TAILIEUCHUNG - Classical Mechanics - 3rd ed. - Goldstein, Poole & Safk Episode 2 Part 6

Tham khảo tài liệu 'classical mechanics - 3rd ed. - goldstein, poole & safk episode 2 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 370 Chapter 9 Canonical Transformations a point transformation of configuration space correspondingly Eqs. define a point transformation of phase space. In developing Hamiltonian mechanics only those transformations can be of interest for which the new Q p are canonical coordinates. This requirement will be satisfied provided there exists some function K Q p t such that the equations of motion in the new set are in the Hamiltonian form V dPi DQi The function K plays the role of the Hamiltonian in the new coordinate set. It is important for future considerations that the transformations considered be problem-independent. That is to say Ổ- B must be canonical coordinates not only for some specific mechanical systems but for all systems of the same number of degrees of freedom. Equations must be the form of the equations of motion in the new coordinates and momenta no matter what the particular initial form of H. We may indeed be incited to develop a particular transformation from ợ p to Ổ P to handle say a plane harmonic oscillator. But the same transformation must then also lead to Hamilton s equations of motion when applied for example to the two-dimensional Kepler problem. As was seen in Section if Qi and A are to be canonical coordinates they must satisfy a modified Hamilton s principle that can be put in the form 3 rt2 t Pi Qi- K Q P f dt 0 where summation over the repeated index i is implied . At the same time the old canonical coordinates of course satisfy a similar principle Ò r Í1 Piqi - H q p t dt 0. The simultaneous validity of Eqs. and does not mean of course that the integrands in both expressions are equal. Since the general form of the modified Hamilton s principle has zero variation at the end points both statements will be satisfied if the integrands are connected by a relation of the form . . . dF X piqi - H PiQi - K . at Here F is any function of the phase space coordinates with continuous second derivatives and

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