1 | % This is LLNCS.DEM the demonstration file of |
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2 | % the LaTeX macro package from Springer-Verlag |
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3 | % for Lecture Notes in Computer Science, version 1.1 |
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4 | \documentstyle{llncs} |
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5 | % |
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6 | \begin{document} |
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7 | |
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8 | \title{Hamiltonian Mechanics} |
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9 | |
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10 | \author{Ivar Ekeland\inst{1} and Roger Temam\inst{2}} |
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11 | |
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12 | \institute{Princeton University, Princeton NJ 08544, USA |
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13 | \and |
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14 | Universit\'{e} de Paris-Sud, |
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15 | Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\ |
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16 | F-91405 Orsay Cedex, France} |
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17 | |
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18 | \maketitle |
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19 | |
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20 | \begin{abstract} |
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21 | The abstract should summarize the contents of the paper |
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22 | using at least 70 and at most 150 words. It will be set in 9-point |
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23 | font size and be inset 1.0 cm from the right and left margins. |
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24 | There will be two blank lines before and after the Abstract. \dots |
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25 | \end{abstract} |
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26 | % |
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27 | \section{Fixed-Period Problems: The Sublinear Case} |
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28 | % |
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29 | With this chapter, the preliminaries are over, and we begin the search |
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30 | for periodic solutions to Hamiltonian systems. All this will be done in |
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31 | the convex case; that is, we shall study the boundary-value problem |
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32 | \begin{eqnarray*} |
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33 | \dot{x}&=&JH' (t,x)\\ |
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34 | x(0) &=& x(T) |
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35 | \end{eqnarray*} |
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36 | with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when |
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37 | $\left\|x\right\| \to \infty$. |
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38 | |
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39 | % |
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40 | \subsection{Autonomous Systems} |
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41 | % |
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42 | In this section, we will consider the case when the Hamiltonian $H(x)$ |
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43 | is autonomous. For the sake of simplicity, we shall also assume that it |
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44 | is $C^{1}$. |
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45 | |
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46 | We shall first consider the question of nontriviality, within the |
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47 | general framework of |
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48 | $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In |
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49 | the second subsection, we shall look into the special case when $H$ is |
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50 | $\left(0,b_{\infty}\right)$-subquadratic, |
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51 | and we shall try to derive additional information. |
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52 | % |
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53 | \subsubsection{ The General Case: Nontriviality.} |
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54 | % |
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55 | We assume that $H$ is |
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56 | $\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic at infinity, |
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57 | for some constant symmetric matrices $A_{\infty}$ and $B_{\infty}$, |
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58 | with $B_{\infty}-A_{\infty}$ positive definite. Set: |
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59 | \begin{eqnarray} |
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60 | \gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\ |
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61 | \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \ |
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62 | J \frac{d}{dt} +A_{\infty}\ . |
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63 | \end{eqnarray} |
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64 | |
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65 | Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value |
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66 | problem: |
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67 | \begin{equation} |
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68 | \begin{array}{rcl} |
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69 | \dot{x}&=&JH' (x)\\ |
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70 | x(0)&=&x (T) |
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71 | \end{array} |
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72 | \end{equation} |
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73 | has at least one solution |
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74 | $\overline{x}$, which is found by minimizing the dual |
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75 | action functional: |
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76 | \begin{equation} |
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77 | \psi (u) = \int_{o}^{T} \left[\frac{1}{2} |
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78 | \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt |
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79 | \end{equation} |
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80 | on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ |
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81 | with finite codimension. Here |
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82 | \begin{equation} |
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83 | N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right) |
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84 | \end{equation} |
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85 | is a convex function, and |
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86 | \begin{equation} |
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87 | N(x) \le \frac{1}{2} |
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88 | \left(\left(B_{\infty} - A_{\infty}\right) x,x\right) |
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89 | + c\ \ \ \forall x\ . |
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90 | \end{equation} |
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91 | |
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92 | % |
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93 | \begin{proposition} |
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94 | Assume $H'(0)=0$ and $ H(0)=0$. Set: |
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95 | \begin{equation} |
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96 | \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ . |
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97 | \label{eq:one} |
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98 | \end{equation} |
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99 | |
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100 | If $\gamma < - \lambda < \delta$, |
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101 | the solution $\overline{u}$ is non-zero: |
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102 | \begin{equation} |
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103 | \overline{x} (t) \ne 0\ \ \ \forall t\ . |
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104 | \end{equation} |
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105 | \end{proposition} |
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106 | % |
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107 | \begin{proof} |
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108 | Condition (\ref{eq:one}) means that, for every |
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109 | $\delta ' > \delta$, there is some $\varepsilon > 0$ such that |
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110 | \begin{equation} |
