Lagrangian and Hamiltonian Mechanics

Lagrange has perhaps done more than any other to give extent and harmony to such deductive researches by showing that the most varied consequences … may be derived from one radical formula, the beauty of the method so suiting the dignity of the results as to make his great work a kind of scientific poem.

                                                                                              W. R. Hamilton

According to Newton's laws, the incremental work dW done by a force f on a particle moving an incremental distance dx, dy, dz in 3-dimensional space is given by the dot product

Now suppose the particle is constrained in such a way that its position has only two degrees of freedom. In other words, there are two generalized position coordinates X and Y such that the position coordinates x, y, and z of the particle are each strictly functions of these two generalized coordinates. We can then define a generalized force F with the components FX and FY such that

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Joseph-Louis Lagrange

William Rowan Hamilton