Perturbation Methods in Celestial Mechanics

Abstract

A concise, not too technical account of the main results of perturbation theory is presented, paying particular attention to the mathematical development of the last 60 years, with the work of Kolmogorov on one hand and of Nekhoroshev on the other hand. The main theorems are recalled with the aim of providing some insight on the guiding ideas, but omitting most details of the proofs that can be found in the existing literature.

The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait. (G. K. Chesterton)

Appendix: A Short Overview on Lie Series Methods

Here I recall a few notions concerning Lie series and Lie transforms that are used in the text. Throughout the appendix all functions will be assumed to be holomorphic.

1.1 8 Lie Series

For a given generating function χ(p, q) the Lie series operator is defined as the exponential of the Lie derivative L _χ⋅ = {⋅, χ}, namely

$$\displaystyle \begin{aligned} \exp(\epsilon L_{\chi}) = \sum_{s\ge 0} {{\epsilon^s}\over{s!}}L_{\chi}^s\ . {} \end{aligned} $$

(18)

This is actually the autonomous flow of the canonical vector field generated by χ(p, q). The flow at time ε is used in order to produce a one-parameter family of canonical transformations that is written as

$$\displaystyle \begin{aligned} p &=\exp(\epsilon L_{\chi}) p^{\prime} =p^{\prime}-\epsilon \left.{{\partial\chi}\over{\partial q}}\right|{}_{p^{\prime},q^{\prime}} + {{\epsilon^2}\over{2}} L_{\chi} \left.{{\partial\chi}\over{\partial q}}\right|{}_{p^{\prime},q^{\prime}} -\ldots \\ q &=\exp(\epsilon L_{\chi}) q^{\prime} =q^{\prime}+\epsilon \left.{{\partial\chi}\over{\partial p}}\right|{}_{p^{\prime},q^{\prime}} + {{\epsilon^2}\over{2}} L_{\chi} \left.{{\partial\chi}\over{\partial p}}\right|{}_{p^{\prime},q^{\prime}} +\ldots\ ; \end{aligned} $$

As an operator acting on holomorphic functions the exponential operator is linear and invertible, and has the remarkable properties of distributing over the products and the Poisson brackets of functions, i.e., $\exp (L_{\chi })(f\cdot g) =\big (\exp (L_{\chi })f\big )\cdot \big (\exp (L_{\chi })g\big )$ and $\exp (L_{\chi })\lbrace f,g\rbrace =\lbrace \exp (L_{\chi })f,\exp (L_{\chi })g\rbrace$. The inverse of $\exp \big (\epsilon L_{\chi }\big )$ is $\exp \big (\epsilon L_{-\chi }\big )$, for the flow is autonomous.

The most useful property of the exponential operator has been named exchange theorem by Gröbner [30]. It is stated (in a somehow puzzling form) as

$$\displaystyle \begin{aligned} f(p,q)\Big\vert_{p=\exp(\epsilon L_{\chi}) p^{\prime},q=\exp(\epsilon L_{\chi}) q^{\prime}} = \exp(\epsilon L_{\chi}) f\Big\vert_{p^{\prime},q^{\prime}}\ . \end{aligned}$$

The meaning is that an operation of substitution of a near the identity transformation followed by an expansion on the parameter (left side) is replaced by a direct application of the exponential operator to the function (right side): substitutions are avoided.

The application of the operator to a function f(p, q) = f ₀(p, q) + εf ₁(p, q) + ε ² f ₂(p, q) + … expanded in power series of the parameter ε is nicely represented by the triangular diagram for Lie series of Table 4. Terms of the same order in ε are aligned on the same row. Remark that the triangle is generated by columns: every column may be calculated separately once the upper term is known. If the function f is known, then the coefficients of the ε expansion of the transformed function $g=\exp (L_{\chi })f$ are calculated by adding up all terms on the same line. The result may be expressed by the formula

$$\displaystyle \begin{aligned} g_0 = f_0\ ,\quad g_s = \sum_{j=0}^{s} {{1}\over{j!}} L_{\chi_1}^j f_{s-j}\ ,\quad s \ge 1\ .\end{aligned} $$

A generating function χ ₂ of order ε ² generates a similar triangle, which, however, will contain many empty cells, as represented in Table 5.

Table 4 The triangular diagram for Lie series

Table 5 The triangular diagram for a generating function of order ε ²

A general formula for the transformation of a function with a generating function of order ε ^r is

$$\displaystyle \begin{aligned} g_0 &= f_0\ ,\ldots,\ g_{r-1} = f_{r-1}\ , \\ g_s &=\sum_{j=0}^k {{1}\over{j!}} L_{\chi_r}^j f_{s-jr}\ ,\quad k=\left\lfloor{{s}\over{r}}\right\rfloor\ ,\quad s \ge r \end{aligned} $$

Remark that the first change occurs at order r + 1.

