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3 The variational principle

It is an importan realization that solutions of the Lagrange equation 5 solve an extreme path problem between two points in the configuration space. The problem can be stated as that of finding the path q(t) , t0tt1 , such that the integral

 S = L(q(t),(t),t) dt (14)

is minimal. The classical variational calculus studies the variation of this integral under perturbations of the path q(t) . We substitute the initial path q(t) with the new path

 q(t) = q(t) + q(t) (15)

where q(t) is an arbitrary vector-valued function on the segment [t0,t1] . We define the variation of integral 14 under the perturbation q(t) to be (see Figure 2):

 S = = 0L(q(t),(t),t) dt. (16)

The classical calculation yields
 = 0L(q(t),(t),t) dt = = 0L(q(t),(t),t) dt = dt. (17)
Using the fact that

 (t) = (q(t)) (18)

and integration of the second term by parts yield
 S = q(t) dt + (19)
This equation implies that if the ends of the perturbation path are clamped at the ends, i.e. q(t0) = q(t1) = 0 then the second summand drops out. Moreover, If S = 0 for all perturbations then the Lagrange equations 5 must be satisfied.

The above extreme property of the solutions of the Lagrange equations 5 shows the invariance of these equations under coordinate changes: if we use a time-dependent substitution q = F(Q,t) where F : x is a change of variables then the new Lagrangian with respect to coordinates Q is

 K(Q,,t) = L(F(Q),DF(Q,t) + (Q,t),t) (20)

where DF(Q,t) is the derivative (Jacobi matrix) of F at (Q,t) with respect to Q and = . This formula allowes us to choose coordinates in a convenient manner, for instance, to express the motion of a body in a rotating coordinate system, as it will be done in section 5.

Next: 4 Hamiltonian mechanics Up: Lagrangian and Hamiltonian mechanics Previous: 2 Lagrangian mechanics
Marek Rychlik
9/2/1997