Energy - Conservative System and Energy Conservation Shina...

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Unformatted text preview: Conservative System and Energy Conservation Shina Tan In a previous lecture, we have learned that the change of a system’s total kinetic energy is equal to the work done on the system. The work done on the system is given by the integral of the force dot the displacement, summed over all particles in the system: W ( t 1 ,t 2 ) = Z t 2 t 1 X α F α ( t ) · d r α ( t ) . (1) For many systems, the forces do not directly depend on time. They may depend on the positions of the particles: F α = F α ( r 1 , ··· , r n ) . (2) For example, the gravitational force exerted on the earth by the sun does not depend on time directly: it depends only on the position of the earth relative to the sun. When the earth gets 1% closer to the sun the force becomes roughly 2% stronger, and it is always in the direction toward the sun, no matter this year or last year. Because each particle has 3 coordinates, we need 3 n coordinates to describe the config- uration of the system. Let the Cartesian components of r α be x α 1 , x α 2 , and x α 3 . We may introduce a 3 n-dimensional vector: r ≡ ( r 1 , r 2 , ··· , r n ) = ( x 11 ,x 12 ,x 13 ; x 21 ,x 22 ,x 23 ; ··· ; x n 1 ,x n 2 ,x n 3 ) . (3) Likewise, the n forces acting on the system form another 3 n-dimensional vector: F ≡ ( F 1 , F 2 , ··· , F n ) = ( F 11 ,F 12 ,F 13 ; F 21 ,F 22 ,F 23 ; ··· ; F n 1 ,F n 2 ,F n 3 ) . (4) With this notation, Eq. (2) can be concisely written as F = F ( r ) , (5) which may be called a force field , since at each point r , a vector F is defined. Each point in this space correponds to a configuration of the n-particle system. Instead of n particles moving in the 3-dimensional space, we may alternatively regard the system as one “particle” moving in the 3 n-dimensional configuration space. If we define a scalar product between any two 3 n-dimensional vectors: A · B ≡ n X α =1 3 X i =1 A αi B αi , (6) 1 the work done on the system can be simply written as W 12 = Z 2 1 F ( r ) · d r , (7) which takes the same form as in single-particle physics! For comparison, you should turn to Eq.(2.84) on page 78 of Thornton and Marion, 5th edition. If the force field satisfies the condition that the work done on the system does not depend on the path taken in the configuration space, but only on the initial and final configurations of the system (see Figure 2-13 on page 79 of the same book), we say the system is conservative ....
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This note was uploaded on 01/22/2012 for the course PHYS 3202 taught by Professor Tan during the Spring '11 term at Georgia Tech.

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Energy - Conservative System and Energy Conservation Shina...

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