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78 now consider a conservative force field in one

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Now consider a conservative force field in one dimension described by a potential V ( x ) so that F = - V ( x ). Then ˙ x = y, ˙ y = - 1 m V ( x ) . Equilibrium points occur where V ( x 0 ) = 0 and y 0 = 0 (so they are all on the x -axis). At such a point, J = 0 1 - V ( x 0 ) /m 0 , so T = 0 and Δ = V ( x 0 ) /m . Hence, using the summary table in the previous section, all equilibrium points must be either saddles (if V ( x 0 ) < 0) or centres (if V ( x 0 ) > 0). This analysis agrees with the stability analysis of § 4.2, with the following phase plane diagrams locally (i.e., close to the equilibrium point): We also have the energy equation 1 2 my 2 + V ( x ) = E. This equation, for different values of the constant E , defines the trajectory curves in the ( x, y )-plane. Trajectories for this system are therefore symmetric in y (which enables us to draw the diagrams above, knowing that the directions given by the eigenvectors must also be reflectionally symmetric in the x -axis). On a given trajectory, at any value of x there are just two values of y , one positive and one negative. If a trajectory is bounded then it must have y = 0 at each end, at which points it joins up. Trajectories for a conservative system must therefore be either closed (i.e., they come back to where they started) or unbounded . Example: the Duffing oscillator for a particle of unit mass, ¨ x = x - x 3 . This corresponds to V ( x ) = - 1 2 x 2 + 1 4 x 4 . We will have a saddle at x = 0 and centres at x = ± 1. The phase portrait is as follows: 79
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Example: a simple pendulum of length l (as in § 2.5) has ¨ θ = - g l sin θ. Let y = ˙ θ , so that ˙ θ = y, ˙ y = - g l sin θ. The stable equilibria are at θ = 2 and the unstable ones at θ = (2 n + 1) π ( n Z ), by considering J = 0 1 - ( g/l ) cos θ 0 . The phase diagram has this form: Around stable equilibria, the pendulum oscillates back and forth (e.g., curve A ). This motion is known as libration . If the pendulum has sufficiently large energy then it can instead undergo rotation (e.g., curve C ) where it always has the same sign of ˙ θ . The curves which join the saddle points (e.g., B ) are known as separatrices because they separate the phase plane into regions containing these two different kinds of motion. They correspond physically to the (highly unlikely) motion where the pendulum starts vertically upwards then executes precisely one revolution, ending vertically upwards again. 80
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11.4 Damped Systems Consider a simple pendulum with damping: ¨ θ = - g l sin θ - k ˙ θ (11.3) where k is a small positive constant. Letting y = ˙ θ we have ˙ θ = y, ˙ y = - g l sin θ - ky. The equilibrium points are still where sin θ = 0; we have J = 0 1 - ( g/l ) cos θ - k . At θ = 2 , T = - k and Δ = g/l , corresponding to a stable focus (from the table in § 11.2, assuming that k is small enough that k 2 < 4 g/l ). At θ = (2 n + 1) π , T = - k and Δ = - g/l so we have a saddle. Because this system is not conservative (the force depends on ˙ θ as well as θ , prohibiting the existence of a potential V ( θ )), the solution curves are not symmetric in y and are not closed. The phase portrait is as follows: Note that if we define energy in the same way as for an undamped pendulum, E = 1 2 ml 2 ˙ θ 2 - mgl cos θ, then d E d t =
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