Velocity, Acceleration and Equations of Motion in the Elliptical Coordinate System

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1 Aailable online at Archies of Physics Research, 018, 9 (): ( ISSN CODEN (USA): APRRC7 Velocity, Acceleration and Equations of Motion in the Elliptical Coordinate System Popoola Abduljelili 1 * and Ogunwale Bisi Bernard 1 Department of Physics, Osun State Uniersity, Osogbo, Nigeria Department of Physics, Uniersity of Ibadan, Nigeria *Corresponding Author: Popoola Abduljelili, Department of Physics, Osun State Uniersity, Osogbo, Nigeria, Popoolaabduljeleel000@gmail.com ABSTRACT The canonical coordinate systems (rectangular, polar and spherical) are sometimes not the best for studying the trajectories of some forms of motions. For example, motion of objects in an elliptical orbit being described by polar or spherical coordinates may not be accurate. It is due to this that we hae deried the position ectors, elocity ectors, acceleration ectors, simple representation of magnitude of the elocity and equations of motion in the elliptical coordinate system. An attempt was also made towards soling the deried equations of motion. The general algorithm for conersion among coordinate systems was also proided. Keywords: Elliptical coordinate system, Canonical coordinate systems, Equations of motion, Position, Velocity and acceleration INTRODUCTION The position, the instantaneous elocity and acceleration of objects are often studied in classical mechanics using rectangular, polar or spherical coordinate system. It is howeer obious that these three coordinate system are far too small for the description of all trajectory of particles [1,]. That is, there are some motion of bodies that this coordinate system cannot fully describe e.g. planetary motion of planets about the sun as the focus, elliptical motion of comets about the sun and many other arbitrary trajectories of bodies that are somehow beyond that which can be described by rectangular, polar, cylindrical and spherical coordinate system. Also, though not a classical problem, the motion of the electron about the nucleus is not also fully circular and neither follows closely any of the mentioned usual coordinate systems. Due to this reason, many types of coordinate system hae been formulated to describe the trajectories of moing bodies. Other than those already mentioned, the rest are parabolic cylindrical, oblate spheroidal, elliptical cylindrical, elliptical, paraboloidal, prolate spheroidal, bipolar, toroidal, conical, confocal ellipsoidal, confocal paraboloidal, etc [3,4]. Efforts are being made by researchers in calculating the properties such as position, elocity, acceleration, diergence, gradients, curl, etc in these new coordinate systems. These deriations are necessary and sufficient for expressing all mechanical quantities (linear momentum, kinetic energy, Lagrangian and Hamiltonian) in terms of the new coordinate system coordinates system. The results also pae way for expressing all dynamical laws of motion (Newton s laws, Lagrange s law, Hamiltonian s law, Einstein s Special Relatiistic Law of Motion and Schrödinger Law of Quantum Mechanics) entirely in the new coordinate coordinates system [5]. Classically, description of motion of bodies along an arbitrary trajectory is of paramount importance. This is because; many other characteristic of the body can be depicted from its equations of motion. Hence, determining the positions, elocities and accelerations in the arious coordinate system interest researchers. Many researchers hae work on this areas and their works hae been on the parabolic coordinate system (Omonile, et al.,) elliptical cylindrical coordinate system (Omaghali, et al.), prolate spheroidal coordinate system (Omonile, et al.), rotational spheroidal coordinate system (Omonile, et al,) [1,,5,6]. 10

