Lagrangian and Hamiltonian Mechanics

The continued exploration of the mathematical underpinnings of Newtonian mechanics, as examined in different physical contexts, led to its reformulation–along with some new mathematics, as well. In the late 1700s, for example, Joseph Louis-Lagrange suggested a new, so-called “Lagrangian” way of framing of mechanics, which produced the same physical results as Newtonian mechanics but was based on the “principle of least action” instead of force. Action is an integral that can be defined for every possible path a particle could take from its starting point to an ending point, given by the equation, S=\int (K-V)\,dt , where K is the kinetic energy of the particle and V is the potential energy of the particle along the path (see Fig. 3 ).