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Physics LibreTexts

13: Lagrangian Mechanics

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  • Sometimes it is not all that easy to find the equations of motion and there is an alternative approach known as lagrangian mechanics which enables us to find the equations of motion when the newtonian method is proving difficult. In lagrangian mechanics we start, as usual, by drawing a large, clear diagram of the system, using a ruler and a compass. But, rather than drawing the forces and accelerations with red and green arrows, we draw the velocity vectors (including angular velocities) with blue arrows, and, from these we write down the kinetic energy of the system. If the forces are conservative forces (gravity, springs and stretched strings), we write down also the potential energy. That done, the next step is to write down the lagrangian equations of motion for each coordinate. These equations involve the kinetic and potential energies, and are a little bit more involved than \(F=ma\), though they do arrive at the same results.

    Thumbnail: Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f(x,y)=0, the constraint force is C, and the one degree of freedom can be described by one generalized coordinate (here the angle theta). Image used with permission (Public Domain; Maschen).