Lie Derivatives and Poisson Brackets

Let p(z) be an arbitrary smooth, scalar-valued function of the phase variables (assumed to lie in R ). At any point z = in phase space assume that there is a unique solution z(t, such that z(t) = f(z(t)), z(0) = that is globally defined for all t. Now differentiate (r) = (z(0) with respect to time, using the chain rule, to see how 0(z(O) s changing as t is varied  

The right hand side suggests a shorthand notation. If we define the Lie derivative / by 

The Taylor series expansion of (z(0) along a solution of the differential equation can therefore be written as 

This gives a concise formula for the evolution of any function of the phase variables, including, in particular, any solution component z,-. Thus the flow map of the system can be represented by exp(t /). When one writes — exp(t /), what is actually meant is that the individual components satisfy 

By analogy with the derivation of the matrix exponential, it is tempting to express the exponential as a series expansion in powers of t  

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