Orthogonal Trajectories and Bernoulli's Equation An orthogonal trajectory is a trajectory whose direction of motion is perpendicular to the direction of a v...
Orthogonal Trajectories and Bernoulli's Equation
An orthogonal trajectory is a trajectory whose direction of motion is perpendicular to the direction of a vector field at any point in the trajectory. These trajectories are often found in applications related to systems of differential equations, such as those used to model fluid flow and heat conduction.
The equation governing the motion of an object subject to a force field is known as Bernoulli's equation:
where:
(\vec{v}) is the velocity vector
(\vec{x}) is the position vector
(f(\vec{x})) is the potential function
(\nabla) is the gradient operator
In this equation, (\nabla f(\vec{x})) represents the negative gradient of the potential function. This equation describes the direction of the object's motion and provides an implicit description of the trajectory.
An orthogonal trajectory is a solution to Bernoulli's equation if the potential function is constant along the trajectory. In other words, the potential is constant, and the object's motion is purely radial.
Examples:
In the context of fluid flow, orthogonal trajectories correspond to the flow patterns of fluids at rest.
In the context of heat conduction, orthogonal trajectories correspond to the directions of heat flow.
In the context of a pendulum, orthogonal trajectories correspond to the directions of motion of the pendulum bob