VectorFieldsFlows copy

# VectorFieldsFlows copy - 1.3 Vector Fields and Flows 1.3 21...

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1.3 Vector Fields and Flows. 21 1.3 Vector Fields and Flows. This section introduces vector felds on Euclidean space and the Fows they determine. This topic puts together and globalizes two basic ideas learned in undergraduate mathematics: the study o± vector felds on the one hand and di²erential equations on the other. Defnition 1.3.1. Let r 0 be an integer. A C r vector feld on R n is a mapping X : U R n of class C r from an open set U R n to R n . The set of all C r vector Felds on U is denoted by X r ( U ) and the C vector Felds by X ( U ) or X ( U ) . We think o± a vector feld as assignning to each point x U a vector X ( x ) based (i.e., bound) at that same point. Example. Consider the ±orce feld determined by Newton’s law o± gravi- tation. Here the set U is R 3 minus the origin and the vector feld is defned by F ( x,y,x )= - mMG r 3 r , where m is the mass o± a test body, M is the mass o± the central body, G is the constant o± gravitation, r is the vector ±rom the origin to ( x,y,z ), and r =( x 2 + y 2 + z 2 ) 1 / 2 ; see ³igure 1.3.1 . Figure 1.3.1. The gravitational force Feld. ±

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22 1. Basic Theory of Dynamical Systems Consider a general physical system that is capable of assuming various “states” described by points in a set Z . For example, Z might be R 3 × R 3 and a state might be the position and momentum ( q , p ) of a particle. As time passes, the state evolves. If the state is z 0 Z at time t 0 and this changes to z at a later time t , we set F t,t 0 ( z 0 )= z and call F t,t 0 the evolution operator ; it maps a state at time t 0 to what the state would be at time t ; i.e., after time t - t 0 has elapsed. “Determin- ism” is expressed by the law F t 2 ,t 1 F t 1 ,t 0 = F t 2 ,t 0 F t,t = identity , sometimes called the Chapman-Kolmogorov law . The evolution laws are called time independent when F t,t 0 depends only on t - t 0 ; i.e., F t,t 0 = F s,s 0 if t - t 0 = s - s 0 . Setting F t = F t, 0 , the preceding law becomes the group property : F τ F t = F τ + t ,F 0 = identity . We call such an F t a fow and F t,t 0 a time-dependent fow , or an evo- lution operator. If the system is de±ned only for t 0, we speak of a semi-fow . It is usually not F t,t 0 that is given, but rather the laws oF motion . In other words, di²erential equations are given that we must solve to ±nd the ³ow. In general, Z is a manifold (a generalization of a smooth surface), but we con±ne ourselves here to the case that Z = U is an open set in some Euclidean space R n . These equations of motion have the form dx dt = X ( x ) ,x (0) = x 0 where X is a (possibly time-dependent) vector ±eld on U . Example. The motion of a particle of mass m under the in³uence of the gravitational force ±eld is determined by Newton’s second law: m d 2 r dt 2 = F ; i.e., by the ordinary di²erentatial equations m d 2 x dt 2 = - mMGx r 3 ; m d 2 y dt 2 = - mMGy r 3 ; m d 2 z dt 2 = - mMGz r 3 ;
1.3 Vector Fields and Flows. 23 Letting q =( x,y,z ) denote the position and p = m ( d r /dt ) the momentum, these equations become d q dt = p m ; d p dt = F ( q ) The phase space here is the open set U R 3 \{ 0 } ) × R 3 . The right-hand side of the preceding equations deFne a vector Feld by X ( q , p ) = (( q , p ) , ( p /m, F (

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VectorFieldsFlows copy - 1.3 Vector Fields and Flows 1.3 21...

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