www.ijcrt.org © 2022 IJCRT | Volume 10, Issue 2 February 2022 | ISSN: 2320-2882 IJCRT2202373 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org d171 Geometrical Representation and Analysis of Lissajous Figures using wxMaxima Vijay Jadhav 1 , Aparna Jaware 1 , Harish Kumar Dubey 1 , Dattatray E. Kshirsagar 1* 1 Department of Physics, B. K. Birla College, Kalyan(MS),India . Abstract The undergraduate students find it difficult to solve trigonometric functions used in Maxwell's equation because they lack in mathematical skills. Therefore, the use of mathematical tools and graphs for plotting will make this task easy and interesting. Nowadays, wxMaxima is open-source mathematical software for algebraic equations, calculus operations, and graph plotting used by many students. wxMaxima is used in a physical system to imagine the results of various parameters. It can act as a tool for simulation. We use wxMaxima in this article to solve an example of Lissajous figures and a damped harmonic oscillator. Keywords: Lissajous figures, harmonic motion, wave equations, wxMaxima. 1. Introduction: For electronic communications, the exact calibration must be carried out. The device is used to compare a very exact crystal- controlled oscillator with the oscillator of the transmitter. This is achieved by a Lissajous figure and the perfect circle is obtained [1]. It is very interesting for students to study Lissajous figures, the nature and shape of the figures with their phases. Proving it with complex equations is truly a herculean task. students learn more with illustrations, images, and simulations, i.e. visual effects improve students’ ability to comprehend physics. wxMaxima is free software for simplifying and solving complex equations and a perfect tool for graphics modeling and display. Here we will use this software as an instrument to help students to begin their understanding of physics. A lot of researchers used wxMaxima to solve problems in physics. M Ikhsan and Suhartati et al used wxMaxima for the study of the vectors [2]. D Afriza1 et al used wxMaxima in electrodynamics to solve the Gauss problems [3]. Timberlake, T.K., and J.W. Mixon have solved various problems with wxMaxima through his lecture notes on physics and classical mechanics [4]. SK Velten et al performed simulation and developed a mathematical model using wxMaxima [5]. F. L. Braga, Wesley Spalenza, Kc Lemos Filhot conducted orbital simulations for a celestial mechanical system using the Lagrangean relativistic formalism using wxMaxima [6]. Lissajous figures can be obtained only by solving wave equations. Due to various complex wave equations, diagramming them on board is difficult. With wxMaxima, these equations can be solved and simplified. However, still, it is hard to imagine and understand the plot with one wave equation on X-axis and the other on Y-axis. wxMaxima plots that with certain commands easily. The parameters can also be easily modified in the instruction, and the effect of changing the parameters can be easily displayed on the screen itself. 2. Method: This paper investigated the effectiveness of using wxMaxima, and a set of commands used to display Lissajous figures. The waveform and the Lissajous patterns can be easily illustrated on the computer by adjusting parameters. When two simple harmonic motions of the same frequencies are acting along the X-axis and Y-axis respectively and if their displacements at any time t are x and y, then they can be represented by amplitude and frequency ratios.  = ( 1  + ) ……(1) and  = ( 2  + ) ……(2) where A and B are the amplitudes of the component vibrations having frequencies  1 and  2 the phase of the first vibration is  and the second vibration is . The first vibration is ahead of the second by a phase angle ( − ), i.e.the phase difference between the two vibrations is ( − ). When  1 is equal to  2 , the above equations can be solved using wxMaxima and we can obtain,  2  2 +  2  2 −2   cos( − ) =  2 ( − ) ….. (3) ฀ Corresponding author at: Department of Physics, B. K. Birla College, Kalyan(MS),India Tel.: +91 8652 431 228. E-mail address: kesagar2008@gmail.com (Dattatray E. Kshirsagar).