PHY.7:4.3 - State what may vary and what must remain fixed

Specify the interval, endpoint conditions, constraints and regularity used in the comparison. For the ordinary fixed-endpoint principle, take q_epsilon=q+epsilon*eta, with eta(t0)=eta(t1)=0. A position constraint requires a family that preserves that constraint, or a justified multiplier formulation. MATH.10 distinguishes a finite admissible family from a tangent calculation valid only to first order.

The endpoint values used to derive an equation need not be known future observations. Fixing them in the variation removes its endpoint contribution. After deriving the local evolution equation, an initial-value use supplies compatible initial position and velocity; it does not add a guessed final position as another condition.

Distinguish position constraints from restrictions on velocity that cannot be integrated into position constraints. For such a restriction, varying entire constrained histories and requiring zero work of ideal reactions on selected instantaneous virtual displacements can give different equations. A virtual displacement here is a comparison of configurations at one fixed time used to state that reaction condition. Obtain the reaction or allowed-variation rule from the physical constraint model before using either construction.

For example, an ideal reaction model for A(q,t)*qdot+b(q,t)=0 may prescribe zero reaction work on displacements satisfying A*delta_q=0. Generalized forces Q are defined by their virtual work, delta W=Q dot delta_q. The Lagrange-d’Alembert equations then have the form d/dt(L_qdot)-L_q=Q+A^T*lambda, together with the velocity constraint; here Q contains the other forces and lambda determines the reactions. This result uses the ideal-reaction premise. Simply putting the velocity constraint into an action with a multiplier is a different construction and need not reproduce it. Use PHY.6 when the balance and reaction description is the sufficient route.