In celestial mechanics, the Lagrangian points (/ləˈɡrɑːndʒiən/ also Lagrange points,^{[1]} or L-points) are the points near two large bodies in orbit where a smaller object will maintain its position relative to the large orbiting bodies. At other locations, a small object would go into its own orbit around one of the large bodies, but at the Lagrangian points the gravitational forces of the two large bodies, the centripetal force of orbital motion, and (for certain points) the Coriolis acceleration all match up in a way that cause the small object to maintain a stable or nearly stable position relative to the large bodies.
There are five such points, labeled L_{1} to L_{5}, all in the orbital plane of the two large bodies, for each given combination of two orbital bodies. For instance, there are five Lagrangian points L_{1} to L_{5} for the Sun–Earth system, and in a similar way there are five different Lagrangian points for the Earth–Moon system. L_{1}, L_{2}, and L_{3} are on the line through the centers of the two large bodies, while L_{4} and L_{5} each act as the third vertex of an equilateral triangle formed with the centers of the two large bodies. L_{4} and L_{5} are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.
Several planets have trojan satellites near their L_{4} and L_{5} points with respect to the Sun. Jupiter has more than a million of these trojans. Artificial satellites have been placed at L_{1} and L_{2} with respect to the Sun and Earth, and with respect to the Earth and the Moon.^{[2]} The Lagrangian points have been proposed for uses in space exploration.
The three collinear Lagrange points (L_{1}, L_{2}, L_{3}) were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two.^{[3]}^{[4]}
In 1772, Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.^{[5]}
The five Lagrangian points are labeled and defined as follows:
The L_{1} point lies on the line defined by the two large masses M_{1} and M_{2}, and between them. It is the point where the gravitational attraction of M_{2} partially cancels M_{1}'s gravitational attraction.^{[clarification needed]} An object that orbits the Sun more closely than Earth would normally have a shorter orbital period than Earth, but that ignores the effect of Earth's own gravitational pull. If the object is directly between Earth and the Sun, then Earth's gravity counteracts some of the Sun's pull on the object, and therefore increases the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L_{1} point, the orbital period of the object becomes exactly equal to Earth's orbital period. L_{1} is about 1.5 million kilometers from Earth, or 0.01 au, 1/100th the distance to the Sun.^{[6]}
The L_{2} point lies on the line through the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L_{2}. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than that of Earth. The extra pull of Earth's gravity decreases the orbital period of the object, and at the L_{2} point that orbital period becomes equal to Earth's. Like L_{1}, L_{2} is about 1.5 million kilometers or 0.01 au from Earth.
The L_{3} point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the L_{3} point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly further from the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' barycenter, which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little further from the Sun in order to have the same 1-year period. It is at the L_{3} point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit.
The L_{4} and L_{5} points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind (L_{5}) or ahead (L_{4}) of the smaller mass with regard to its orbit around the larger mass.
The triangular points (L_{4} and L_{5}) are stable equilibria, provided that the ratio of M_{1}/M_{2} is greater than 24.96.^{[note 1]}^{[7]} This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, kidney bean-shaped orbit around the point (as seen in the corotating frame of reference).
In contrast to L_{4} and L_{5}, where stable equilibrium exists, the points L_{1}, L_{2}, and L_{3} are positions of unstable equilibrium. Any object orbiting at L_{1}, L_{2}, or L_{3} will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ station keeping in order to maintain their position.
It is common to find objects at or orbiting the L_{4} and L_{5} points of natural orbital systems. These are commonly called "trojans". In the 20th century, asteroids discovered orbiting at the Sun–Jupiter L_{4} and L_{5} points were named after characters from Homer's Iliad. Asteroids at the L_{4} point, which leads Jupiter, are referred to as the "Greek camp", whereas those at the L_{5} point are referred to as the "Trojan camp".
