CLASSICAL THEORY in .NET framework
CHAPTER Reading PDF 417 In .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Paint PDF 417 In Visual Studio .NET Using Barcode maker for VS .NET Control to generate, create PDF417 2d barcode image in Visual Studio .NET applications. CLASSICAL THEORY
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Barcode Generation In ObjectiveC Using Barcode creation for iPhone Control to generate, create bar code image in iPhone applications. Painting Barcode In None Using Barcode generation for Word Control to generate, create barcode image in Word applications. action. This stationary value is a unique minimum if q1, t1 and q2, t2 are close enough. The action is therefore to be considered as a functional of all regular functions q(t) satisfying the boundary conditions q(t 1) = q1, q(t2) = q2' If Q(t) is the actual trajectory, a nearby one is written q(t) = Q(t) + bq(t). Expanding the action in powers of bq as Code 39 Extended Drawer In .NET Framework Using Barcode creator for ASP.NET Control to generate, create Code 39 image in ASP.NET applications. DataMatrix Printer In ObjectiveC Using Barcode encoder for iPhone Control to generate, create ECC200 image in iPhone applications. l(q) I(Q) + Jtl
r dt bq(t) (Q)bq(t) + ... M
(12) we express the principle by setting
(13) To compare this expression with the EulerLagrange equations we note that since
bq(t) = :t bq(t) (14) the variation of (11) reads
r M r ['OL oL d ] Jtl dt bq(t) bq(t) = Jtl dt oq(t) bq(t) + oq(t) dt bq(t) t2 t2
An integration by parts of the last term, taking into account boundary conditions, yields the familiar form M _ oL d oL _ 0 bq(t) = oq(t)  dt oq(t) (15) which is to be interpreted as a vector equation in the case of several degrees of freedom. In the simplest cases L is the difference between a kinetic part quadratic in the velocities and a potential part. The equations are unchanged if we add to L a total derivative with respect to time, and the action is only modified by contributions depending on the boundary conditions. This observation shows that the Lagrange function is not an intrinsic object to determine the motion and leads to a more abstract formalism, developed in particular by E. Cartan. The hamiltonian formulation is obtained through the definition of conjugate momenta .) Pi= oL (q,q ~ uqi
(16) Assume that we are able to invert this equation, i.e., to express the velocities in terms of the momenta and the positions. We shall see later what happens if the jacobian of this transformation vanishes. Hamilton's function is then obtained by a Legendre transformation H(p, q) = Piqi(P, q)  L[q, q(p, q)] (17) CLASSICAL THEORY
where summation over dummy indices is understood. By differentiation
. [ qi
OL)] 04j :+ 04j ( Pj  : dPi + [OL   (OL  Pj)] dqi  0Pi oqj Oqi Oqi oqj
We see, using the expression of conjugate momenta, that the EulerLagrange equations take the form . oH . oH p.  q.(18) , Oqi , Pi More generally the variation of a function defined on phase space (the space of the 2N variables p, q) is df = of dt ot
+ oH of _ oH of
0Pi Oqi Oqi 0Pi
= of
+ {H f} (19) We have introduced the Poisson bracket notation
{j, g} = of og _ of og
, 0Pi Oqi Oqi 0Pi
(110) It follows from (19) that a function without an explicit dependence on time, and such that its Poisson bracket with H vanishes, is a constant of the motion. It is remarkable that Hamilton's Eqs. (18) also follow from a principle of stationarity. Indeed, let us insert L(q, 4) dt
P dq  H(p, q) dt
in (11) so that the action reads
It2 P dq t, H dt
(111) and can be thought of as a functional of 2N independent functions q(t) and p(t). Assume again that q(td = ql, q(t2) = q2 without any restriction on p(td or P(t2)' We then have [bP(4 

