Lagrangian Mechanics: An Alternative to the Traditional Newtonian Formulation
In this article, I will introduce a field in classical mechanics known as Lagrangian mechanics and one application of this formulation.
What is Lagrangian Mechanics?
First introduced by Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics that can act as an alternative to the traditional Newtonian formulation.
Lagrangian mechanics relies on an important principle known as Hamilton’s Principle and is as follows:
The actual path which a particle follows from two points 1 and 2 in a given time interval, t₁ to t₂ is such that the following action integral is stationary when taken along the actual path:
In the above expression, S is known as known as the action and the fancy L is known as the Lagrangian and is defined as the kinetic energy minus the potential energy. q(t), in this case refers to a certain ‘degree of freedom’ that the object is moving along, whether that be the x-axis, y-axis, or any other coordinate system. Also, if you aren’t familiar with the notation, the dot on top of the q indicates a single derivative with respect to time.
