This essay was written by upper-sixth former Archie Forsyth, and shortlisted for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.
Estimated read time of abstract: 2 minutes
Estimated read time of essay: 20 minutes
In my ILA I explain the premises and applications of a variation of Newtonian mechanics called ‘Lagrangian’ mechanics and then go on to derive a differential equation that describes the motion of a double-sprung pendulum in three dimensions and then obtain its path of motion through coding a fourth order Runge-Kutta numerical solver.
In short, Lagrangian mechanics uses Newton’s three laws of motion and slightly alters their form in order to make them more applicable to more complicated systems, including systems such as pendulums which are best described using polar coordinates. What makes this possible is the application of the Euler-Lagrange (E-L) equation, which involves differentiation, to the Lagrangian of the system. The Lagrange is simply the kinetic energy of the system minus the potential energies and when the E-L equation is applied to it, these energies are converted into an equation for the forces acting on the masses in the system and thus, dividing by the mass, a differential equation that describes the motion of the system.
The next section of my ILA is a brief explanation of what a chaotic system is and that it is not always necessarily a complex system, for instance, take a double pendulum, i.e. a pendulum attached to the base of another pendulum. This is a very simple system however it is extremely chaotic, which is to say that the slightest change in its initial conditions, in this case angles from the equilibrium positions of the pendula, length of pendula and the greatness of the masses, greatly affect the system’s path of motion.
From here the first major section of work begins as I derive a differential equation that describes the motion of first a double pendulum in polar coordinates (in 2-D) and then a double sprung pendulum in Cartesian coordinates (in 3-D). To demonstrate the effectiveness of Lagrangian mechanics, I derived the equation describing the sum of the forces acting on the masses in the double-sprung pendulum using both ordinary Newtonian mechanics, i.e. vector forces, and Lagrangian mechanics to show how much faster and more reliable it is to use the Lagrangian method.
The final part of my ILA is the code that I made to find the path of the double-sprung pendulum’s motion. It is, as described before, a fourth order Runge-Kutta numerical solver, which uses an iterative method based on setting a fixed time difference and calculating the acceleration and velocities at the end of each of those time differences, which can be substituted back into the equations for acceleration to give a new coordinate, repeating this process many times so that you can find the coordinates of the masses after a given time. The code achieves this and allows for the choosing of the initial conditions, which are the spring constants, the unstretched lengths of the springs, the initial coordinates of the masses and the magnitude of the masses attached to the springs.
To view Archie’s full article, follow this link below.