The evolution of the dynamical variables  is given by
ẋ = ax

ṗ = -p

where a is a constant. The trajectory in (x, p) space for -1 < a < 0 is best described by

Correct option is A

Given:

• dx/dt = ax,
• dp/dt = -p,
  where "a" is a constant, and -1 < a < 0. Determine the trajectory in the (x, p) space.

Solution:

1. Behavior of x(t):

   • The equation dx/dt = ax implies that x changes exponentially over time.
   • Since a is negative (-1 < a < 0), x decays exponentially with time, approaching zero as t → ∞.

2. Behavior of p(t):

   • The equation dp/dt = -p describes exponential decay for p as well.
   • This means that p(t) also approaches zero as t → ∞.

3. Coupled dynamics:

   • The coupled system implies that as both x and p decay, the relationship between the two will result in trajectories moving directly toward the origin without spiraling.
   • Specifically, the trajectories in the (x, p) space will converge directly toward the origin, creating straight lines that represent exponential decay in both variables.

4. Identify the correct trajectory:

   • The correct depiction of this behavior is shown in option (a), where all trajectories point directly toward the origin.

Conclusion:
The correct answer is (a). The trajectories converge directly to the origin without spiraling.