Max | Min | $ S_1 $ | $ S_2 $ | $ S_3 $ | NCP |
1 | 1 | 0 | 0 | 0 | 2 |
2 | 2 | 2 | 0 | 0 | 6 |
3 | 3 | 4 | 0 | 0 | 10 |
3 | 3 | 0 | 2 | 0 | 8 |
4 | 4 | 6 | 0 | 0 | 14 |
4 | 4 | 2 | 2 | 0 | 12 |
4 | 4 | 0 | 0 | 2 | 10 |
Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.
Citation: |
Figure 2. Numerical integration of the motion generated by the potential (10) with the choice $ {\beta} = 1 $ for initial conditions near to normal modes. In all cases, initial data correspond to zero speed and position at $ r = 1 $ along eigenvectors, with an offset of 0.001 from the latter. The simulation show the outcome, for $ t \in (0,100) $, for initial data: (a) near the eigenvector $ \theta = 0 $, (b) near the eigenvector $ \theta = \pi $, (c) near the eigenvector $ \theta = \pi/4 $, (d) near the eigenvector $ \theta = - \pi/4 $
Table 1.
Different possibilities for the number and type of critical points in the case of a cubic potential in three dimensions; here "Max"and "Min" represent the number of maxima and minima, while "
Max | Min | $ S_1 $ | $ S_2 $ | $ S_3 $ | NCP |
1 | 1 | 0 | 0 | 0 | 2 |
2 | 2 | 2 | 0 | 0 | 6 |
3 | 3 | 4 | 0 | 0 | 10 |
3 | 3 | 0 | 2 | 0 | 8 |
4 | 4 | 6 | 0 | 0 | 14 |
4 | 4 | 2 | 2 | 0 | 12 |
4 | 4 | 0 | 0 | 2 | 10 |
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The potential
Numerical integration of the motion generated by the potential (10) with the choice
The potential