Ivancevic_Applied-Diff-Geom

Introduction 23Such a force is independent of third– or higher–order derivatives of ⃗r, soNewton’s Second Law forms a set of 3 second–order ODEs. Therefore,the motion of the particle can be completely described by 6 independentvariables, or degrees of freedom (DOF). An obvious set of variables is theCartesian components of ⃗r and their time derivatives, at a given instant oftime, that is position (x, y, z) and velocity (v x , v y , v z ).More generally, we can work with a set of generalized coordinates,q i , (i = 1, ..., n), and their time derivatives, the generalized velocities, ˙q i .The position vector ⃗r is related to the generalized coordinates by sometransformation equation: ⃗r = ⃗r(q i , t). The term ‘generalized coordinates’is really a leftover from the period when Cartesian coordinates were thedefault coordinate system. In the q i −coordinates the Lagrange’s equationsread:∂L∂q i = d ∂Ldt ∂ ˙q i ,where L = T − V is the system’s Lagrangian.The time integral of the Lagrangian L, denoted S is called the action: 28∫S = L dt.Let q 0 and q 1 be the coordinates at respective initial and final times t 0 andt 1 . Using the calculus of variations, it can be shown the Lagrange’s equationsare equivalent to the Hamilton’s principle: “The system undergoesthe trajectory between t 0 and t 1 whose action has a stationary value.” Thisis formally written:δS = 0,where by ‘stationary’, we mean that the action does not vary to first–orderfor infinitesimal deformations of the trajectory, with the end–points (q 0 , t 0 )28 The action principle is an assertion about the nature of motion, from which thetrajectory of a dynamical system subject to some forces can be determined. The pathof an object is the one that yields a stationary value for a quantity called the action.Thus, instead of thinking about an object accelerating in response to applied forces,one might think of them picking out the path with a stationary action. The action isa scalar (a number) with the unit of measure for Action as Energy × Time. Althoughequivalent in classical mechanics with Newton’s laws, the action principle is better suitedfor generalizations and plays an important role in modern physics. Indeed, this principleis one of the great generalizations in physical science. In particular, it is fully appreciatedand best understood within quantum mechanics. Richard Feynman’s path integralformulation of quantum mechanics is based on a stationary–action principle, using pathintegrals. Maxwell’s equations can be derived as conditions of stationary action.