math4b-15 - Math 4B Lecture 15 Doug Moore The rest of the...

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Math 4B Lecture 15 May 20, 2013 Doug Moore
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The rest of the course will be concerned with systems of dif- ferential equations. Such systems arise in the first applications of calculus, to the solution of the Kepler problem. If F is the gravitational force which the sun (of mass M and located at 0 , 0 , 0)) exerts on a planet (of mass m and located at the point x = ( x, y, z )), then F = - GMm x 2 + y 2 + z 2 x i + y j + z k q x 2 + y 2 + z 2 = - GMm k x k 2 x k x k , where G is Newton’s gravitation constant. Combining with the second law of motion yields the second-order vector differential equation m d 2 x dt 2 = - GMm k x k 2 x k x k .
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This is a system of three nonlinear second order differential equa- tions for the unknowns x ( t ), y ( t ) and z ( t ): m d 2 x dt 2 = - GMmx ( x 2 + y 2 + z 2 ) 3 / 2 , m d 2 y dt 2 = - GMmy ( x 2 + y 2 + z 2 ) 3 / 2 , m d 2 z dt 2 = - GMmx ( x 2 + y 2 + z 2 ) 3 / 2 . By the end of the course we will have studied many of the tech- niques needed to solve this system. Other techniques will be presented in Math 6A and 6B.
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The basic objects of study in the theory of ordinary differential equations are first order systems , which are also called dynam- ical systems . Indeed, any higher order equation or system of equations can be reduced to a first order system by “reduction of order.” A first order system is said to be in standard form if it is written as follows: dx 1 /dt = f 1 ( x 1 , x 2 , . . . x n , t ) , dx 2 /dt = f 2 ( x 1 , x 2 , . . . x n , t ) , · · · · · · (1) · · · · · · dx n /dt = f n ( x 1 , x 2 , . . . x n , t ) , where f 1 , f 2 , . . . , f n are well-behaved functions.
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The crux of the theory of dynamical systems is the fundamental existence and uniqueness theorem: Picard’s Theorem. Suppose that f 1 , f 2 , . . . , f n are continuous and have continuous first partial derivatives on a region D of ( x 1 , x 2 , . . . x n , t )-space. Then given any point c = ( c 1 , c 2 , . . . , c n , t 0 ) in D , there exists a solution x 1 = x 1 ( t ) , x 2 = x 2 ( t ) , . . . , x n = x n ( t ) to the above dynamical system in standard form which is defined for t in some open interval, t 0 - < t < t 0 + , and which satisfies the initial conditions x 1
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