548-551] Lagmnges Equations 481 



(502), etc. In this way we obtain a system of values for B0 l} &0 2 , ... &0 n 

 which is permitted by the constraints of the system. 



The m multipliers X, //., ... are at our disposal: let these be supposed to be 

 chosen so that the m equations 



...... (504), 



ou s dt 



are satisfied. Then equation (503) reduces to 



(gUxa, + /B 6. + ...J8*.= ............ (505), 



dt \d0 8 ' 



and since arbitrary values have been assigned to B0 m+lt ... S0 n , it follows that 

 each coefficient in this equation must vanish separately. Combining the 

 system of equations so obtained with equations (504), we obtain the complete 

 system of equations 



...... (506). 



dv s 



Lagranges Equations for Non-conservative Forces. 



551. If the system of forces is not a conservative system, we cannot 

 replace the expression 



in 545 by 8W where W is the potential energy. We may, however, still 

 denote this expression for brevity by }8 W], no interpretation being assigned 

 to this symbol, and equation (496) will assume the form 



(BT-{8W})dt = (507). 



By the transformation used in 548, we may replace I BTdt by 



Jo 



Now {STF} is, by definition, the work done in moving the system from 

 the configuration l ,0 2) ... 6 n to the configuration 6 l + S# 1} 2 + &0 2 , . . . O n + 80 n . 

 It is therefore a linear function of 8^, 8# 2 , &&n, and we may write 



where lt 2 , . .. n are functions of lt 2 , ... O n . 

 We now have equation (507) in the form 



31 



