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Unformatted text preview: Classical Dynamics and Fluids P 73 I NTRODUCTION TO N ORMAL M ODES • For a general Newtonian system, the equations of motion take the form d 2 d t 2 variables = F variables , d d t variables • These equations are often very complicated (e.g. rigid body motion). • However, we are often interested in small displacements of a system from equilibrium . • If we expand the variables (by long tradition we denote them as { q i } or q ) about their equilibrium values q eq : variables ≡ q ≈ q eq + δ q we obtain the approximate equations ¨ q = ∂F ∂ q eq · δ q + ∂F ∂ ˙ q eq · ˙ q i.e. linear equations • In general, small displacements about equilibrium lead to linear equations . • This in turn leads to the superposition of solutions . [i.e. we can analyse the problem in pieces and add them all up afterwards. . . ] Classical Dynamics and Fluids P 74 R EMINDER : P REAMBLE TO N ORMAL M ODES x U(x) x U • Particle in a potential well U ( x ) . • Total energy is conserved. E = T + U = 1 2 m ˙ x 2 + U ( r ) • d E d t = 0 ⇒ ˙ x ( m ¨ x + d U d x ) = 0 • The resulting equation of motion m ¨ x + d U d x = 0 may well be nonlinear. • Suppose there is an equilibrium position at x where d U d x = 0 . • To study small oscillations about x , expand U ( x ) in a Taylor series: U ( x ) = U + 1 2 d 2 U d x 2 x ( x x ) 2 + ... • Defining the small displacement ξ ≡ x x and denoting d 2 U d x 2 x ≡ U 00 ( i.e. a constant ) we get the linear equation of motion m ¨ ξ + U 00 ξ = 0 which is SHM at angular frequency ω 2 = U 00 m . Classical Dynamics and Fluids P 75 D EFINITION OF A N ORMAL M ODE • We are going to study the free (unforced) small oscillations of manyparticle systems about equilibrium. The system will oscillate, but a system with many dynamical variables can oscillate in many different ways. The important concept is that of the normal mode of oscillation. • Definition of a normal model: A normal mode of a system is an oscillation that has a single frequency . • In a normal mode all parts of the system share the same periodic time dependence. • All the more general free oscillations of the system can be expressed in terms of these simple normal modes. • It is worth emphasising once more that we are considering free oscillations in the absence of external forces. • Example: a system of two masses and 3 springs: m m x x 2 1 k k k ( x 1 ,x 2 ) are the displacements of the masses from equilibrium. The masses are both m and the spring constants are all k . • The general motion is rather complicated. • However, the system has a symmetry as the masses and spring constants are equal. Classical Dynamics and Fluids P 76 D YNAMICS OF A T WOM ASS S YSTEM k k k m m x 1 x 2 =0 • Suppose the mass at x 1 is displaced from equilibrium at t = 0 , but x 2 = 0 ....
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 Spring '07
 SFGull
 normal modes, classical dynamics, Classical Dynamics and Fluids

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