Ngoài ra còn có các bài tập như được nêu ra vài các chương sau. Ấn bản đầu tiên sẽ có một bộ đầy đủ các bài tập cho mỗi chương. Tác giả đón chào những chỉnh sửa, ý kiến, và những lời chỉ trích. vấn đề. Điều này là đúng đối với một phối hợp ignorable đà tương ứng sẽ trở thành một tham số thời gian liên tục, và phối hợp biến mất khỏi các vấn đề còn lại. | 44 CHAPTER 2. LAGRANGE S AND HAMILTON S EQUATIONS where we used and to get the third line. Plugging in the expressions we have found for the two terms in D Alembert s Principle X d@T_@L_ .- 0 j dtdqj @Cj j_ Qj We assumed we had a holonomic system and the q s were all independent so this equation holds for arbitrary virtual displacements Sqj and therefore d@T- - @T - Qj 0- dt dq_j @qj Now let us restrict ourselves to forces given by a potential with Fi -ViU r t or X d i . dU q t Qj ãĩj iU - t Notice that Qj depends only on the value of U on the constrained surface. Also U is independent of the qỉs so d dT_ - dT dU 0 d d T - U - d T - U dt dq_j dqj dqj dt dq_j dqj or d@L - @L 0- dt dq_j @Cj This is Lagrange s equation which we have now derived in the more general context of constrained systems. Some examples of the use of Lagrangians Atwood s machine consists of two blocks of mass m1 and m2 attached by an inextensible cord which suspends them from a pulley of moment of inertia I with frictionless bearings. The kinetic energy is T U 2 mi X2 2 2Il 2 m-1gx m2g K - x m1 - m2 gx const . LAGRANGIAN MECHANICS 45 where we have used the fact that the sum of the heights of the masses is a constant K. We assume the cord does not slip on the pulley so the angular velocity of the pulley is x r and L 2 mi m2 I r2 x2 m2 - mi gx and Lagrange s equation gives d@L dt di 0 m1 m2 I r2 x m2 mi g. Notice that we set up our system in terms of only one degree of freedom the height of the first mass. This one degree of freedom parameterizes the line which is the allowed subspace of the unconstrained configuration space a three dimensional space which also has directions corresponding to the angle of the pulley and the height of the second mass. The constraints restrict these three variables because the string has a fixed length and does not slip on the pulley. Note that this formalism has permitted us to solve the problem without solving for the forces .
