HISTORICAL SURVEY Wilhelm von Leibniz (1646-1716): Leibniz s approach to mechanics was based on the use of mathematical operations with the scalar quantities of energy, as opposed to the vector quantities of force and acceleration in Newton mechanics. Johann Bernoulli (1667-1748)he established the principle of the so-called virtual work for static systems. Jean LeRond D Alembert (1717-1783): the principle of virtual work was extened to describing the motion of dynamical system (D Alembert principle) Joseph Louis Lagrange (1736-1813): he used the D Alembert principle for the derivation of the dynamical equations of motion that, in his honor, now bear his name: Lagrange s equations. This new mechanics has been called the analytical mechanics. Sir William Rowan Hamilton (1805-1865): he has announced in 1834 hypothesis of global economy: The basic idea of this concept is that the Mother Nature, given choices, always dictates to any objects which make up the physical universe to evolve in space and time in accordance with some extrema principles. Newton s laws themselves can be obtained using this postulate in the form known as Hamilton s variational principle.
HISTORY OF EXTREMUM PROBLEMS 1. Shortest distance between two points on a plane or on a sphere; 2. Dido problem : Find the figure bounded by a line which has a maximum area for a given perimeter. Solution: Semicircle. 3. Isoperimetric problem : Find a closed plane curve of a given length which encloses the greatest Area. Solution: Circle. 4. Isovolume problem : Find the surface enclosing the maximum volume per unit surface area Solution: Sphere. 5. Heron (Alexandria): light ray transfer, later Fermat has derived the law of refraction. 6. Brachistochrone problem : find the curve of quickest descent. It has been solved by J. Bernoulli, Newton and Leibniz. All these problems were solved using geometrical methods. Euler and Lagranger have discovered the differential equation to solved these problems and have developed the Calculus of Variation.
CALCULUS OF VARIATIONS: Find an as-yet unknown curve y(x) which minimizes the integral x 2 S = f[y(x), y (x), x] dx, (1) x 1 at the condition { y(x1 ) = y 1 y(x 2 ) = y 2 (2) where f[y(x), y (x), x] is known function of three variables y, y and x.
HAMILTON S PRINCIPLE In mechanics, a typical example is given by Hamilton s principle for a single particle that states: The actual path which a particle follows between two points 1 and 2 in a time interval from t 1 till t 2 is such that the integral t 2 I = L dt, (3) t 1 is stationary when taken along the actual path, that is t 2 δ L dt = 0. (4) t 1 Here L is the Lagrangian function, or just Lagrangian of a particle, defined as L = T V, (5) where T and V are the kinetic and potential energy of a particle.
EULER-LAGRANGE EQUATION The procedure for using the Euler-Lagrange equation contains three stages: (1) Set up the problem so that the quantity whose correct path you seek is expressed as an integral in the standard form x 2 S = f[y(x), y (x), x] dx, (6) x 1 where f[y(x), y (x), x] is the function appropriate to the problem at hand. (2) Write down the Euler-Lagrange equation in terms of the function f[y(x), y (x), x]: d f dx y f = 0. (7) y (3) Solve (if possible) the Euler-Lagrange equation for the function y(x) that defines the required path. EULER-LAGRANGE EQUATION: GENERALIZATION The generalization of this equation to an arbitrary number of dependent variables is straightforward, and doesn t need to be spelled out in detail. So we represent at once the result: The set of n functions y 1, y 2,..., y n which supplies a stationary value for the integral I = b a f(y 1, y 2,..., y n ; y 1, y 2,..., y n; x)dx (8) must satisfy the system of the Euler-Lagrange equations d f dx y i f y i = 0 (i = 1, 2,..., n). (9)
The concept of the configuration space: we associate the n numbers q 1, q 2,..., q n, with a point P in n-dimensional space. We treat the equations q 1 = q 1 (t)...... q n = q n (t) (10) as representing the motion of a point along some curve in this n- dimensional space called configuration space. For the sake of brevity, the points and curves in the configuration space which symbolize the position and the motion of the mechanical system, are sometimes reffered to us C-point and C-curve. C-point is frequently called also the system point. SUMMARY: Note that the entire mechanical system is pictured as a single point in the configuration space of no matter how numerous the particles constituting a system may be, or how complicated are the relations existing between them. This concept makes it possible to extend the mechanics of a single mass point to arbitrarily complicated manyparticle mechanical systems. It is necessary only to remaind that the space which carries this point is no longer the ordinary physical space. It is an abstract space with as many dimensions as the nature of problem requires.
We formulate the Hamilton s principle mathematically as follows: The motion of an arbitrary mechanical system occurs in such a way that the definite line integral in the configuration space t 2 I = L(q 1, q 2,..., q n ; q 1, q 2,..., q n ; t)dt (11) t 1 has a stationary value for the actual path of motion of a system point provided the initial and final configurations of the system are prescribed. The function L is defined as the excess of kinetic energy T and potential energy V, L = T V. (12) This function which frequently reffered to as the Lagrangian function or the Lagrangian is the most fundamental quantity in the analytical mechanics and its applications in modern branches of physics such as relativity and quantum theories.