Mathematics of Physics and Engineering

86 The Lagrange-Hamilton Method

(2.3.7)

Similarly,

d / dx \ dx d ( dy\ _ dy

dt\dqi) dqi' dt \dqiJ dqi'

d ( dx \ _ dx d f dy\ _ dy

dt \dq 2 J dq 2 dt \dq 2 J dq 2 '

EXERCISE 2.3.1.C Verify (2.3.7).

By definition, the kinetic energy £K of m is

£K = j(i^2 + y^2 ). (2-3.8)

Replacing x and y by their functions of qi,q 2 as given by (2.3.3), we obtain

the function

£if = £/c(9i,92,91,92)- (2.3.9)

From (2.3.8), again by differentiation and using (2.3.4),

d£K 9£K dx Q£K dy. dx. dy

dq\ dx dq\ dy dqi dq\ dqi

Differentiating with respect to t and applying (2.3.7), we get

TJTd (d£ \-^r- )= mx-— + my— +K\ ..dx ..dy. dx. dy mi- \-my-. (2.3.10) /^0 q1f^

dt \ dqi J dqi dq\ dqi dqi

From (2.3.8) we also get

9£K. dx .dy ,OQ1l,

—— = mx-^— +my—. (2.3.11)

dqi dqi dqi

Subtracting (2.3.11) from (2.3.10), we find that

d fd£K\ d£K ..dx ...dy ,„,,„>

dt \ dqi J dqi dqi dqi — mx- \-my-—. (2.3.12)

EXERCISE 2.3.2.c Verify that

d (d£K\ d£K ..dx ..dy ,„,.„,

-77 [-XT- --5— =mx— +my—. (2.3.13)

dt \ dq 2 ) dq 2 dq 2 dq 2

To continue our derivation, we use equations (2.3.1) to rewrite (2.3.12)

and (2.3.13) as

±(^)-%«=F 1 ^ + F* j = 1>2. (2.3.14)

dt \ dqj J dqj dqj dqj