Other examples of natural objects orbiting at Lagrange points:
Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common barycenter, there are five positions in space where a third body, of comparatively negligible mass, could be placed so as to maintain its position relative to the two massive bodies. As seen in a rotating reference frame that matches the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined providing the centripetal force at the Lagrangian points, allowing the smaller third body to be relatively stationary with respect to the first two.^{[15]}
The location of L_{1} is the solution to the following equation, gravitation providing the centripetal force:
where r is the distance of the L_{1} point from the smaller object, R is the distance between the two main objects, and M_{1} and M_{2} are the masses of the large and small object, respectively. Solving this for r involves solving a quintic function, but if the mass of the smaller object (M_{2}) is much smaller than the mass of the larger object (M_{1}) then L_{1} and L_{2} are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:
This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M_{2} in the absence of M_{1}, is that of M_{2} around M_{1}, divided by √3 ≈ 1.73:
The location of L_{2} is the solution to the following equation, gravitation providing the centripetal force:
with parameters defined as for the L_{1} case. Again, if the mass of the smaller object (M_{2}) is much smaller than the mass of the larger object (M_{1}) then L_{2} is at approximately the radius of the Hill sphere, given by:
The location of L_{3} is the solution to the following equation, gravitation providing the centripetal force:
with parameters M_{1,2} and R defined as for the L_{1} and L_{2} cases, and r now indicates the distance of L_{3} from the position of the smaller object, if it were rotated 180 degrees about the larger object. If the mass of the smaller object (M_{2}) is much smaller than the mass of the larger object (M_{1}) then:^{[16]}
The reason these points are in balance is that, at L_{4} and L_{5}, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycenter of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of system. (Indeed, the third body need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work on the three-body problem.
The radial acceleration a of an object in orbit at a point along the line passing through both bodies is given by:
where r is the distance from the large body M_{1} and sgn(x) is the sign function of x. The terms in this function represent respectively: force from M_{1}; force from M_{2}; and centrifugal force. The points L_{3}, L_{1}, L_{2} occur where the acceleration is zero — see chart at right.
Although the L_{1}, L_{2}, and L_{3} points are nominally unstable, there are (unstable) periodic orbits called "halo" orbits around these points in a three-body system. A full n-body dynamical system such as the Solar System does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous-curve trajectories. These quasi-periodic Lissajous orbits are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time. Also, for Sun–Earth-L_{1} missions, it is preferable for the spacecraft to be in a large-amplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit around L_{1} than to stay at L_{1}, because the line between Sun and Earth has increased solar interference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around L_{2} keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
The L_{4} and L_{5} points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25^{[note 1]} times the mass of the secondary body (e.g. the Moon).^{[17]}^{[18]} The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth^{[19]}). Although the L_{4} and L_{5} points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position, Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map)^{[18]} curves the trajectory into a path around (rather than away from) the point.^{[18]}^{[20]}
This table lists sample values of L_{1}, L_{2}, and L_{3} within the solar system. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. Distances are measured from the larger body's center of mass with L_{3} showing a negative location. The percentage columns show how the distances compare to the semimajor axis. E.g. for the Moon, L_{1} is located 326400 km from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% in front of the Moon; L_{2} is located 448900 km from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L_{3} is located −381700 km from Earth's center, which is 99.3% of the Earth–Moon distance or 0.7084% in front of the Moon's 'negative' position.
