5 Conclusion This result would in fact be completely unexpected and surprising were it not for the discovery of actual instances of astrological bodies at these points in our Solar system.Indeed,by looking at the second partial derivatives in (4.2)we find that U=U=-2<0 and U=U =-2<0. This would indicate that L4 and L5 are at peaks(local maxima)of the potential in the z-y plane and would thus imply that that these points are extremely unstable.What gives L4 and Ls their stability is simply the coriolis force discussed previously.Initially,a body at L4 and Ls start moving away from the equilibrium point but,as the body pick up speed,the coriolis force takes effect sending the body into an effective orbit around the Lagrange point. Because of this effect,the areas around L4 and Ls that are effectively stable are in fact quite large as is illustrated by Figure 1.Indeed,this is the reason that so many trojan asteroids exist, some more than 5 off of 60,where the L4 and L5 are located.No satellites have been placed at these locations (unlike LI and L2 hosting the SOHO satellite [7]and the WMAP satellite [6] respectively despite their inherent instability),however,they were visited in 2009 by the STEREO satellites.As the only stable Lagrange points,L4 and L5 are unique phenomena in the solar system and the areas around them have been and are of great interest(for example as possible places of origin of the moon or locations from which to better observe solar storms [5])to the astrophysics community. References [1]E.M.Lifshitz L.D.Landau.Course of Theoretical Physics,volume 1.Elsevier Inc.,3rd edition, 2011. [2]Joseph Lagrange.Le Probleme des trois corps.1772. [3]Richard Montgomery.A new solution to the three-body problem.Technical report,Notices Amer.Math Soc,2001. [4]Robert L.Devaney Morris W.Hirsch,Stephen Smale.Differential Equations,Dynamical Sys- tems,and an Introduction to Chaos.Number 978-0-12-382010-5.Elsevier Inc.,3rd edition, 2013. [5]NASA.Stereo.http://www.nasa.gov/mission_pages/stereo/news/gravity_parking. html. [6]NASA.Wilkinson microwave anisotropy probe.http://map.gsfc.nasa.gov/. [7]NASA.Solar and heliospheric observatory.http://sohowww.nascom.nasa.gov/home.html, January 2014. [8 Seth B.Nicholson.The trojan asteroids.Leaflet of the Astronomical Society of the Pacific, 8:239.1961. [9]Milovan Dmitrasinovic.V.Suvakov.Three classes of newtonian three-body planar periodic orbits.Phys.Rev.Lett.,110:114301,Mar 2013. 95 Conclusion This result would in fact be completely unexpected and surprising were it not for the discovery of actual instances of astrological bodies at these points in our Solar system. Indeed, by looking at the second partial derivatives in (4.2) we find that Uxx = U 0 xx = − 3 4Ω 2 < 0 and Uyy = U 0 yy = − 9 4Ω 2 < 0. This would indicate that L4 and L5 are at peaks (local maxima) of the potential in the x-y plane and would thus imply that that these points are extremely unstable. What gives L4 and L5 their stability is simply the coriolis force discussed previously. Initially, a body at L4 and L5 start moving away from the equilibrium point but, as the body pick up speed, the coriolis force takes effect sending the body into an effective orbit around the Lagrange point. Because of this effect, the areas around L4 and L5 that are effectively stable are in fact quite large as is illustrated by Figure 1. Indeed, this is the reason that so many trojan asteroids exist, some more than 5◦ off of 60◦ , where the L4 and L5 are located. No satellites have been placed at these locations (unlike L1 and L2 hosting the SOHO satellite [7] and the WMAP satellite [6] respectively despite their inherent instability), however, they were visited in 2009 by the STEREO satellites. As the only stable Lagrange points, L4 and L5 are unique phenomena in the solar system and the areas around them have been and are of great interest (for example as possible places of origin of the moon or locations from which to better observe solar storms [5]) to the astrophysics community. References [1] E. M. Lifshitz L. D. Landau. Course of Theoretical Physics, volume 1. Elsevier Inc., 3rd edition, 2011. [2] Joseph Lagrange. Le Probl`eme des trois corps. 1772. [3] Richard Montgomery. A new solution to the three-body problem. Technical report, Notices Amer. Math Soc, 2001. [4] Robert L. Devaney Morris W. Hirsch, Stephen Smale. Differential Equations, Dynamical Sys￾tems, and an Introduction to Chaos. Number 978-0-12-382010-5. Elsevier Inc., 3rd edition, 2013. [5] NASA. Stereo. http://www.nasa.gov/mission_pages/stereo/news/gravity_parking. html. [6] NASA. Wilkinson microwave anisotropy probe. http://map.gsfc.nasa.gov/. [7] NASA. Solar and heliospheric observatory. http://sohowww.nascom.nasa.gov/home.html, January 2014. [8] Seth B. Nicholson. The trojan asteroids. Leaflet of the Astronomical Society of the Pacific, 8:239, 1961. [9] Milovan & Dmitra˘sinovi´c. V. Suvakov. Three classes of newtonian three-body planar periodic ˘ orbits. Phys. Rev. Lett., 110:114301, Mar 2013. 9