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Exponential Estimates – part 4

March 25, 2013

4. Nekhoroshev  Theorem

We conclude with an overview about Nekhoroshev Theory, which deals again with Hamiltonian systems of the form

\displaystyle H(I,\varphi) = h(I) + \epsilon f(I,\varphi),

but with suitable non-isochronous {h}.
The aim is to get exponential estimates like in the isochronous case: we’ll have to assume again the analicity of H, but we’ll have to replace the Diophantine conditions on {\omega} with a suitable geometric assumption on {h}.
This last point is very delicate: to get exponential estimates we need something stronger than pure anisochronicity ({det (h'') \neq 0}), as one can easily see by choosing

\displaystyle H(I_1,I_2,\varphi_1,\varphi_2) = \frac{1}{2} (I_1^2-I_2^2) + \epsilon \sin(\varphi_1+\varphi_2)
The previous is an anisochronous Hamiltonian which admits the motion

\displaystyle I_1(t) = I_2(t) = I^0+\epsilon t

\displaystyle \varphi_1(t) = -\varphi_2(t) = \varphi^0+I^0t+\frac{1}{2}\epsilon t^2

that cannot satisfy any exponential estimate.
This problem leads to very subtle notions (such as quasi-convexity and steepness), but for simplicity we’ll state an easy sufficient condition for exponential estimates to hold.

Proposition 6 Consider the Hamiltonian

\displaystyle H(I,\varphi) = h(I) + \epsilon f(I,\varphi),

with {(I,\varphi) \in B \times \mathbb{T}^n}, and assume that

  • H is analytic in a complex neighborhood {D_\rho} of the real domain {D:=B \times \mathbb{T}^n} ;
  • h is convex ({h''} is positive definite).

Then there exists constants {C_1,C_2,\epsilon_0 > 0} s.t. for {\epsilon < \epsilon_0} any motion satisfies

\displaystyle \|I(t)-I(0)\| < C_1 (\epsilon/\epsilon_0)^{1/(2n)}

for {|t| < C_2 exp( (\epsilon_0/\epsilon)^{1/(2n)} )}.

Example 2 A typical example in which Nekhoroshev Theorem holds is a set of rotators coupled by positional forces,

\displaystyle H(I,\varphi) = \sum_{j=1}^n \frac{p_j^2}{2} + \epsilon f(\varphi).

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