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2.1. ·The Principle of Least Action and Lagrangian Mechanics Let us take as our prototype of the Newtonian scheme a point particle of mass

m moving along the x axis under a potential V(x). According to Newton's Second

Law,

(2.1.1)

If we are given the initial state variables, the position x(tt) and velocity x(tt), we can calculate the classical trajectory Xcr (t) as follows. Using the initial velocity and acceleration [obtained from Eq. (2.1.1)] we compute the position and velocity at a time t 1+ At. For example,

Having updated the state variables to the time t1+At, we can repeat the process again to inch forward to t1+2At and so on.

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Figure 2.1. The Lagrangian formalism asks what distinguishes the actual path xc1 (1) taken by the particle from all possible paths connecting the end points (x,, t;) and (x1 , t1 ).

The equation of motion being second order in time, two pieces of data, x(t;) and x(t;), are needed to specify a unique xc1 (t). An equivalent way to do the same, and one that we will have occasion to employ, is to specify two space-time points (x;, t;) and (xf, t1 ) on the trajectory. The above scheme readily generalizes to more than one particle and more than one dimension. If we use n Cartesian coordinates (x 1 , X2, . . . , Xn) to specify the positions of the particles, the spatial configuration of the system may be visualized as a point in an n-dimensional configuration space. (The term "configuration space" is used even if then coordinates are not Cartesian.) The motion of the representative point is given by (2.1.2) where mj stands for the mass of the particle whose coordinate is xj. These equations can be integrated step by step, just as before, to determine the trajectory. In the Lagrangian formalism, the problem of a single particle in a potential V(x) is posed in a different way: given that the particle is at X; and x1 at times t; and t1 , respectively, what is it that distinguishes the actual trajectory xc1 (t) from all other trajectories or paths that connect these points? (See Fig. 2.1.) The Lagrangian approach is thus global, in that it tries to determine at one stroke the entire trajectory Xc1 (t), in contrast to the local approach of the Newtonian scheme, which concerns itself with what the particle is going to do in the next infinitesimal time interval. The answer to the question posed above comes in three parts: (l) Define a function !£',called the Lagrangian, given by!£= T-V, T and V being the kinetic and potential energies of the particle. Thus !£ = !f(x, x, t). The explicit t dependence may arise if the particle is in an external time-dependent field. We will, however, assume the absence of this t dependence. (2) For each path x(t) connecting (xio t;) and (x1 , t1 ), calculate the action S[x(t)] defined by

f,,

it

S[x{t)] =

!f'(x, x) dt

(2.1.3)

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REVIEW OF CLASSICAL MECHANICS

Figure 2.2. If xc1 (t) minimizes S, then 8S 11 > =0 if we go to any nearby path xc1 (t) + TJ(I).

We use square brackets to enclose the argument of S to remind us that the