A Proof of the Constraint Force Equation

Consider the constrained Hamiltonian with the Fixman potential 

This Hamiltonian is distinct from jg since it is a function of (x p.j (set of Cartesian coordinates) whereas jgf is a function of (q, p,) (generalized coordinates). 

We use the Lagrangian formulation since the constraint = o can be naturally expressed in this setting. The Lagrangian Jz F is associated with 

Using (4.26) for d/l/d(, we now have an expression for the derivative which involves the Lagrange multiplier AF and the Lagrangian  

The last step consists of proving that ( (a5 F/d ))F = 0 For Hamiltonian systems, it is generally true that the phase-space average of the time derivative of any function of q and p is equal to zero. For an arbitrary function /(q, p) and a Hamiltonian function 34  

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