Hamilton-Lagrange and Gibbs-Appel Equations

Dynamics of particles stems from theoretical physics where most of the pioneering work started with Isaac Newton and Leibniz’s discovery of calculus. Calculus of variation was then introduced by Leonard Euler which became an important tool in physics. Joseph-Louis Lagrange introduced the Lagrange function L ( q , q ) which depends on a set of generalized coordinates and velocities of a system. The general work of Euler and Lagrange also led to the principle of least action, where differential equations are obtained by minimizing the action over an interval of time. The Hamiltonian formalism of mechanics was then introduced in 1830, where the Hamiltonian function represents the total energy of the system. The Hamilton principle is related to the Lagrange through a transformation called the Legendre transformation. Equations of motion can be derived from the Hamilton principle. While the history of mechanics was based on particle dynamics, Lagrange functions are extended to continuous system- or rigid-body dynamics. Gibbs-Appell introduced a function that deals with acceleration, velocity and position from which the equations of motion are derived. This chapter introduces the formulation for the various ways of deriving the equations of motion and highlights the benefits of each. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.