HW4_prob_4_GPS17_G16

# HW4_prob_4_GPS17_G16 - Goldstein 2-16(3rd ed 2.17 The...

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Goldstein 2-16 (3 rd ed. 2.17) The kinetic energy T and potential energy V for a system of n degrees of freedom are expressed as T = X i f i ( q i ) · q i 2 ,V = X i V i ( q i ) , (1) where q i and · q i are the i th generalized coordinate and velocity, respectively, and the functions f i ( q i ) and V i ( q i ) each depend on only a single generalized coordinate. The Lagrangian for this system is L = T V = X i [ f i ( q i ) · q i 2 V i ( q i )] . (2) The Lagrange EOM are de f ned as d dt L · q j L q j =0 . (3) Note that it is highly advisable to use a dummy index ( j )thatd i f ers from the one ( i )usedinde f ning L . To get a more explicit form for the EOM we need the partial derivatives L · q j =2 f j ( q j ) · q j , (4) and L q j = · q 2 j f j ( q j ) q j V j ( q j ) q j . (5) Please note that in Eqs.(4) and (5) there is only one term from Eq.(2) that survives the di f erentiation: the one for which the index

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## This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

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HW4_prob_4_GPS17_G16 - Goldstein 2-16(3rd ed 2.17 The...

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