Comment 4 v1
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A short Word document containing an edited version of a comment, marked as obsolete, from Phil's notes on Lagrange multipliers. It explains that the function H = f + Σλiai is sometimes called the Lagrangian, but corresponds more closely to the action S, a time integral of the mechanics Lagrangian L. It compares dH = 0 with δS = 0 under holonomic constraints, which gives modified Euler-Lagrange equations (referenced to Appendix D). Some equations are garbled in extraction.
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Edit of Comment 4.
this is obs
4. The function H in (2.1) is sometimes referred to as the Lagrangian and is written 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)
We must now rename the constraint functions using one of the methods from Comment 2 above, and this means for the moment that we give up the notation that ai ≡ ∂a/∂xi. Setting S-1 = C and defining a1 = a, a2 = b. .... , aC = q , the above equation for H becomes
H(r,λ) ≡ f(r) + Σi=1Cλiai(r) . (2.10)
We require that
dH = 0 (2.11)
from which we conclude in (2.4) that ai(r) = 0 and
= + Σi=1C λi = 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 holonomic constraints ai(qi,t) = 0, we require, similar to (2.11), that
δS = 0 . (2.14)
This produces a set of modified Euler-Lagrange equations (D.54) which closely resemble the set (2.12),
( – ) + Σi=1C λi = 0 k = 1,2...N (2.15)
Here the role played by in (2.12) is assumed by ( – ) in (2.15) .