Lie series operators of increasing order may be formally composed as follows. Let χ = {χ ₁(p, q), χ ₂(p, q), …} be a sequence of generating functions of increasing orders ε, ε ², … ; the composition is formally defined as

$$\displaystyle \begin{aligned} S_{\chi} = \ldots\circ\exp\big(L_{\chi_3}\big) \circ\exp\big(L_{\chi_2}\big) \circ\exp\big(L_{\chi_1}\big) \end{aligned}$$

We may also use the recursive definition of a sequence of operator S ₁, S ₂, S ₃, …

$$\displaystyle \begin{aligned} S_1 = \exp\big(L_{\chi_1}\big)\ ,\quad S_r = \exp\big(L_{\chi_r}\big)\circ S_{r-1}\ ,\quad \end{aligned}$$

considering S _χ as the limit (in formal sense) of the latter sequence for r →∞.

Compositions of Lie series are unavoidable in view of the following property: every near the identity canonical transformation of coordinates

$$\displaystyle \begin{aligned} p = p^{\prime} + \varphi_1(p^{\prime},q^{\prime}) + \varphi_2(p^{\prime},q^{\prime}) + \ldots \ ,\quad q = q^{\prime} + \psi_1(p^{\prime},q^{\prime}) + \psi_2(p^{\prime},q^{\prime}) + \ldots \end{aligned}$$

may be represented by a composition of Lie series. In general this is untrue for a single Lie series. For this reason the composition of Lie series is often replaced by the algorithm of Lie transform, introduced independently by Hori [34] and Deprit [15]. The two methods are formally equivalent. However, the composition of Lie series is in definitely better position as regards the convergence question (for instance in the case of Kolmogorov’s theorem). If the reader tries to reformulate the control of small divisors in the present notes using the Lie transform he or she will likely fail.

1.2 8 An Algorithm for Lie Transform

Contrary to Lie series, Lie transform can be constructed in a number of different ways. Here I present one of the formulations. Given a sequence {χ ₁, χ ₂, …} of generating functions define the Lie transform operator as

$$\displaystyle \begin{aligned} T_{{\chi}} = \sum_{s\ge 0} E_s {} \end{aligned} $$

(19)

with the sequence E _s of linear operators recursively defined as

$$\displaystyle \begin{aligned} E_0 = {\textsf{1}}\ ,\quad E_s = \sum_{j=1}^{s} {{j}\over{s}}L_{{\chi}_j} E_{s-j}\ . {} \end{aligned} $$

(20)

The operator may be seen as a generalization of the exponential operator of Lie series. A straightforward remark is that if we choose the generating sequence χ = {χ ₁, 0, 0, …} then $T_{{\chi }} = \exp (L_{{\chi }_1})$. Moreover T _χ has the same properties of the exponential operator of Lie series: it is linear and invertible, and distributes over products and Poisson brackets, i.e., T _χ(f ⋅ g) = T _χ f ⋅ T _χ g and T _χ {f, g} = {T _χ f, T _χ g} for any pair f, g of functions. The inverse requires some care: it has an elaborate expression which requires a second sequence of operators:

$$\displaystyle \begin{aligned} T_{\chi}^{-1} = \sum_{s\ge 0} G_j\ ,\quad G_0 = {\textsf{1}}\ ,\quad G_s = -\sum_{j=1}^{s} {{j}\over{s}} G_{s-j} L_{\chi_j}\ . {} \end{aligned} $$

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However, using the latter formula for an actual calculation is not recommended: we shall see in a short a more effective method. The formula is useful for analytical convergence estimates. It should be remarked that the inverse is not elementary because the Lie transform may be interpreted as generated by the flow of a non autonomous vector field, which can not be inverted by a mere change of sign of the vector field (as it happens for Lie series). Precisely the latter idea is developed in the paper of Deprit [15].

Finally, T _χ possesses the property expressed by the exchange theorem, namely

$$\displaystyle \begin{aligned} f(p,q)\Big\vert_{p=T_{\chi} p^{\prime},q=T_{\chi} q^{\prime}} = T_{\chi} f\Big\vert_{p^{\prime},q^{\prime}}\ . \end{aligned}$$

The scheme of application of T _χ may also be represented by a triangular diagram similar to that of Lie series, as represented in Table 6. Here too the triangle is filled in by columns, and a function g = T _χ f is found by adding up all contributions on the same line. The diagram also provides a straightforward method for calculating the inverse $f=T_{\chi }^{-1}g$. Just proceed as follows: from the first line get f ₀ = g ₀ , and fill the column for f ₀ ; from the second line get f ₁ = g ₁ − E ₁ f ₀ , and fill the column for f ₁ ; from the third line get f ₂ = g ₂ − E ₁ f ₁ − E ₂ f ₀ , and fill the column for f ₂ , and so on.

