Variational Problem with Non-holonomic Constraint

The following is a local and integrated rewrite of two replies the author posted to sci.math. Subject line of the thread is "Variational Problem with Derivative Constraint", and the start date was October 11, 2001. For general remarks concerning these local copies, see the home page of this web site.

The two replies were dated October 12, 2001 and November 9, 2001. There were some notational problems in the first reply, these were corrected in the second. In this local file, the two replies have been integrated; this local file is the preferred version.

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An article posted in sci.math asked about solving variational problems 
with non-holonomic constraints.  The basic result here is that the 
familiar Lagrange multiplier method for variational problems with 
holonomic constraints carries over to non-holonomic constraints.

We have the following result:

IF

{
(X, Y) is a minimum path for the functional
(Xv, Yv) into Int[a to b: G(Xv, Xv', Yv, Yv', I)] subject to the
constraints
(1) F(Xv, Xv', Yv, Yv', I) = 0, and
(2) (a,b) end-points of Xv, and Yv are fixed;

}

THEN

there exists an L such that:
{
if [(a) both (Xv,Yv)-> P4(F)(Xv,Xv',Yv,Yv',I) and
        (Xv,Yv)->P2(F)(Xv,Xv',Yv,Yv',I) are uniformly zero          
        throughout some neighborhood of (X, Y),
        OR
    (b) either (Xv,Yv)-> P4(F)(Xv,Xv',Yv,Yv',I) or
        (Xv,Yv)->P2(F)(Xv,Xv',Yv,Yv',I) is uniformly non-zero    
        throughout some neighborhood of (X, Y)],
then
[(1) (d/dt)[P2(G+L(t)*F)(X(t),X'(t),Y(t),Y'(t),t)]
         -  P1(G+L(t)*F)(X(t),X'(t),Y(t),Y'(t),t) = 0   and
 (2) (d/dt)[P4(G+L(t)*F)(X(t),X'(t),Y(t),Y'(t),t)]
         -  P3(G+L(t)*F)(X(t),X'(t),Y(t),Y'(t),t) = 0   and
 (3) F(X,X',Y,Y',I) = 0]

}      

Notes:

1) P2(function) means: partial with respect to the 2nd argument of      
   function, etc.
2) function I is defined by: I(t) = t.
3) Xv, Yv represent comparison functions which lie within a certain
   neighborhood of the minimizing functions: X, Y.
4) X' means: derivative of X, etc.
5) L is a real into real mapping.
6) In an expression such as P1(G+L(t)*F), L(t) is a number, not a
   function, and thus the partial is equal to P1(G) + L(t)*P1(F).

For a careful exposition of variational calculus and the non-holonomic
situation, see Section 11.6-2 and precedents in Korn, G., & Korn, T.,
“Mathematical Handbook for Scientists and Engineers”, Dover, 2000,
ISBN 0-486-41147-8.   

David Ziskind
http://davidziskind.org/

  

Rev. 12/15/11