2 Popoola, et al., Archies of Physics Research, 018, 9():10-16 Hence, in this work, we shall work on the deriations of position, elocity and acceleration in elliptical coordinate system. This coordinate system was chosen specifically due to its important applicability in description of motion of planets, comets and asteroids about the sun in the solar system. To ease paths for other researchers, we shall also gie an algorithm for deriing the position, elocity and accelerations in different coordinate systems. We hope to use the elocity deried in this coordinate system to derie the equations of motions of objects under a central force potential by employing the Euler-Langrange relations. We then make comparisons between the obtained equations of motions in different coordinate systems (Cartesian, polar and elliptical coordinate system). MATHEMATICAL FORMULATIONS The most common definition of elliptical coordinate system (u, ) is x acosh usin ; y asinh usin ; [1]... 1 Where u is a non-negatie real number and ϵ [0,π] The elliptical coordinates unit ectors are expressed in terms of the cartesian units ectors as [1] sinhu cos i+ cosh usin j u (coshusin i+ sinhucos j) ( sinh u+ sin ) 1/ ( sinh u+ sin ) 1/ We can express () as: sinhu cos cosh usin 1/ 1/ u ( sinh u+ sin ) ( sinh u+ sin ) i cosh usin sinhu cos j 1/ 1/ ( sinh u+ sin ) ( sinh u+ sin ) After soling equation (3) for i and j, the alues of cartesian unit ectors are:... 3 ( sinh u sin ) 1/ + i sinhucos ucoshusin cosh usin sinh ucos 4 ( sinh u sin ) 1/ + j coshusin u+ sinhucos cosh usin sinh ucos The position ector is gien by r xi+ y j Substituting (1), (4) and (5) in to (6), we get: ( sinh sin ) 1/ a u+ r coshusinhu ( sin cos u ) sincos ( cosh usinh u ) cosh usin sinh ucos 5 6 (7) The equation aboe gies the position in the elliptical coordinate system The elocity is gien by xi+ y j 8 Before we deried the expression for the elocity, let us state the following substitution we shall use to simplify the results 11

3 Popoola, et al., Archies of Physics Research, 018, 9():10-16 ( sinh + sin ) 1/ a u Qu (, ) cosh usin sinh ucos By substituting the first deriaties of equation (1) into (8), we get: Qu (, ) uu+ Where; ( cosh sin sinh sin cos ) coshusinhu ( sincos cos ) u u u + ucoshusinhu sin cos sin + sinh ucos cosh sin cos Equation (9) is the elocity ector equation in the elliptical coordinate system [7,8] In the application of Euler-Lagrange equation, the magnitude of elocity is howeer needed and it is gien by x + y 1 After simplification of form obtained from (1), we get; 1 ( cosh usin ) u + ( sin + sinh u) + ucosh usin 13 The acceleration can also be obtained from a xi+ y j 14 By substituting the second deriatie of equation (1), (4) and (5) into (14), we get the acceleration as a Qu (, ) au u a 15 We shall make use of the following substitution to write the acceleration in a concise form A 1 ( u, ) cosh u sin sinh u sin cos 16 A ( u, ) sinh u coshu ( sin cos cos ) 17 A 3 ( u, ) sinh u coshu ( sin sin cos ) 18 A 4 ( u, ) cosh u cos sin sinh u cos 19 Hence, the acceleration in the elliptical coordinate system is gien by a Q( u, ) A1u+ A + A3 u + A3u A3u+ A4 + A1 u + Au 0 ALGORITHM FOR CONVERSION AMONG THE VARIOUS COORDINATE SYSTEMS To help researchers interested in moing from one coordinate system to another, we present general steps for conersion between the arious coordinate system. The methods works for any coordinate system proided one has been fed with or has obtained the representation of the interested coordinate system in terms of the Cartesian coordinate system. 1