Body pair | Semimajor axis (SMA) | L_{1} | 1 − L_{1}/SMA (%) | L_{2} | L_{2}/SMA − 1 (%) | L_{3} | 1 + L_{3}/SMA (%) |
---|---|---|---|---|---|---|---|
Earth–Moon | 3.844×10^{8} m | 3.2639×10^{8} m | 15.09 | 4.489×10^{8} m | 16.78 | −3.8168×10^{8} m | 0.7084 |
Sun–Mercury | 5.7909×10^{10} m | 5.7689×10^{10} m | 0.3806 | 5.813×10^{10} m | 0.3815 | −5.7909×10^{10} m | 0.000009683 |
Sun–Venus | 1.0821×10^{11} m | 1.072×10^{11} m | 0.9315 | 1.0922×10^{11} m | 0.9373 | −1.0821×10^{11} m | 0.0001428 |
Sun–Earth | 1.496×10^{11} m | 1.4811×10^{11} m | 0.997 | 1.511×10^{11} m | 1.004 | −1.496×10^{11} m | 0.0001752 |
Sun–Mars | 2.2794×10^{11} m | 2.2686×10^{11} m | 0.4748 | 2.2903×10^{11} m | 0.4763 | −2.2794×10^{11} m | 0.00001882 |
Sun–Jupiter | 7.7834×10^{11} m | 7.2645×10^{11} m | 6.667 | 8.3265×10^{11} m | 6.978 | −7.7791×10^{11} m | 0.05563 |
Sun–Saturn | 1.4267×10^{12} m | 1.3625×10^{12} m | 4.496 | 1.4928×10^{12} m | 4.635 | −1.4264×10^{12} m | 0.01667 |
Sun–Uranus | 2.8707×10^{12} m | 2.8011×10^{12} m | 2.421 | 2.9413×10^{12} m | 2.461 | −2.8706×10^{12} m | 0.002546 |
Sun–Neptune | 4.4984×10^{12} m | 4.3834×10^{12} m | 2.557 | 4.6154×10^{12} m | 2.602 | −4.4983×10^{12} m | 0.003004 |
Sun–Earth L_{1} is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978 International Sun Earth Explorer 3 (ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances.^{[21]} Since June 2015, DSCOVR has orbited the L_{1} point. Conversely it is also useful for space-based solar telescopes, because it provides an uninterrupted view of the Sun and any space weather (including the solar wind and coronal mass ejections) reaches L_{1} a few hours before Earth. Solar telescopes currently located around L_{1} include the Solar and Heliospheric Observatory and Advanced Composition Explorer.
Sun–Earth L_{2} is a good spot for space-based observatories. Because an object around L_{2} will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's umbra,^{[22]} so solar radiation is not completely blocked at L_{2}. Spacecraft generally orbit around L_{2}, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L_{2}, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark-side can allow the telescope to cool passively to around 50 K – this is especially helpful for infrared astronomy and observations of the cosmic microwave background. The James Webb Space Telescope is due to be positioned at L_{2}.
Sun–Earth L_{3} was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Once space-based observation became possible via satellites^{[23]} and probes, it was shown to hold no such object. The Sun–Earth L_{3} is unstable and could not contain a natural object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this L_{3} every 20 months).
A spacecraft orbiting near Sun–Earth L_{3} would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a 7-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth L_{3} would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L_{3} were studied and several designs were considered.^{[24]}
Missions to Lagrangian points generally orbit the points rather than occupy them directly.
Another interesting and useful property of the collinear Lagrangian points and their associated Lissajous orbits is that they serve as "gateways" to control the chaotic trajectories of the Interplanetary Transport Network.
Earth–Moon L_{1} allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a half-way manned space station intended to help transport cargo and personnel to the Moon and back.
Earth–Moon L_{2} has been used for a communications satellite covering the Moon's far side, for example, Queqiao, launched in 2018^{[25]}, and would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.^{[26]}
Scientists at the B612 Foundation were^{[27]} planning to use Venus's L_{3} point to position their planned Sentinel telescope, which aimed to look back towards Earth's orbit and compile a catalogue of near-Earth asteroids.^{[28]}
In 2017, Nasa proposed the idea of positioning a magnetic dipole shield at the Sun–Mars L_{1} point for use as an artificial magnetosphere for Mars.^{[29]} The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.
International Sun Earth Explorer 3 (ISEE-3) began its mission at the Sun–Earth L_{1} before leaving to intercept a comet in 1982. The Sun–Earth L_{1} is also the point to which the Reboot ISEE-3 mission was attempting to return the craft as the first phase of a recovery mission (as of September 25, 2014 all efforts have failed and contact was lost).^{[30]}
Solar and Heliospheric Observatory (SOHO) is stationed in a halo orbit at L_{1}, and the Advanced Composition Explorer (ACE) in a Lissajous orbit. WIND is also at L_{1}.
