Page 532 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Appen. E Lagrange’s Equation 519

or
I T V ■ (E -5 )

W e now form the differential of 2 7 from the preceding equation by using the
product rule in calculus:

(E -6 )

Z- 1
By subtracting Eq. (E-3) from this equation, the second term with dq, is elim i­
nated. By shifting the scalar quantity dt , the term d { d T / d q - ) q - becomes
{ d / d t ) { d T / d q ^ ) dq^ and the result is
N
ÍTT_
d T = Z dt ( dq- dq, dq, (E -7 )
1
From Eq. (E -2) the differential of U is

d U = L ^ dq, (E -8 )
Z-1
Thus, Eq. (E -1) for the invariance of the total energy becomes
irr_ w
d { T + Í7) = E dq, = 0 (E -9 )
dt ( dq. dq, ^ dq.
; = I
Because the N generalized coordinates are independent of one another, the dq^
can assume arbitrary values. Therefore, the previous equation is satisfied only if
dT dU ^
dq.
d t \ dq. n— ^ n— “ 0 i = 1 , 2 , . . . , N (E-1Ü )
dq,
This is Lagrange’s equation for the case in which all forces have a potential U.
They can be somewhat modified by introducing the Lagrangian L = (T - U).
=
Because dU/dq^ 0, Eq. (E-10 ) can be written in terms of L as
= Ü i = 1 , 2 , . . . , jV ( E -1 1 )
dt [ dq. dq.

Nonconservative systems. W hen the system is also subjected to given
forces that do not have a potential, we have instead of Eq. (E -1)

d ( T + U ) = 8 W „ ^ (E -12 )
where is the work of the nonpotential forces when the system is subjected to
an arbitrary infinitesim al displacem ent. Expressing in terms of the general-