The development of dynamics theory began with the work of Isaac Newton.
In his theory the most basic law of classical mechanics is f = ma, which
describes the motion n in IR. of a point of mass m under the action of a
force f by giving the acceleration a. If n the position of the point is
taken to be a point x E IR., and if the force f is supposed to be a
function of x only, Newton's Law is a description in terms of a
second-order ordinary differential equation: J2x m dt = f(x). 2 It makes
sense to reduce the equations to first order by defining the velo city
as an extra n independent variable by v =: i; = E IR. . Then x = v, mv =
f(x). L. Euler, J. L. Lagrange and others studied mechanics by means of
an analytical method called analytical dynamics. Whenever the force f is
represented by a gradient vector field f = - \lU of the potential energy
U, and denotes the difference of the kinetic energy and the potential
energy by 1 L(x, v) = 2'm(v, v) - U(x), the Newton equation of motion is
reduced to the Euler-Lagrange equation are used as the variables, the
Euler-Lagrange equation can be If the momenta y written as . 8L y= 8x'
Further, W. R.