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lecture_28 (dragged) 2

# lecture_28 (dragged) 2 - (c that the motion is not a±ected...

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MA 36600 LECTURE NOTES: FRIDAY, APRIL 3 3 Then there exists a unique solution on the interval I . This is known as the Existence and Uniqueness Theorem for Linear Systems . The statement for linear systems follows from that for nonlinear systems because if we denote G i ( t, x 1 , x 2 , . . . , x n ) = n j =1 p ij ( t ) x j + g i ( t ) = G i x j = p ij ( t ) . Note the similarity here with the Existence and Uniqueness Theorem for second order linear equations presented in Lecture #14 on Wednesday, October 1. Applications Spring-Mass System: One Mass, One Spring. We discuss several examples of oscillations with springs, each increasing in complexity. For simplicity, we assume that (a) the original length of the spring is negligible to the motion of the mass, so that Hooke’s Law states that the restoring force is F s = k u in terms of a displacement u ; (b) all motion takes place on a frictionless track so that we can ignore any frictional forces; and
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Unformatted text preview: (c) that the motion is not a±ected of the force of gravity. First, say that we have a mass m attached to a spring with constant k . Assume that the mass is under the in²uence of an external force F ( t ). Denote u = u ( t ) as the position of the mass at time t . Equilibrium is seen to be at u = 0, so this also denotes the displacement from equilibrium of the mass at time t . Hooke’s Law states that the restoring force of the spring on the mass is F s = − k u. Newton’s Second Law of Motion states that F = m a is the sum of the forces on the mass: m d 2 u dt 2 = − k u + F ( t ) . This can be expressed as a system of ³rst order di±erential equations. Indeed, if we denote x 1 = u x 2 = u ° = ⇒ x ° 1 = x 2 m x ° 2 = − k x 1 + F ( t )...
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