ \mu / \Delta 2$. The direction of gravitational force from

*m*

_{2}depends on whether

*x*> 1 (L

_{2}) or

*x*< 1 (L

_{1}). In this and the following analyses, the upper sign will be associated with the L

_{2}branch and the lower sign with the L

_{1}branch. The specific form of the dimensionless part of Eq. (13) evaluated with

*θ*= 0 is

_{3}solution. A first-order Taylor series expansion of the first and third terms yields

The Sun–Earth system has $ ( m E / 3 m S ) 1 / 3 \u2248 1 / 100$ to an accuracy of better than one part in a thousand. It follows that the distance of the L_{1} and L_{2} points from Earth is very close to $ 1 / 100 th$ of the Earth–Sun distance (about 150 × 10^{6} km), yielding the 1.5 × 10^{6} km value that was cited in the introduction. At the Sun–Earth L_{2} point, the angle subtended by the Sun is approximately 9.3 mrad, whereas that of the Earth is approximately 8.5 mrad. It follows that the visible fraction of the Sun's area at L_{2} is about $ 1 \u2212 ( 8.5 / 9.3 ) 2$, or roughly 20% of the total, though the actual fraction of solar radiation received there is somewhat less than this due to solar limb darkening.^{22}

### B. Numerical analysis

An alternative to symbolic analysis is the discovery of solutions through a search procedure that calculates a finite table of solutions. The results of such a numerical search are presented as the solid color lines in Fig. 4. Many different numerical methods and tools can be used for such purposes. Two possible approaches are described here. Figure 4 was generated using a simple Python script with a nested loop structure. The first loop evolved *μ* through a specified set of values, and a secondary loop then scanned *x* values for each value of *μ* to find the roots of the relevant polynomials (presented subsequently) for the L_{1}, L_{2}, and L_{3} Lagrange points. A simple and convenient way to explore the solution space can be done using graphing software, like Desmos (https://www.desmos.com/), with a first equation that specifies the range of *μ* values to be explored with a second equation defining the relevant polynomial. The roots can then be inspected by hand and a graph constructed from a table of solutions. Other tools, such as root-finding functions that are common in many programming languages, can be used to the same effect.

### C. Higher-order approximations and exact solutions

The following subsections present higher-order, quasi-analytic approximations for the L_{1}, L_{2}, and L_{3} points, and derive exact solutions for the L_{4} and L_{5} points.

#### 1. Case A: θ = 0

This case considers Lagrange points L_{1} and L_{2} that exist along the axis extending from *m*_{1} through *m*_{2}. The search for analytic solutions proceeds best when using an alternate form of Eq. (13) that seeks zeros of the numerator after a common denominator has been extracted. The numerator becomes a fifth-order polynomial, which is described as $ f ( x ; \mu ) = \u2211 n = 0 n = 5 f n x n$, where the coefficients are given in Table I. Examples of the two forms of $ f ( x ; \mu )$, calculated for the specific case of $ \mu = 0.5$, are presented in Fig. 5. A single real-valued and positive root exists for each branch. Since all of the denominators of Eq. (13) are squared, and therefore positive, the sign of the effective radial force has been preserved in the transformations leading to $ f ( x ; \mu )$. This means that the sign of the curves in Fig. 5 can be interpreted as the direction of the net radial force. This also illustrates the unstable nature of these equilibria since a positive displacement will lead to a positive force in both cases.

*x*to Δ using $ x = 1 + \Delta $ results in the cancellation of many terms in

*f*, yielding the simplified form

*μ*, a logical way to order this equation would be to group terms by powers of

*μ*and Δ and then ignore second-order and higher terms. A naive approach to generating solutions is to begin by keeping only the lowest order terms by ignoring the $ \Delta 3$ term and everything multiplying it in Eq. (20). The only way to balance the equation in this limit is by requiring $ \Delta \u2248 \u2212 ( 1 + \mu )$. However, this is incorrect since it does not provide positive solutions (required for L

_{2}), does not allow Δ to go to zero as

*μ*vanishes, and does not reproduce anything like the approximate form found in the prior section. This approach fails because the approximation scheme is inconsistent if Δ is proportional to a fractional power of

*μ*since it would then be improper to ignore higher powers of Δ as these can be of the same or lower order as

*μ*. The only way to resolve this inconsistency is to balance the lowest order components of the first term in Eq. (20), being $ \u2213 \mu $, with the lowest order component of the second term with square brackets, $ 3 \Delta 3$. Doing so yields the approximate equation $ \u2213 \mu + 3 \Delta 3 \u2248 0$, which gives $ \Delta \u2248 \xb1 ( \mu / 3 ) 1 / 3$, in agreement with the conclusions based on the physical reasoning described in Sec. III A. This illustrates the importance of beginning with simpler solutions that can help put the full analysis on the right path (Figs. 6 and 7).