4 Popoola, et al., Archies of Physics Research, 018, 9(): Express the coordinates of the new coordinate system in terms of the Cartesian coordinate system. Express the unit ectors of the new coordinates system in terms of the unit ectors in the Cartesian coordinate system 3. Perform the inersion of equation in step (). This is done to express the Cartesian unit ectors in terms of the new coordinate system unit ectors. This can be done mostly by soling equation in the step () the resulting n n matrix The position in the new coordinate system can therefore be gien by r xi+ y j+ zk. Where all the dependent ariables hae been determined from step (1), () and (3). Hence, elocity, acceleration, the Lagrangian and Hamiltonian in the new coordinate system can be determined once the position is known. THE EQUATIONS OF MOTION OF OBJECTS IN AN ELLIPTICAL ORBIT The kinetic energy in the elliptical coordinate system is gien by 1 1 T m( cosh usin ) u + ( sin + sinh u) + ucosh usin 1 The potential energy is gien as shown below if the distance between the sun of mass M and any object of mass m orbiting the elliptical path with the sun as focus is gien by: r x + y Where; Hence, r x y + which is equialent to ( ) 1 r a u GMm 1 V cos + sinh u a Hence, the lagrangian in this coordinate system is gien by 1 1 L m( cosh usin ) u + ( sin + sinh u) + ucosh usin + GMm 1 cos + sinh u a Applying the Euler-Lagrange equation: cos + sinh in the elliptical coordinate system. 3 4 d L L dt u u d L L dt 5 6 The equations of motion of the particle in the elliptical orbit can be written as: cosh u usin + sin u sin sinh u sinh u + cosh u sinh ucos 1 GM sinh u + usin sinh ucosh u 3/ a cos + sinh u ( ) 7 13

5 Popoola, et al., Archies of Physics Research, 018, 9():10-16 u 1 3 ( sin + sinh u) + cosh usin u sin cosh usinh u sin 4 GM sin + u( cosh ucos sinh u) + 3/ a cos sinh u ( + ) 8 Equations (7) and (8) gie the equation of objects moing in a purely elliptical orbit about a larger mass M. Equations (7) and (8) can also be written for the equation of motion of electron of mass m e about the nucleus of mass M n as: cosh u usin + sin u sin sinh u sinh u + cosh u sinh ucos 1 keqsinh u + usin sinh ucosh u 3/ am cos + sinh u e ( ) 9 cosh u usin + sin u sin sinh u sinh u + cosh u sinh ucos 1 keqsinh + usin sinh u cosh u + 3/ am cos + sinh u e ( ) 30 Hence, equations (7) and (8) can be written as the equation of motion for any object of mass m in an elliptical orbit around a larger object of mass M by changing just the potential energy. Attempted Solution of 7, 8 Equations 7 and 8 are ery difficult to sole analytically. The best one can do is to try to simplify or sole them numerically. Equations 7 and 8 are coupled non-linear partial differential equations. We can simplify the coupled equations by making use of the following substitutions 31 and 3. The result being the elimination of time-dependency from equations 7 and 8. du u d 31 du u d Let there be a constraint such that the rate of change of angular displacement kis constant. 3 That is, k and 0 33 Substituting 31, 3 and 33 into 7, we get: d u du tanh u du 6 ( 4 tanh u) 4 cot + d d sin d cosh u GM 3/ ( cos + sinh u) 0 ak cosh u Substituting 31, 3 and 33 into 8, we get:

6 Popoola, et al., Archies of Physics Research, 018, 9():10-16 d u sinh u du du sinh u cos tanh u ( cot cos ec ) d cosh u d d sin sin GM tanh u 3/ + ( cos + sinh u) 0 ak sin Equations 34 minus 35 gie sinh u du 4 tanh u 1 du 3tanh u+ + cot 4cot + cosh u d sin sin d 6 sinh u cos GM 3/ 4 tanh u ( cos + sinh u) 0 cosh u sin sin ak cosh u sin Equation 36 is a non-linear ordinary differential equation which ordinarily is ery easy to sole analytically. Howeer, it has lots of transcendental and hyperbolic coefficients that will make its solution almost impossible either numerically or analytically. This difficulty in soling the equation of motion obtained probably explains why the literature rather choose the polar coordinate for the representation of objects naturally in the elliptical orbit, though using the polar coordinate for the description may not be as accurate using the elliptical coordinate system. Since we hae not been able to sole the resulting equations of motion 7 and 8, it is important to compare with the equations of motion in some other known coordinate systems. The equations of motion of object of mass m under a central force potential in the Cartesian coordinate system is gien by [11] as: d x x dt GM x y ( + ) 3/ 37 d y y dt GM x y ( + ) 3/ [9] 38 By keen inspection of 7, 8 and 37, 38, one notices some similarities but the most important of the similarities is the fact that the equations in both coordinate system cannot be soled analytically because they are both non-linear. Howeer, 37 and 38 can be linearized by conersion to polar coordinates using the transformation: x rcosθ and y rsinθ and obtain: d r dθ GM r dt dt r 39 d r dr dθ r dt dt dt + 0 [9] 40 Equations 39 and 40 describe the trajectory of a planet in polar coordinates. Howeer, these equations are still nonlinear. To linearize further, we employ the conseration of angular momentum for bodies under the influence of mr θ L d r L 1 GM 3 3 dt m r r 41 central force. That is, 15

