Mathematics of Physics and Engineering

88 The Lagrange-Hamilton Method

and the generalized coordinates are

(9i,92,93,.-.,9fc,9i,---,9fc)>

two for each degree of freedom. Equations of the type (2.3.1)-(2.3.8) can
be written for Xi, qi, Xi, qi. The kinetic energy is then given by

n

£K = J2(mi/^2 )(x^2 i+yf + zf) or £K = £*(?i,. • • ,9fc,9i, • • •,«*)• (2-3.18)

i=l

With forces Fi = (•Fi,»,i:2,i,-p3,i), i = l,---,n, equations (2.3.1) become

Fi,j = mjZj, F 2 ,j = rriij/i, F3)i = m,^, i = 1,..., n,

equation (2.3.12) becomes

T7 "^v- --5— = VrajZi—+ > mm-5—+> m^t-S—> (2.3.19) d (d£K\ d£K <A .. dxi ^ ~ <9?/i , v^ •• ^ /o o m\

d* V dqj ) dqj j^ dqj f^ dqj £> dqj

equation (2.3.14) becomes

Tf (iT-) - IT = £ (F"P + F^7T + F>.*jr) • <^2 -^3 -^20 )
dt V d?j y dq, f^ \ dqj dqj dqj J
j = 1,..., fc, and equation (2.3.15),

n k

dW = J2

F

i-

dr

i = J2 Qi

d(

ir (2.3.21)

i=l j=l

The generalized forces are

Qi = £ (F^P + ^ + ^) = £* • P> (2-3-22)

j — l,...,k. Thus, there are exactly k Lagrange's equations, one for each
degree of freedom:

(%)-%->• '-^1 k- <2'323)
Let us now look at the particular case of CONSERVATIVE FORCE FIELDS.
By definition, we say that the force F is conservative or defines a
conservative force field if the vector F is a gradient of some scalar