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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) $\in$ $\reals^{n}_{}$ , t0$\le$t$\le$t1 , such that the integral

 
S = $\displaystyle\int_{t_0}^{t_1}$L(q(t),$\displaystyle\dot{q}$(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$\scriptstyle\epsilon$(t) = q(t) + $\displaystyle\epsilon$$\displaystyle\delta$q(t) (15)

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

$\displaystyle\delta$S = $\displaystyle{\frac{d}{d\epsilon}}$$\displaystyle\bigg\vert$$\scriptstyle\epsilon$ = 0$\displaystyle\int_{t_0}^{t_1}$L(q$\scriptstyle\epsilon$(t),$\displaystyle\dot{q}_{\epsilon}^{}$(t),t) dt. (16)


 
Figure 2:  The variation of a path
\begin{figure}
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\begin{center}

\setlength {\unitlengt...
 ...{14.4}{rm}$q(t)+\epsilon\delta q(t)$}}}}}\end{picture}}
\end{center}\end{figure}

The classical calculation yields
$\displaystyle{\frac{d}{d\epsilon}}$$\displaystyle\bigg\vert$$\scriptstyle\epsilon$ = 0$\displaystyle\int_{t_0}^{t_1}$L(q$\scriptstyle\epsilon$(t),$\displaystyle\dot{q}_{\epsilon}^{}$(t),t) dt = $\displaystyle\int_{t_0}^{t_1}$$\displaystyle{\frac{d}{d\epsilon}}$$\displaystyle\bigg\vert$$\scriptstyle\epsilon$ = 0L(q$\scriptstyle\epsilon$(t),$\displaystyle\dot{q}_{\epsilon}^{}$(t),t) dt   
  = $\displaystyle\int_{t_0}^{t_1}$$\displaystyle\left(\frac{\partial L}{\partial q}\delta q(t)+
\frac{\partial L}{\partial\dot q}\delta\dot q(t)
\right)\,$dt. (17)
Using the fact that

$\displaystyle\delta$$\displaystyle\dot{q}$(t) = $\displaystyle{\frac{d}{dt}}$($\displaystyle\delta$q(t)) (18)

and integration of the second term by parts yield
$\displaystyle\delta$S = $\displaystyle\int_{t_0}^{t_1}$$\displaystyle\left[\frac{\partial L}{\partial q}-
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot
q}\right)\right]\delta$q(t) dt   
  + $\displaystyle\left[\frac{\partial L}{\partial\dot q}(q(t_1),\dot q(t_1),t_1)\de...
 ...\frac{\partial L}{\partial\dot q}(q(t_0),\dot q(t_0),t_0)\delta
q(t_0)
\right].$ (19)
This equation implies that if the ends of the perturbation path are clamped at the ends, i.e. $\delta$q(t0) = $\delta$q(t1) = 0 then the second summand drops out. Moreover, If $\delta$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 : $\reals^{n}_{}$ x $\reals$$\to$$\reals^{n}_{}$ is a change of variables then the new Lagrangian with respect to coordinates Q is

 
K(Q,$\displaystyle\dot{Q}$,t) = L(F(Q),DF(Q,t)$\displaystyle\dot{Q}$ + $\displaystyle\dot{F}$(Q,t),t) (20)

where DF(Q,t) is the derivative (Jacobi matrix) of F at (Q,t) with respect to Q and $\dot{F}$ = ${\frac{\partial F}{\partial t}}$ . 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 up previous
Next: 4 Hamiltonian mechanics Up: Lagrangian and Hamiltonian mechanics Previous: 2 Lagrangian mechanics
Marek Rychlik
9/2/1997