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111 | \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le |
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112 | \frac{\delta '}{2} \left\|x\right\|^{2}\ . |
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113 | \end{equation} |
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114 | |
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115 | It is an exercise in convex analysis, into which we shall not go, to |
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116 | show that this implies that there is an $\eta > 0$ such that |
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117 | \begin{equation} |
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118 | f\left\|x\right\| \le \eta |
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119 | \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '} |
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120 | \left\|y\right\|^{2}\ . |
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121 | \label{eq:two} |
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122 | \end{equation} |
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123 | |
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124 | \begin{figure} |
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125 | \vspace{2.5cm} |
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126 | \caption{This is the caption of the figure displaying a white eagle and |
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127 | a white horse on a snow field} |
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128 | \end{figure} |
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129 | |
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130 | Since $u_{1}$ is a smooth function, we will have |
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131 | $\left\|hu_{1}\right\|_\infty \le \eta$ |
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132 | for $h$ small enough, and inequality (\ref{eq:two}) will hold, |
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133 | yielding thereby: |
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134 | \begin{equation} |
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135 | \psi (hu_{1}) \le \frac{h^{2}}{2} |
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136 | \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2} |
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137 | \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ . |
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138 | \end{equation} |
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139 | |
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140 | If we choose $\delta '$ close enough to $\delta$, the quantity |
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141 | $\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$ |
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142 | will be negative, and we end up with |
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143 | \begin{equation} |
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144 | \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ . |
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145 | \end{equation} |
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146 | |
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147 | On the other hand, we check directly that $\psi (0) = 0$. This shows |
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148 | that 0 cannot be a minimizer of $\psi$, not even a local one. |
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149 | So $\overline{u} \ne 0$ and |
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150 | $\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$. \qed |
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151 | \end{proof} |
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152 | % |
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153 | \begin{corollary} |
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154 | Assume $H$ is $C^{2}$ and |
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155 | $\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. Let |
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156 | $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$ be the |
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157 | equilibria, that is, the solutions of $H' (\xi ) = 0$. |
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158 | Denote by $\omega_{k}$ |
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159 | the smallest eigenvalue of $H'' \left(\xi_{k}\right)$, and set: |
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160 | \begin{equation} |
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161 | \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ . |
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162 | \end{equation} |
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163 | If: |
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164 | \begin{equation} |
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165 | \frac{T}{2\pi} b_{\infty} < |
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166 | - E \left[- \frac{T}{2\pi}a_{\infty}\right] < |
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167 | \frac{T}{2\pi}\omega |
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168 | \label{eq:three} |
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169 | \end{equation} |
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170 | then minimization of $\psi$ yields a non-constant $T$-periodic solution |
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171 | $\overline{x}$. |
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172 | \end{corollary} |
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173 | % |
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174 | |
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175 | We recall once more that by the integer part $E [\alpha ]$ of |
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176 | $\alpha \in \bbbr$, we mean the $a\in \bbbz$ |
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177 | such that $a< \alpha \le a+1$. For instance, |
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178 | if we take $a_{\infty} = 0$, Corollary 2 tells |
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179 | us that $\overline{x}$ exists and is |
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180 | non-constant provided that: |
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181 | |
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182 | \begin{equation} |
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183 | \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi} |
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184 | \end{equation} |
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185 | or |
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186 | \begin{equation} |
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187 | T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ . |
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188 | \label{eq:four} |
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189 | \end{equation} |
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190 | |
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191 | % |
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192 | \begin{proof} |
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193 | The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The |
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194 | largest negative eigenvalue $\lambda$ is given by |
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195 | $\frac{2\pi}{T}k_{o} +a_{\infty}$, |
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196 | where |
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197 | \begin{equation} |
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198 | \frac{2\pi}{T}k_{o} + a_{\infty} < 0 |
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199 | \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ . |
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200 | \end{equation} |
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201 | Hence: |
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202 | \begin{equation} |
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203 | k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ . |
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204 | \end{equation} |
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205 | |
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206 | The condition $\gamma < -\lambda < \delta$ now becomes: |
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207 | \begin{equation} |
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208 | b_{\infty} - a_{\infty} < |
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209 | - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty} |
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210 | \end{equation} |
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211 | which is precisely condition (\ref{eq:three}).\qed |
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212 | \end{proof} |
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213 | % |
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214 | |
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215 | \begin{lemma} |
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216 | Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and |
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217 | that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local |
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218 | minimizer $\widetilde{x}$ of $\psi$ has minimal period $T$. |
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219 | \end{lemma} |