Deep Space Climate Observatory (DSCOVR), launched on 11 February 2015, began orbiting L_{1} on 8 June 2015 to study the solar wind and its effects on Earth.^{[31]} DSCOVR is unofficially known as GORESAT, because it carries a camera always oriented to Earth and capturing full-frame photos of the planet similar to the Blue Marble. This concept was proposed by then-Vice President of the United States Al Gore in 1998^{[32]} and was a centerpiece in his film An Inconvenient Truth.^{[33]}
LISA Pathfinder (LPF) was launched on 3 December 2015, and arrived at L_{1} on 22 January 2016, where, among other experiments, it tested the technology needed by (e)LISA to detect gravitational waves. LISA Pathfinder used an instrument consisting of two small gold alloy cubes.
Spacecraft at the Sun–Earth L_{2} point are in a Lissajous orbit until decommissioned, when they are sent into a heliocentric graveyard orbit.
Unflown or planned mission | Mission en route or in progress (including mission extensions) | Mission at Lagrangian point completed successfully (or partially successfully) |
Mission | Lagrangian point | Agency | Description |
---|---|---|---|
International Sun–Earth Explorer 3 (ISEE-3) | Sun–Earth L_{1} | NASA | Launched in 1978, it was the first spacecraft to be put into orbit around a libration point, where it operated for four years in a halo orbit about the L_{1} Sun–Earth point. After the original mission ended, it was commanded to leave L_{1} in September 1982 in order to investigate comets and the Sun.^{[39]} Now in a heliocentric orbit, an unsuccessful attempt to return to halo orbit was made in 2014 when it made a flyby of the Earth–Moon system.^{[40]}^{[41]} |
Advanced Composition Explorer (ACE) | Sun–Earth L_{1} | NASA | Launched 1997. Has fuel to orbit near L_{1} until 2024. Operational as of 2019^{[update] }.^{[42]} |
Deep Space Climate Observatory (DSCOVR) | Sun–Earth L_{1} | NASA | Launched on 11 February 2015. Planned successor of the Advanced Composition Explorer (ACE) satellite. In safe mode as of 2019^{[update] }, but is planned to reboot.^{[43]} |
LISA Pathfinder (LPF) | Sun–Earth L_{1} | ESA, NASA | Launched one day behind revised schedule (planned for the 100th anniversary of the publication of Einstein's General Theory of Relativity), on 3 December 2015. Arrived at L_{1} on 22 January 2016.^{[44]} LISA Pathfinder was deactivated on 30 June 2017.^{[45]} |
Solar and Heliospheric Observatory (SOHO) | Sun–Earth L_{1} | ESA, NASA | Orbiting near L_{1} since 1996. Operational as of 2020^{[update] }.^{[46]} |
WIND | Sun–Earth L_{1} | NASA | Arrived at L_{1} in 2004 with fuel for 60 years. Operational as of 2019^{[update] }.^{[47]} |
Wilkinson Microwave Anisotropy Probe (WMAP) | Sun–Earth L_{2} | NASA | Arrived at L_{2} in 2001. Mission ended 2010,^{[48]} then sent to solar orbit outside L_{2}.^{[49]} |
Herschel Space Telescope | Sun–Earth L_{2} | ESA | Arrived at L_{2} July 2009. Ceased operation on 29 April 2013; will be moved to a heliocentric orbit.^{[50]}^{[51]} |
Planck Space Observatory | Sun–Earth L_{2} | ESA | Arrived at L_{2} July 2009. Mission ended on 23 October 2013; Planck has been moved to a heliocentric parking orbit.^{[52]} |
Chang'e 2 | Sun–Earth L_{2} | CNSA | Arrived in August 2011 after completing a lunar mission before departing en route to asteroid 4179 Toutatis in April 2012.^{[37]} |
ARTEMIS mission extension of THEMIS | Earth–Moon L_{1} and L_{2} | NASA | Mission consists of two spacecraft, which were the first spacecraft to reach Earth–Moon Lagrangian points. Both moved through Earth–Moon Lagrangian points, and are now in lunar orbit.^{[53]}^{[54]} |
WIND | Sun–Earth L_{2} | NASA | Arrived at L_{2} in November 2003 and departed April 2004. |
