International Journal of Astronomy and Astrophysics, 2017, 7, 91-111 http://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717 Periodic Orbits of the First Kind in the Autonomous Four-body Problem with the Case of Collision M. R. Hassan 1 , Md. Aminul Hassan 2 , Payal Singh 3 , Vinay Kumar 4 , R. R. Thapa 5 1 Department of Mathematics, S. M. College, T. M. Bhagalpur University, Bhagalpur, India 2 GTE, Bangalore, India 3 Research Scholar, T. M. Bhagalpur University, Bhagalpur, India 4 Department of Mathematics, Zakir Hussain College, University of Delhi, New Delhi, India 5 Department of Mathematics, P. G. Campus, Tribhuvan University, Biratnagar, Nepal Abstract In this manuscript, the existence of periodic orbits of collision of the first kind has been discussed on the model of Autonomous Four-body Problem by the method of analytic continuation given by Giacaglia [1] and Bhatnagar [2] [3]. For the existence of periodic orbits, Duboshin’s criterion [4] has been satisfied and it has been confirmed by analyzing the Poincare surfaces of section (PSS) [5]. Also it has been shown that the case of collision given by Levi-Civita [6] [7] is conserved by the method analytic continuation. In all sections of this manuscript, equilateral triangular configuration given by Ceccaroni and Biggs [8] has been considered. In this model, third primary of inferior mass (in comparison of the other primaries) is placed at the equilibrium point 4 L of the R3BP. Keywords Autonomous Four-Body Problem, Regularization, Periodicity, Poincare Surfaces of Section, Collision Orbit, Zero Velocity Curves 1. Introduction We know that the four most popular methods of proving the existence of peri- odic orbits are: (i) the method of analytic continuation, (ii) the process of equating Fourier coefficients of equal frequencies, (iii) the application of fixed point theorem given by Poincare, How to cite this paper: Hassan, M.R., Hassan, Md. A., Singh, P., Kumar, V. and Thapa, R.R. (2017) Periodic Orbits of the First Kind in the Autonomous Four-body Problem with the Case of Collision. Inter- national Journal of Astronomy and Astro- physics, 7, 91-111. https://doi.org/10.4236/ijaa.2017.72008 Received: April 1, 2017 Accepted: June 9, 2017 Published: June 12, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access DOI: 10.4236/ijaa.2017.72008 June 12, 2017