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New Insights on Energy Conserved Planar Motion

Maruthi R. Akella1, Sofokli Cakalli2

Professor and Myron L. Begeman Fellow in Engineering, Department of Aerospace Engineeringand Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, USA. E-mail:
Undergraduate Research Assistant, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, USA. E-mail:

Computer Modeling in Engineering & Sciences 2016, 111(2), 119-127.


The planar motion of a particle within an arbitrary potential field is considered. The particle is additionally subject to an external force wherein the applied thrust-acceleration is constrained to remain normal to the velocity vector. The system is thus non-conservative but since the thrust force is non-working, the total energy is a conserved quantity. Under this setting, a major result of fundamental importance is established in this paper: that the flight direction angle (more precisely, the sine of the angle between the position and velocity vectors) is shown to always satisfy a linear first-order differential equation with variable coefficients that depend upon the underlying potential function. As a consequence, an analytical solution for the flight direction angle can be obtained directly in terms of the particle's distance from the center of the field for a significant number of special cases for the potential function. In the case of J2 perturbed spacecraft motion within equatorial orbits, the problem is reduced to that of solving an incomplete elliptic integral. Another important implication of the main result established here is that motion problems subject to velocity-normal thrusting can always be reduced to the study of equivalent single degree-of-freedom conservative systems with an effective potential function. The paper concludes with various examples of both academic and practical interest including the study of bounded two-body Keplerian orbits and hodograph interpretetions.


Cite This Article

APA Style
Akella, M.R., Cakalli, S. (2016). New insights on energy conserved planar motion. Computer Modeling in Engineering & Sciences, 111(2), 119-127.
Vancouver Style
Akella MR, Cakalli S. New insights on energy conserved planar motion. Comput Model Eng Sci. 2016;111(2):119-127
IEEE Style
M.R. Akella and S. Cakalli, "New Insights on Energy Conserved Planar Motion," Comput. Model. Eng. Sci., vol. 111, no. 2, pp. 119-127. 2016.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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