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220 | % |
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221 | \begin{proof} |
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222 | We know that $\widetilde{x}$, or |
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223 | $\widetilde{x} + \xi$ for some constant $\xi |
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224 | \in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system: |
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225 | \begin{equation} |
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226 | \dot{x} = JH' (x)\ . |
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227 | \end{equation} |
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228 | |
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229 | There is no loss of generality in taking $\xi = 0$. So |
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230 | $\psi (x) \ge \psi (\widetilde{x} )$ |
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231 | for all $\widetilde{x}$ in some neighbourhood of $x$ in |
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232 | $W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$. |
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233 | |
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234 | But this index is precisely the index |
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235 | $i_{T} (\widetilde{x} )$ of the $T$-periodic |
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236 | solution $\widetilde{x}$ over the interval |
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237 | $(0,T)$, as defined in Sect.~2.6. So |
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238 | \begin{equation} |
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239 | i_{T} (\widetilde{x} ) = 0\ . |
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240 | \label{eq:five} |
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241 | \end{equation} |
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242 | |
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243 | Now if $\widetilde{x}$ has a lower period, $T/k$ say, |
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244 | we would have, by Corollary 31: |
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245 | \begin{equation} |
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246 | i_{T} (\widetilde{x} ) = |
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247 | i_{kT/k}(\widetilde{x} ) \ge |
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248 | ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ . |
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249 | \end{equation} |
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250 | |
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251 | This would contradict (\ref{eq:five}), and thus cannot happen.\qed |
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252 | \end{proof} |
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253 | % |
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254 | \paragraph{Notes and Comments.} |
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255 | The results in this section are a |
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256 | refined version of \cite{clar:eke}; |
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257 | the minimality result of Proposition |
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258 | 14 was the first of its kind. |
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259 | |
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260 | To understand the nontriviality conditions, such as the one in formula |
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261 | (\ref{eq:four}), one may think of a one-parameter family |
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262 | $x_{T}$, $T\in \left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$ |
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263 | of periodic solutions, $x_{T} (0) = x_{T} (T)$, |
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264 | with $x_{T}$ going away to infinity when $T\to 2\pi \omega^{-1}$, |
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265 | which is the period of the linearized system at 0. |
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266 | |
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267 | \begin{table} |
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268 | \caption{This is the example table taken out of {\it The |
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269 | \TeX{}book,} p.\,246} |
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270 | \vspace{2pt} |
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271 | \begin{tabular}{r@{\quad}rl} |
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272 | \hline |
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273 | \multicolumn{1}{l}{\rule{0pt}{12pt} |
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274 | Year}&\multicolumn{2}{l}{World population}\\[2pt] |
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275 | \hline\rule{0pt}{12pt} |
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276 | 8000 B.C. & 5,000,000& \\ |
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277 | 50 A.D. & 200,000,000& \\ |
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278 | 1650 A.D. & 500,000,000& \\ |
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279 | 1945 A.D. & 2,300,000,000& \\ |
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280 | 1980 A.D. & 4,400,000,000& \\[2pt] |
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281 | \hline |
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282 | \end{tabular} |
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283 | \end{table} |
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284 | % |
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285 | \begin{theorem} [(Ghoussoub-Preiss)] |
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286 | Assume $H(t,x)$ is |
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287 | $(0,\varepsilon )$-subquadratic at |
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288 | infinity for all $\varepsilon > 0$, and $T$-periodic in $t$ |
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289 | \begin{equation} |
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290 | H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t |
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291 | \end{equation} |
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292 | \begin{equation} |
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293 | H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x |
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294 | \end{equation} |
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295 | \begin{equation} |
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296 | H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \ |
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297 | {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty |
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298 | \end{equation} |
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299 | \begin{equation} |
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300 | \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\ |
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301 | H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ . |
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302 | \end{equation} |
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303 | |
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304 | Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite |
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305 | everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of |
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306 | $kT$-periodic solutions of the system |
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307 | \begin{equation} |
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308 | \dot{x} = JH' (t,x) |
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309 | \end{equation} |
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310 | such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with: |
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311 | \begin{equation} |
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312 | p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ . |
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313 | \end{equation} |
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314 | \qed |
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315 | \end{theorem} |
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316 | % |
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317 | \begin{example} [{\rm(External forcing)}] |
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318 | Consider the system: |
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319 | \begin{equation} |
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320 | \dot{x} = JH' (x) + f(t) |
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321 | \end{equation} |
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322 | where the Hamiltonian $H$ is |
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323 | $\left(0,b_{\infty}\right)$-subquadratic, and the |
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324 | forcing term is a distribution on the circle: |
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325 | \begin{equation} |
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326 | f = \frac{d}{dt} F + f_{o}\ \ \ \ \ |
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327 | {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ , |
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328 | \end{equation} |
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329 | where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance, |
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330 | \begin{equation} |
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331 | f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ , |
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332 | \end{equation} |
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333 | where $\delta_{k}$ is the Dirac mass at $t= k$ and |
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334 | $\xi \in \bbbr^{2n}$ is a |
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335 | constant, fits the prescription. This means that the system |
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336 | $\dot{x} = JH' (x)$ is being excited by a |
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337 | series of identical shocks at interval $T$. |
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338 | \end{example} |
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339 | % |
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340 | \begin{definition} |
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341 | Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric |
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342 | operators in $\bbbr^{2n}$, depending continuously on |
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343 | $t\in [0,T]$, such that |
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344 | $A_{\infty} (t) \le B_{\infty} (t)$ for all $t$. |
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345 | |
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346 | A Borelian function |
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347 | $H: [0,T]\times \bbbr^{2n} \to \bbbr$ |
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348 | is called |
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349 | $\left(A_{\infty} ,B_{\infty}\right)$-{\it subquadratic at infinity} |
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350 | if there exists a function $N(t,x)$ such that: |
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351 | \begin{equation} |
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352 | H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x) |
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353 | \end{equation} |
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354 | \begin{equation} |
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355 | \forall t\ ,\ \ \ N(t,x)\ \ \ \ \ |
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356 | {\rm is\ convex\ with\ respect\ to}\ \ x |
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357 | \end{equation} |
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358 | \begin{equation} |
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359 | N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \ |
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360 | {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty |
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361 | \end{equation} |
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362 | \begin{equation} |
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363 | \exists c\in \bbbr\ :\ \ \ H (t,x) \le |
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364 | \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ . |
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365 | \end{equation} |
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366 | |
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367 | If $A_{\infty} (t) = a_{\infty} I$ and |
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368 | $B_{\infty} (t) = b_{\infty} I$, with |
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369 | $a_{\infty} \le b_{\infty} \in \bbbr$, |
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370 | we shall say that $H$ is |
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371 | $\left(a_{\infty},b_{\infty}\right)$-subquadratic |
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372 | at infinity. As an example, the function |
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373 | $\left\|x\right\|^{\alpha}$, with |
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374 | $1\le \alpha < 2$, is $(0,\varepsilon )$-subquadratic at infinity |
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375 | for every $\varepsilon > 0$. Similarly, the Hamiltonian |
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376 | \begin{equation} |
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377 | H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha} |
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378 | \end{equation} |
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379 | is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$. |
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380 | Note that, if $k<0$, it is not convex. |
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381 | \end{definition} |
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382 | % |
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383 | |
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384 | \paragraph{Notes and Comments.} |
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385 | The first results on subharmonics were |
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386 | obtained by Rabinowitz in \cite{rab}, who showed the existence of |
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387 | infinitely many subharmonics both in the subquadratic and superquadratic |
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388 | case, with suitable growth conditions on $H'$. Again the duality |
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389 | approach enabled Clarke and Ekeland in \cite{clar:eke:2} to treat the |
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390 | same problem in the convex-subquadratic case, with growth conditions on |
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391 | $H$ only. |
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392 | |
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393 | Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar}) |
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394 | have obtained lower bound on the number of subharmonics of period $kT$, |
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395 | based on symmetry considerations and on pinching estimates, as in |
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396 | Sect.~5.2 of this article. |
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397 | |
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398 | % |
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399 | % ---- Bibliography ---- |
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400 | % |
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401 | \begin{thebibliography}{5} |
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402 | % |
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403 | \bibitem {clar:eke} |
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404 | Clarke, F., Ekeland, I.: |
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405 | Nonlinear oscillations and |
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406 | boundary-value problems for Hamiltonian systems. |
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407 | Arch. Rat. Mech. Anal. {\bf 78} (1982) 315--333 |
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408 | % |
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409 | \bibitem {clar:eke:2} |
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410 | Clarke, F., Ekeland, I.: |
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411 | Solutions p\'{e}riodiques, du |
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412 | p\'{e}riode donn\'{e}e, des \'{e}quations hamiltoniennes. |
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413 | Note CRAS Paris {\bf 287} (1978) 1013--1015 |
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414 | % |
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415 | \bibitem {mich:tar} |
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416 | Michalek, R., Tarantello, G.: |
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417 | Subharmonic solutions with prescribed minimal |
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418 | period for nonautonomous Hamiltonian systems. |
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419 | J. Diff. Eq. {\bf 72} (1988) 28--55 |
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420 | % |
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421 | \bibitem {tar} |
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422 | Tarantello, G.: |
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423 | Subharmonic solutions for Hamiltonian |
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424 | systems via a $\bbbz_{p}$ pseudoindex theory. |
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425 | Annali di Matematica Pura (to appear) |
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426 | % |
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427 | \bibitem {rab} |
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428 | Rabinowitz, P.: |
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429 | On subharmonic solutions of a Hamiltonian system. |
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430 | Comm. Pure Appl. Math. {\bf 33} (1980) 609--633 |
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431 | \end{thebibliography} |
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432 | % |
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433 | \end{document} |
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