Tham khảo tài liệu 'classical mechanics - 3rd ed. - goldstein, poole & safk episode 2 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 320 Chapter 7 The Classical Mechanics of the Special Theory of Relativity not as a parameter but as an additional generalized coordinate. Since 0 must be a monotonic function of t as measured in some Lorentz frame we have dx dx dff x dt dỡ dt c x ữ Hence the action integral is transformed as c 2 . 1 . x y L x7 t XJ dt - L I xẠ c 77 lx do. Jit c J6l x Il would seem therefore that a recipe for a suitable A is given by the relation x 0 A xm x m L c The Lagrangian obtained this way is however a strange creature unlike any Lagrangian we have so far met. Note that no matter what the functional form of L the new Lagrangian A is a homogeneous function of the generalized velocities in the first degree AU Lnx ízA x x x m . This is not a phenomenon of relativistic physics per se it is a mathematical consequence of enlarging configuration space to include Í as a dynamical variable and using some other parameter to mark the system-point s travel through the space. A Lagrangian obeying Eq. is often called somewhat misleadingly a homogeneous Lagrangian and the corresponding homogeneous problem of the calculus of variations requires special treatment. The most serious of the resulting difficulties will arise in the Hamiltonian formulation but we can glimpse some of them by noting that in consequence the energy function h according to Eq. is identically zero. It follows from Euler s theorem on homogeneous functions that if A is homogeneous to first degree in X1 1. then SA A X . dx - We can then show cf. Derivation 10 at the end of this chapter that as a result the function A identically satisfies the relation pA a A _d0 9xM _ x M 0. Thus if any three of the Lagrangian Eqs. are satisfied it will follow solely as a consequence of the homogeneous property of A that the fourth is satisfied identically. Being thus forewarned to tread carefully so to speak let US carry out this transformation for a free particle. From Eq. . the relativistic