The exact values of the coefficients of the final terms are $ \u2212 220 / 81$ and $ + 212 / 81$, respectively. These terms have been modified from their actual values to better match the solutions calculated through numerical analysis. These quasi-analytic forms work well over the entire range of *μ* and have average deviations, measured over the range $ 0 \u2264 \mu \u2264 1$, from the numerical solutions of less than $ 10 \u2212 2$. Exact expansions out to sixth-order can be found in Appendix B and derivations of these terms are presented in the supplementary material.

This pair of solutions possess interesting symmetry properties. The magnitudes of the coefficients of *x*_{1} and *x*_{2} are symmetric in the first three terms and then deviate starting at the fourth term, which begins with the introduction of the $ \mu 2$ term at that order in the expansion. In this light, the solutions to Eq. (20) possess qualities that may inspire thoughts of explicit symmetry breaking that is frequently encountered in quantum mechanics, as in Zeeman splitting.^{23}

#### 2. Case B: $ \theta = \pi $

Similar to the prior case, there are two sub-cases for $ \theta = \pi $, depending on whether the radial location of *m*_{3} is greater than or less than *r*_{1}, which are illustrated in Fig. 6 for the case of μ = 0.5. However, given that the linear approximations of Sec. III A established that there is only one viable branch for $ \theta = \pi $, only the $ x > \mu $ branch is analyzed here. A full analysis of this case, including both branches, can be found in the supplementary material. Proceeding as for the prior case by changing variables from *x* to Δ, the fifth-order polynomial for the L_{3} point is $ g ( \Delta ; \mu ) = \u2211 n = 0 n = 5 g n \Delta n$ whose coefficients of are given in Table II.

_{1}and L

_{2}that required expansion in fractional powers of

*μ*, the solution for the L

_{3}point is solved with integer powers of

*μ*. A quasi-analytic approximation for

*x*

_{3}with average deviation from the numerical solutions of less than $ 10 \u2212 3$ is

#### 3. Case C: $ r 1 \u2032 = r 2 \u2032$

The third case is defined by the condition that the tertiary be equidistant from the primaries and gives rise to the L_{4} and L_{5} points. By itself, this condition requires that the triangle formed by the locations of the masses be isosceles, with the tertiary at the point of symmetry. As will be shown, the additional constraint imposed by the requirement that the radial derivative of the effective potential vanish leads to the conclusion that the locations of the masses form an equilateral triangle for all values of *μ*.

*x*= 0 and the value of

*x*that causes the quantity in square brackets to vanish. The former is the limiting value of the L

_{1}branch when

*μ*= 1. The latter provides the solution for the radius of the L

_{4}and L

_{5}Lagrange points, which is given by

Figure 7 presents Eq. (25) with μ = 0.5. Equation (26) admits two solutions, which produce the well-known angular positions of $ \xb1 \pi / 3$ ( $ \xb1 60 \xb0$) in the limit of *μ* = 0. Because *θ* is measured from the center of mass location, it is not obvious what geometry between the masses is implied by this solution for other values of *μ*. This is readily resolved by observing that the distance between the primaries is $ 1 + \mu $ and, as established previously, the other two sides have lengths of $ ( x 2 + \mu ) 1 / 2 = ( 1 + 2 \mu + \mu 2 ) 1 / 2 = 1 + \mu $, using Eq. (25). Therefore, the shape formed by the locations of the masses is always an equilateral triangle.

## IV. DISCUSSION

The foregoing analysis has shown how the positions of all Lagrange points can be calculated for arbitrary mass ratios. A relatively simple image of the solutions emerges when one considers the mass ratio as describing a fixed *m*_{1}, such as a star, with the mass of the second primary increasing from something infinitesimal in comparison (e.g., a small planet) to that of an equal mass binary star system. In this representation, all points except for the L_{1} point move outward with increasing *μ*. The solutions presented here stand in contrast to other analyses which, using a different normalization scheme, generate solutions that are non-monotonic or decreasing functions of the mass ratio and whose interpretations require more care.

One of the main contributions of this work has been to clarify the meaning of the curious form of the effective potential described in Eq. (6), which is often used in discussions of the Lagrange points, either directly or indirectly through contour plots. While this form gives the correct equilibrium points upon taking a derivative with respect to *r*, it is a form that is rather devoid of physical meaning since it requires that the angular velocity be treated as a constant of the motion. This cannot be true since departures from stationary points result in epicyclic motion with variable angular velocity even in two-body systems.^{16} Rather than being mere convenience, it seems likely that the use of this unphysical form of the effective potential stems from the fact that contour plots of Eq. (6) conveniently illustrate the correct locations of the Lagrange points, in contrast to Eq. (7), which does not reveal the Lagrange points. This is because, fundamentally, the analysis of the Lagrange points is a force analysis, which means that illustrations of the solution space require a function that represents the spatial structure of the forces. While Eq. (7) is the correct starting place for this analysis, it does not lend itself to visualization of the solution space since the centrifugal term changes sign after taking the radial derivative of Eq. (7), which must be followed by a change of variables from *L* to $ m 3 r 2 \omega 0$. Conversely, Eq. (6) preserves the overall structure of the forces and lends itself to visual representation. A comparison of contour plots generated in Eqs. (6) and (7) is presented in Sec. V of the supplementary material notes.

