Comment 4

Edit of Comment 4. 4.The function H in (2.1) is sometimes referred to as the Lagrangian and is written as L. This Lagrangian differs from that which appears in the Lagrangian formulation of particle motion which we describe in Appendix D. Here we show the connection. In (2.1) the function H is defined by, H(r,λ) ≡ f(r) + λ1a(r) + λ2 b(r) + ...... + λS-1 q(r) . (2.1) Setting S-1 = C and defining a = a(1), b = a(2) and so on, this becomes H(r,λ) ≡ f(r) + Σi=1Cλia(i)(r) . (2.10) We require that dH = 0 (2.11) from which we conclude in (2.4) that a(i)(r) = 0 and Hk = fk + Σi=1Cλiak(i)(r) = 0 . k = 1,2...N (2.12) The N coordinates xk are the components of r = (x1, x2, ....xN). In Appendix D the object that most closely corresponds to H is the action S, where S = !Syntax Error, Idt L(qn(t), n(t), t ) . (2.13) The N coordinates qk are the generalized coordinates of the Lagrange problem. It is OK to refer to the function H in (2.10) as "the Lagrangian", but H really corresponds to S which is in fact a certain time integral of the classical mechanics Lagrangian L. In the presence of C non-holonomic constraints, we require, similar to (2.11), that δS = 0 . (2.14) This produces a set of modified Euler-Lagrange equations (D.53) which vaguely resembles (2.12), ( – ) + Σi=1CλiAik = 0 k = 1,2...N (2.15) Here the role played by fk in (2.12) is assumed by ( – ) in (2.15), while the role played by the holonomic constraint gradients ak(i) is assumed by the non-holonomic constraint functions Aik .