npsm 새물리 New Physics : Sae Mulli

pISSN 0374-4914 eISSN 2289-0041


Research Paper

New Phys.: Sae Mulli 2020; 70: 1003-1008

Published online November 30, 2020

Copyright © New Physics: Sae Mulli.

An Efficient Way to Calculate Lagrange’s Equations of Motion in Classical Mechanics


Department of Physics, Dankook University, Cheonan 31116, Korea


Received: June 23, 2020; Revised: September 21, 2020; Accepted: September 21, 2020


The equation of motion is essential for understanding and predicting the behavior of a dynamics system. In classical mechanics, the equations of motion are obtained using the Newtonian method and the Lagrange method. Because the Newton method treats the force as a vector quantity, obtaining the equation of motion is difficult when the constraint force is complex. On the other hand, the Lagrange method uses the scalar quantities of kinetic energy and potential energy, making the obtaining the equation of motion relatively easy. In this paper, we examined the physical meaning of the Lagrange equation through a coordinate transformation. As a result, Lagrange’s motion equation in the generalized coordinate can be seen to be a linear transformation of the constraint force in Cartesian coordinate into the transpose of the Jacobian matrix. Furthermore, when the x, y, and z coordinates are explicit functions of the time t, the method proposed in this paper is more efficient than the conventional Lagrange method for obtaining the equation of motion.

Keywords: Classical dynamics, Newton, Lagrange, Jacobian matrix

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