First Integrals and Their Preservation Under Discretization

4 First Integrals and Their Preservation Under Discretization 

It is important in this definition that this is an equivalence that holds everywhere (or at least in some open set in W ), so the statement is not just that I z t)) = /(z(0)) for some particular trajectory, but, moreover, I is conserved for all nearby initial conditions. The flow map preserves the first integral, thus I(. i z)) = I(z). For example, in a Hamiltonian system, the flow map conserves the energy  

There may, in specific instances, be additional first integrals present. For example, in a planar 2-body system in central forces, the angular momentum = I 

Now if / R R has continuous partial derivatives, we may conclude directly that 

This means if we know that V/(z) remains bounded, say less than B, in an open domain containing the solution, we could conclude that the error in /-values computed along a numerical trajectory is of order tf. 

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