tailieunhanh - Classical Mechanics Joel phần 7

Bây giờ chúng ta chuyển sự chú ý của chúng tôi trở lại không gian giai đoạn, với một tập hợp các tọa độ kinh điển (khí, pi). Sử dụng tọa độ này, chúng ta có thể xác định cụ thể 1 hình thức ω1 = i pi dqi. Đối với một điểm chuyển đổi Qi = Qi (q1,., Qn, t), chúng tôi có thể sử dụng các Lagrange reexpressed trong các biến mới, tất nhiên. Ở đây, Qi là độc lập của các vận tốc qj, pha ˙ space9 dQi = j (∂ Qi / ∂ qj) dqj. Vận. | 144 CHAPTER 5. SMALL OSCILLATIONS We may think of the last part of this limit lim X a L y xlyxlxyix m j dx L y x y x x if we also dehne a limiting operation . 1 d 6 lim-7T- a 0 a dỹi oy x and similarly for a @y which act on functionals of y x and y x by y xi _ _X y xi _ y xi oy xì wt ố xi - x2 o r oJxT0 syxr - x2 . Here ỗ x x is the Dirac delta function dehned by its integral f x S x x dx f x 7xi for any function f x provided x 2 x1 x2 . Thus P x f. z I dx -py2 x t I dx pỳ x t ỗ x x py x t . oi x J0 2 Jo We also need to evaluate -U. f x Ipk 2 . Oy x L iy x Jo d 2 hd x x For this we need oyu djf dx6 x x 0 x x - which is again dehned by its integral p2 f x ỗ x x dx r f x -@-Ịỗ x x dx Jxi Jxi @x f x ỗ x x ixi dx fo x x @f . dx x . FIELD THEORY 145 where after integration by parts the surface term is dropped because 8 x x 0 for x x which it is for x X1 X2 if x 2 x1 x2 . Thus Ấ .L i dx t@- x o x x T i oy x Jo @x @x2 and Lagrange s equations give the wave equation A @2y n py x t x2 0- Exercises Three springs connect two masses to each other and to immobile walls as shown. Find the normal modes and frequencies of oscillation assuming the system remains along the line shown. Consider the motion in a vertical plane of a double pendulum consisting of two masses attached to each other and to a fixed point by inextensible strings of length L. The upper mass has mass mi and the lower mass m2. This is all in a laboratory with the ordinary gravitational forces near the surface of the Earth. 146 CHAPTER 5. SMALL OSCILLATIONS a Set up the Lagrangian for the motion assuming the strings stay taut. b Simplify the system under the approximation that the motion involves only small deviations from equilibrium. Put the problem in matrix form appropriate for the procedure discussed in class. c Find the frequencies of the normal modes of oscillation. Hint following exactly the steps given in class will be complex but the analogous procedure reversing the order of U and T will .