Table 6 The triangular diagram for Lie transform

1.3 8 Analytical Tools

Here I introduce some basic tools that allow us to discuss the convergence of Lie series and of composition of Lie series. I shall restrict my attention to the case of a phase space endowed with action-angle variables $p\in {\mathcal {G}}\subset \mathbb {R}^n$ and $q\in \mathbb {T}^n$, as considered in the present notes. However, the whole argument is based on the theory of holomorphic functions.

The first step requires introducing a family of complex domains

$$\displaystyle \begin{aligned} {\mathcal{D}}_{(1-d)({\varrho},\sigma)} = \Delta_{(1-d){\varrho}}\times\mathbb{T}^n_{(1-d)\sigma}\end{aligned} $$

with fixed ϱ, σ > 0 and 0 ≤ d < 1; here

$$\displaystyle \begin{aligned} {\Delta}_{{\varrho}} = \big\{p\in{\mathbb{C}}^n\>:\>|p|\le {\varrho}\big\} \ ,\quad {\mathbb{T}}^n_{\sigma} = \big\{q\in{\mathbb{C}}^n\>:\> |\,{\text{Im}}\, {q}| \le \sigma\big\}\ . {}\end{aligned} $$

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In the case of one degree of freedom the domain is represented in Fig. 17. The action domain here is a polydisk Δ_ϱ centered at the origin of $\mathbb {C}^n$, which is enough for the proof of the theorem of Kolmogorov. However, the whole argument may be extended to the case of a complex domain ${\mathcal {G}}_{{\varrho }}=\bigcup _{p\in {\mathcal {G}}}\Delta _{{\varrho }}(p)$ constructed by making the union of all complex disks of radius ϱ centered at every point of the real domain ${\mathcal {G}}$ of the actions.

Fig. 17

Construction of the family of complex domains

The second step is concerned with the extension of Cauchy’s estimates for the derivatives of holomorphic functions to the case of Lie derivatives. For a function f(p, q) which is holomorphic in ${\mathcal {D}}_{({\varrho },\sigma )}$ we shall use the supremum norm

$$\displaystyle \begin{aligned} \big|f \big|{}_{({\varrho},\sigma)} = \sup_{(p,q)\in{\mathcal{D}}_{({\varrho},\sigma)}} |f(p,q)|\ . {}\end{aligned} $$

(23)

We assume that |f|_(ϱ,σ) is finite. Following Cauchy, the derivatives of the function f(p, q) are estimated as

$$\displaystyle \begin{aligned} \left|{{{\partial}{f}}\over{{\partial}{p}}}\right|{}_{(1-d)({\varrho},\sigma)} \le {{1}\over{d{\varrho}}}\big|f \big|{}_{({\varrho},\sigma)}\ ,\quad \left|{{{\partial}{f}}\over{{\partial}{q}}}\right|{}_{(1-d)({\varrho},\sigma)} \le {{1}\over{d\sigma}}\big|f \big|{}_{({\varrho},\sigma)}\ . \end{aligned}$$

Higher order derivatives can be estimated, too. However, for our purposes, it is better to obtain estimates for Lie derivatives. An appropriate approach is the following. Assume that we know the norm |χ|_ϱ,σ of a generating function χ on the whole domain and the norm $\|{f}\|{ }_{(1-d^{\prime })({\varrho },\sigma )}$ in a possibly smaller domain, with 0 ≤ d < 1. Then for d ^′ < d < 1 one gets generally an estimate such as

$$\displaystyle \begin{aligned} \big|{L_{\chi} f}\big|{}_{(1-d)({\varrho},\sigma)} \le {{C}\over{d(d-d^{\prime}){\varrho}\sigma}} |{\chi}|{}_{{\varrho},\sigma} |{f}|{}_{(1-d^{\prime})({\varrho},\sigma)}\ . {}\end{aligned} $$

(24)

with some constant C ≥ 1 depending on the choice of the norm (and on the method of estimate). In the present case of the supremum norm a straightforward calculation gives C = 2n, because the Poisson bracket is expressed by the sum of 2n products of derivatives. However, a more careful estimate, using the fact that we are performing a derivative in a given direction, provides the better value C = 1.

