Question

VI. Complete the derivation of Lagrange's equations from D'Alembert's principle.

Answer 1

DeAlembertis brinciple of virtual work ( Now having - - ) sẽ= 0. a general system virtual displacements for of a N- particles El Fie - miť). Sõi - Fie - Applied force on ith particle. switch to acneralized coordinate system as they are independent. f constraint forces are out of game Now need of additional subscript. Fi zinstead of Fie Il Fin mit ). 85.00 let's take lot teom. je! ZQ;89; ... O Q; = - Generalised force. Dimensions of Qj is not always a force Dimension of Qj.sa; is alway force.

Dond tarm Em7 1 Bit rearrangement in derivatives IO 0 Time & Coordinate derivative can be interchanged Thus and term becomes 3.6t; = 2m ({ $(**)3-,**) ; - (4 11 (24mi) - 5 min2) - (3.), 180 Nip

from eant 2 71 (40), 50, = 3a; saj Since term each reneralized in the indep. coordinates ai summation is are all sooo. OT If all the force are conservative F = -Evi Q; = (Ev) Hence Assum & does not depend on a;, then ou dari so a 3,5T-v)) - Laj, Arjit) = T(&j, 9, 1t)-v(9; it)