One might object that, despite being the more accurate physical representation, the loss of the ability to visualize the problem when using Eq. (7) is a loss of intuition and that Eq. (6) is justified on these grounds. That would be a good argument in favor of continued use of Eq. (6) if it were not for a third path. It is possible to maintain rigor by using Eq. (7) as a starting point for the analysis and presenting a visual illustration of the solution space through a contour plot of the norm of the total force, as is done in Fig. 8. The norm of the total force is a quantity that has minima (zeros) at the Lagrange points and therefore similarly motivates this analysis without any of the misleading aspects of Eq. (6). Of course, all such forms should be taken with a grain of salt since they lose their physical meaning outside of a small neighborhood around each of the Lagrange points where the angular momentum cannot be treated as a constant of the motion.

## V. CONCLUSIONS

The primary goal of this work is a thorough analysis of the restricted three-body problem using the natural geometry described by polar coordinates, culminating in formulas that describe the locations of all Lagrange points. This problem was originally solved as special cases of the more general problem of harmonic motion in the three-body problem, first by Leonhard Euler in 1767 for the L_{1}, L_{2}, and L_{3} points,^{24,25} and subsequently by Joseph Louis Lagrange in 1772 for the L_{4} and L_{5} points.^{26} It is humbling to revisit the studies of the past and see how much was done with so little, and it is remarkable that this problem, now over 250 years old, continues to provide fertile ground for new insights.

A number of interesting puzzles and mathematical curiosities were encountered throughout this work, among them an application of Galois theory of polynomials, symmetry and symmetry-breaking, and the role of normalization in shaping the solutions. A new contribution of this work is a set of quasi-analytic approximations for the radial locations of the L_{1}, L_{2}, and L_{3} points that describe the solution space to high accuracy for all values of *μ*. Perhaps the most important aspect of this work is the clarification regarding the nature of angular momentum. The proper formulation of the problem requires that angular momentum be treated as an invariant near the equilibrium points even though it is not a locally conserved quantity. This observation naturally leads to the recognition that a third kind of conservation law, described here as regional conservation, is needed. Students of general relativity, in particular, may benefit from thinking of this aspect of the Lagrange point problem as they ponder the nature of energy conservation on local and cosmic scales.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### APPENDIX A: LAGRANGIAN ANALYSIS

*z*coordinate and taking

*θ*to be measured in the rotating frame of the

*m*

_{1}-

*m*

_{2}system, the Lagrangian and its associated Euler equations for

*θ*and

*r*are

### APPENDIX B: HIGHER-ORDER EXPANSIONS

*μ*can be carried out, though with progressively more effort required for each additional term due to the nonlinear nature of the generating equations for the series expansions. Further details of this aspect of the analysis are provided in the supplementary material that accompany this text. The sixth-order expansions for the L

_{1}, L

_{2}, and L

_{3}points are

Equilibrium points of a dynamical system are sometimes referred to as stationary points. Less commonly, these particular equilibrium points are known as libration points. Some sources distinguish the points by geometry, with the first three being the colinear libration points and latter two being the triangular or equilateral libration points. Other sources (e.g., Ref. 8) refer to the first three equilibrium points as the Euler points, with L_{4} and L_{5} described as the Lagrange points. The nomenclature used here follows that of many other texts and refers to all equilibrium points of the restricted three-body problem as Lagrange points.

Though the smaller of two massive objects is sometimes referred to as the secondary, the standard of the literature is to refer to both *m*_{1} and *m*_{2} as primaries.

Given this, one might object that the potential is ill-defined because the angular momentum will not, in general, be conserved when integrating from infinity to *r* as is often done when defining potentials. This issue is resolved by taking the reference point to be the Lagrange point itself and adding a constant to Eq. (7), which leaves the subsequent analysis unchanged.

This is easily proven by replacing *μ* with $ \mu * / ( 1 \u2212 \mu * )$ and then calculating a new $ x 3 *$ by scaling *x*_{3} by $ r 2 / ( r 2 + r 1 ) = 1 / ( 1 + \mu ) = 1 \u2212 \mu *$, which gives $ x 3 * = 1 + ( 5 / 12 ) \mu *$.

## REFERENCES

*Theory of Orbits: The Restricted Problem of Three Bodies*

*The Geometry of Celestial Mechanics*

*Orbital Mechanics for Engineering Students*

_{4}and L

_{5}points of the restricted three-body problem

*Analytical Mechanics*

*Classical Mechanics*

*An Introduction to Mechanics*

*Galactic Dynamics*

*Gravitation*

*General Relativity*