The estimate of multiple Lie derivatives is more delicate. Suppose we know |χ|_ϱ,σ and |f|_ϱ,σ on the common domain ${\mathcal {D}}_{{\varrho },\sigma }$. If we want the evaluate $\big |L_{\chi }^s f\big |{ }_{(1-d)({\varrho },\sigma )}$ in a restricted domain we can define δ = d∕s and estimate, in sequence,

$$\displaystyle \begin{aligned} \big|L_{\chi} f\big|{}_{(1-\delta)({\varrho},\sigma)} \>,\ \big|L_{\chi}^2 f\big|{}_{(1-2\delta)({\varrho},\sigma)} \>,\>\ldots\>,\> \>,\ \big|L_{\chi}^s f\big|{}_{(1-s\delta)({\varrho},\sigma)}\ .\end{aligned} $$

To this end we apply by recurrence the estimate (24) for a single derivative, setting step by step d ^′ = 0, δ, …, (s − 1)δ. With some calculations we end up with the estimate (setting C = 1)

$$\displaystyle \begin{aligned} {{1}\over{s!}} \big|L_{\chi}^s f\big|{}_{(1-d)({\varrho},\sigma)} \le {{1}\over{e}} \left({{e}\over{d^2{\varrho}\sigma}}\right)^s |{\chi}|{}_{{\varrho},\sigma}^s |{f}|{}_{(1-d^{\prime})({\varrho},\sigma)}\ . \end{aligned}$$

1.4 8 Convergence of Lie Series and of the Composition of Lie Series

Substituting the latter estimate in the expression of Lie series we prove

Lemma 8

Let χ(p, q) be holomorphic and bounded in ${\mathcal {D}}_{({\varrho },\sigma )}\,$ . If the convergence condition

$$\displaystyle \begin{aligned} \left|\chi\right|{}_{({\varrho},\sigma)} < {{d^2{\varrho}\sigma}\over{2e}}\ ,\quad d < 1/2\end{aligned} $$

is satisfied, then the near the identity canonical transformation

$$\displaystyle \begin{aligned} p^{\prime} = \exp\big(L_{\chi}\big)p\ ,\quad q^{\prime} = \exp\big(L_{\chi}\big)q\end{aligned} $$

is holomorphic in ${\mathcal {D}}_{(1-d)({\varrho },\sigma )}\,$ , and

$$\displaystyle \begin{aligned} |p-p^{\prime}| < d{\varrho}\ ,\quad |q-q^{\prime}| < d\sigma\ . \end{aligned}$$

Fig. 18

The deformation induced by the near the identity transformation of Lemma 8. The flow is denoted by ϕ, with inverse ϕ ⁻¹

By the way, the lemma is actually a reformulation of Cauchy’s theorem on existence and uniqueness of a local flow in the analytic case. The implications of the lemma can be understood looking at Fig. 18 and recalling that the transformation is defined through the flow generated by χ(p, q). The transformation is essentially a deformation of coordinates. Therefore if we consider it as defined on a domain ${\mathcal {D}}_{(1-d)({\varrho },\sigma )}$, with d < 1∕2 then the relation

$$\displaystyle \begin{aligned} {\mathcal{D}}_{(1-2d)({\varrho},\sigma)} \subset {\mathcal{D}}_{(1-d)({\varrho},\sigma)} \subset {\mathcal{D}}_{{\varrho},\sigma} {} \end{aligned} $$

(25)

holds true, so that there is a domain where the transformation is well defined.

Coming to the composition of Lie series, we may intepret it as a composition of flows. Therefore we should check that the relations (25) are still satisfied. The final result is expressed by

Proposition 9

Let the sequence of generating functions χ = {χ ₁, χ ₂, …} be holomorphic and bounded in ${\mathcal {D}}_{({\varrho },\sigma )}\,$ . If the convergence condition

$$\displaystyle \begin{aligned} \sum_{j\ge 1} \left|\chi_j\right|{}_{({\varrho},\sigma)} < {{d^2{\varrho}\sigma}\over{4e}}\ ,\quad d < 1/2 \end{aligned}$$

is satisfied, then the near the identity canonical transformation

$$\displaystyle \begin{aligned} p^{\prime} = S_{\chi} p\ ,\quad q^{\prime} = S_{\chi}q \end{aligned}$$

generated by the infinite composition of Lie series

$$\displaystyle \begin{aligned} S_{\chi} = \ldots\circ\exp\big(L_{\chi_3}\big) \circ\exp\big(L_{\chi_2}\big) \circ\exp\big(L_{\chi_1}\big) \end{aligned}$$

is holomorphic in ${\mathcal {D}}_{(1-d)({\varrho },\sigma )}\,$ , and

$$\displaystyle \begin{aligned} |p-p^{\prime}| < d{\varrho}\ ,\quad |q-q^{\prime}| < d\sigma\ . \end{aligned}$$

Similar results may be obtained also for the algorithm of Lie transform. However, they are not needed for the purposes of the present notes, thus I omit them.