Chapter12-page9 - ! q i However, since we know how to...

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Physics 235 Chapter 12 - 9 - Let us now consider a system with n coupled oscillators. We can describe the state of this system in terms of n generalized coordinates q i . The configuration of the system will be described with respect to the equilibrium state of the system (at equilibrium, the generalized coordinates are 0, and the generalized velocity and acceleration are 0). The evolution of the system can be described using Lagrange's equations: ! L ! q i " d dt ! L ! ! q i = 0 The second term on the left-hand side will contain terms that include the generalized velocity and the generalized acceleration, and is thus equal to 0 at the equilibrium position. Lagrange's equations thus tells us that ! L ! q i 0 = 0 = ! T ! q i 0 " ! U
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Unformatted text preview: ! q i However, since we know how to express the kinetic energy of the system in terms of the generalized coordinates we conclude that ! T ! q i = ! ! q i 1 2 m jk ! q j ! q k j , k " # $ % & ( = where m jk = m ! " x , i " q j " x , i " q k # $ % & ( i ) ) For the potential energy U we conclude that ! U ! q i = ! T ! q i = The potential energy can be expanded around the equilibrium position using a Taylor series and we find that U q 1 , q 2 ,... ( ) = U + ! U ! q k " # $ % & q k k ( + 1 2 ! 2 U ! q j ! q k " # $ % & q j q k j , k ( + .. ) 1 2 ! 2 U ! q j ! q k " # $ % & q j q k j , k ( = 1 2 A jk q j q k j , k ( where...
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