7 Popoola, et al., Archies of Physics Research, 018, 9():10-16 u 1 r [11] 4 Equations 41 and 4 are the popular orbit equation that we are familiar with. 41 is still nonlinear while 4 is the representation of the angular momentum conseration. So, we make use of substitution u 1 r to further linearize equation 41 and we get: du L + u GM dθ m Equation 43 is now a linear differential equation which can be soled analytically. Howeer, in our situation of the elliptical coordinated system, it takes more than just conseration of angular momentum to linearize the differential equations (7 and 8). While the conseration theorem linearizes the equations with respect to, the equations still remains non-linear with respect to u. Another alternatie to linearizing 7 and 8 is to transform the coordinates to polar representation where momentum conseration is enough for linearization. Otherwise, the differential equations (7 and 8) remain non-linear. This is also the case in the Cartesian coordinate system as depicted in equations 37 and 38 where angular momentum is not also enough for linearization. Hence, 37 and 38 cannot also be soled analytically. Only numerical solution is possible for 37 and 38. Proided, non-linearity is the only hindrance to soling the differential equations (7 and 8), we would hae employed a numerical solution straight forwardly. Howeer, the coefficients of the differentials are combinations of hyperbolic and transcendental functions. These make the differential equations highly difficult to sole, een numerically. CONCLUSION So far in this paper, we hae highlighted the algorithms for conerting the parameters of a coordinate system to another. With this, we deried the ectors for the position, elocity and acceleration in the elliptical coordinate system. We also deried a simple way of expressing the magnitude of the elocity in the coordinate system [10,11]. Oerall, we deried the equations of motion of objects in the elliptical orbit for objects under an inerse-square, central, attractie potential. Howeer, the resulting equation remains non-linear despite some attempts to remoe the nonlinearity. Moreoer, the presence of hyperbolic and transcendental coefficients introduce more difficulty in soling the equations of motion in the elliptical coordinate system. Hence, though the natural orbits of planets is elliptical, the description of the trajectories of the planets is possible only in polar coordinate system, at least analytically. REFERENCES [1] Omaghali, N.E.J., et al., 016. Velocity and acceleration in elliptic cylindrical coordinates. Archies of Applied Science Research, 8(3), pp [] Omonile, J.F., et al., 014. Velocity and acceleration in prolate spheroidal coordinates. Archies of Physics Research, 5(1), pp [3] Arfken, G., Mathematical Methods for Physicists, Academic Press, Orlando. [4] Morse, P.M., et al., Methods of Theoretical Physics. McGraw-Hill, New York. [5] Omonile, J.F., et al., 015. Velocity and acceleration in rotational prolate spheroidal coordinates. International Journal of Basic and Applied Sciences, 4(4), pp [6] Omonile, J.F., et al., 015. Velocity and acceleration in parabolic cylindrical coordinates. Pelagia Research Library Adances in Applied Science Research, 6(5), pp [7] Howusu, S.X.K., 003.Vector and tensor analysis, Jos Uniersity Press Ltd, Jos, pp [8] Howusu, S.X.K., 003. Riemannian reolutions in physics and mathematics. Jos Uniersity Press Ltd., Jos. [9] Papadatos, K., 016. The equations of planetary motion and their solution. The general Science Journal, May 4. [10] Spiegel, M.R., Theory and problems of ector analysis and introduction to tensor analysis. MC Graw-Hill, New York. [11] Hilderbrand, F.B., Adanced calculus for applications, Prentice Hall, Englewood-Cliff, pp