Gaia Space Observatory | Sun–Earth L_{2} | ESA | Launched 19 December 2013.^{[55]} Operational as of 2020^{[update] }.^{[56]} |
Chang'e 5-T1 Service Module | Earth–Moon L_{2} | CNSA | Launched on 23 October 2014, arrived at L_{2} halo orbit on 13 January 2015.^{[38]} |
Queqiao | Earth–Moon L_{2} | CNSA | Launched on 21 May 2018, arrived at L_{2} halo orbit on June 14.^{[57]} |
Spektr-RG | Sun–Earth L_{2} | IKI RAN DLR |
Launched 13 July 2019. Roentgen and Gamma space observatory. En route to L_{2} point. |
Mission | Lagrangian point | Agency | Description |
---|---|---|---|
"Lunar Far-Side Communication Satellites" | Earth–Moon L_{2} | NASA | Proposed in 1968 for communications on the far side of the Moon during the Apollo program, mainly to enable an Apollo landing on the far side—neither the satellites nor the landing were ever realized.^{[58]} |
Space colonization and manufacturing | Earth–Moon L_{4} or L_{5} | — | First proposed in 1974 by Gerard K. O'Neill^{[59]} and subsequently advocated by the L5 Society. |
EQUULEUS | Earth–Moon L_{2} | University of Tokyo, JAXA | 6U CubeSat, launch planned in 2019 as a secondary payload onboard SLS Artemis 1.^{[60]} |
James Webb Space Telescope (JWST) | Sun–Earth L_{2} | NASA, ESA, CSA | As of 2018^{[update] }, launch is planned for 2021.^{[61]} |
Euclid | Sun–Earth L_{2} | ESA, NASA | As of 2013^{[update] }, launch is planned in 2020.^{[62]} |
Aditya-L1 | Sun–Earth L_{1} | ISRO | Launch planned for 2021; it will be going to a point 1.5 million kilometers away from Earth, from where it will observe the Sun constantly and study the solar corona, the region around the Sun's surface.^{[63]} |
Demonstration and Experiment of Space Technology for INterplanetary voYage (DESTINY) |
Earth–Moon L_{2} | JAXA | Candidate for JAXA's next "Competitively-Chosen Medium-Sized Focused Mission", possible launch in the early 2020s.^{[64]} |
Exploration Gateway Platform | Earth–Moon L_{2}^{[65]} | NASA | Proposed in 2011.^{[66]} |
Nancy Grace Roman Space Telescope (WFIRST) | Sun–Earth L_{2} | NASA, USDOE | As of 2013^{[update] }, in a "pre-formulation" phase until at least early 2016; possible launch in the early 2020s.^{[67]} |
LiteBIRD | Sun–Earth L_{2}^{[68]} | JAXA, NASA | As of 2015^{[update] }, one of two finalists for JAXA's next "Strategic Large Mission"; would be launched in 2024 if selected.^{[69]} |
Planetary Transits and Oscillations of stars (PLATO) | Sun–Earth L_{2} | ESA | Planned for launch in 2024 for an initial six-year mission.^{[70]} |
Space Infrared Telescope for Cosmology and Astrophysics (SPICA) |
Sun–Earth L_{2} | JAXA, ESA, SRON | As of 2015^{[update] }, awaiting approval from both Japanese and European side, launch proposed for 2025.^{[71]} |
Advanced Telescope for High Energy Astrophysics (ATHENA) |
Sun–Earth L_{2} | ESA | Launch planned for 2028.^{[72]} |
Spektr-M | Sun–Earth L_{2} | Roscosmos | Possible launch after 2027. ^{[73]} |
L_{2} is in deep space far away from any planetary surface and hence the thermal, micrometeoroid, and atomic oxygen environments are vastly superior to those in LEO. Thermodynamic stasis and extended hardware life are far easier to obtain without these punishing conditions seen in LEO. L_{2} is not just a great gateway—it is a great place to store propellants. ... L_{2} is an ideal location to store propellants and cargos: it is close, high energy, and cold. More importantly, it allows the continuous onward movement of propellants from LEO depots, thus suppressing their size and effectively minimizing the near-Earth boiloff penalties.
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Categories: Trojans (astronomy) | Lagrangian mechanics