What Is Analytical Dynamics?

Analytical dynamics is a modern formulation of classical mechanics; it is the branch of physics describing the effects of forces on the motion of physical objects. The theories of Sir Isaac Newton and the calculus he developed for formulating them are the basis of this field. Later scientists, such as Joseph-Louis Lagrange and William Rowan Hamilton, generalized the behavior of physical systems with the use of more advanced and descriptive mathematics. This work was important in the study of field theories, such as electromagnetism, and the later development of quantum mechanics.

In Newton's physics, forces act on the motion of bodies as if the objects were infinitesimally small. Rotating objects were treated as if rigid, or non-deformable due to their motion. These assumptions yield highly accurate approximations of the real world and are particularly amenable to solution by means of Newton's calculus. Mathematically, force was treated as a vector, a quantity having both direction and magnitude. The aim was to calculate, given the initial position and velocity of an object, its position at some arbitrary time in the future.

Analytical dynamics methodology extends the scope of Newtonian mechanics by becoming a more abstract description. Its mathematics does not simply describe the position of objects, but can also apply to general physical systems. Among these are field theories such as those describing electromagnetism and general relativity. Each point in a field can be associated with, among other things, a vector or a scalar, a quantity having only magnitude and not direction. In general, analytic dynamics uses two scalar properties, kinetic and potential energies, to analyze motion rather than vectors.

Lagrangian mechanics, introduced in the late 18th century, combined Newton's second law, the conservation of momentum, with the first law of thermodynamics, the conservation of energy. This formulation of analytical dynamics is powerful and forms the basis of most modern theories. The Lagrangian equations reveal all relevant information about a system and can be used to describe everything from Newtonian mechanics to general relativity.

In 1833, a further refinement to analytical dynamics was presented in the form of Hamiltonian mechanics, which differs from the Lagrangian method in the way it describes the properties of a system. The purpose was not to offer a more convenient method of problem solving, but to provide deeper insight into the nature of complex dynamic systems. With a further generalization, the Hamiltonian equations were later made applicable to describing quantum mechanics as well as classical. The abstraction necessary to deepen the insight of analytical dynamics has also broadened the scope of its inquiry into other areas of science.