Mathematics of Physics and Engineering

Lagrange's Equations^87

Recall that the work W done by the force F is the line integral JcF • dr.
Hence, the differential is dW = F • dr. Since dr — dxi + dyj, we have

dW = Fidx + F 2 dy. (2.3.15)

By the chain rule,

dx = (dx/dqi)dq 1 + {dx/dq 2 )dq 2 , dy = (dy/dq^dqi + (dy/dq 2 )dq 2.

Substituting in (2.3.15), we get dW = Q\dq\ + Q 2 dq 2 , where

n - p dx _u F Qy - T? dr

Ql = r\- V t 2 — = t • ——,

91 Ql qi (2 3 16)

Q2 = Flp. + F2^=F.^L.

oq 2 aq 2 aq 2

The functions Q\ and Q 2 are called generalized forces corresponding to

the generalized coordinates q\,q 2. Substituting in (2.3.14), we obtain the

Lagrange equations of motion,

*(w )-%">'• -

1

'

2

<2

'

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where Qj is given by (2.3.16).

For the point mass moving in cartesian coordinates under the force

F = F\ i + F 2 j, we have q\ = x, q 2 = y, 8K = m(x^2 +y^2 )/2, and equations

(2.3.17) coincide with (2.3.1), so it might seem that we did not accomplish

much. It is clear, though, that (2.3.17) is more general, and covers mo-

tions not only in cartesian, but also polar and any other coordinates that

might exist in R^2. In other words, equation (2.3.17) is at a higher level of

abstraction than (2.3.1), and therefore has a higher mathematical value.

EXERCISE 2.3.3. (a)c Verify that for qi = x, q 2 = y, equations (2.3.17)

indeed coincide with (2.3.1). (b)A Write (2.3.17) in polar coordinates, with

9i =t, q 2 = 0.

Next, we consider a system of n point masses in IR^3 , and assume that

there are k degrees of freedom, k < 3n. In other words, only k out of 3n

coordinates can change independently of one another; the remaining 2>n - k

coordinates are uniquely determined by those k. The state vector is now

{xi,yi,zi, x 2 ,y 2 ,z 2 ,... ,xn,yn,zn,±i,yi,zi,... ,xn,yn,zn)