Question

Sketch the phase portrait and describe the dynamical trajectories for a particle of mass \(m\) moving under the following forces (1) \(\quad F=m\left(\frac{3}{2} x^{2}-x\right)\) (2) \(\quad F=2 m x\left(x^{2}-1\right)\) (3) \(\quad F=-m \sin x\) (4) \(\quad F=-m x \dot{x} /(1+\dot{x})\) In the fourth case, what at happens as the trajectories approach the line \(v=-1\) in the phase plane?

Short answer

Based on the analysis of the phase portraits and dynamical trajectories for a particle of mass m under different forces, describe the main equilibrium points and behaviors of each case, as well as the specific behavior for the fourth case as trajectories approach the line \(v=-1\) in the phase plane.

Step-by-step solution

Determine the equations of motion

According to Newton's second law, \(F = m \ddot{x}\). For each force, we will replace the force expressions to determine the equation of motion. (1) \(F = m\left(\frac{3}{2} x^{2}-x\right)\) \(\Rightarrow m \ddot{x} = m\left(\frac{3}{2} x^{2}-x\right)\) \(\Rightarrow \ddot{x} = \frac{3}{2} x^{2}-x\) (2) \(F = 2 m x\left(x^{2}-1\right)\) \(\Rightarrow m \ddot{x} = 2 m x\left(x^{2}-1\right)\) \(\Rightarrow \ddot{x} = 2 x\left(x^{2}-1\right)\) (3) \(F = -m \sin x\) \(\Rightarrow m \ddot{x} = -m \sin x\) \(\Rightarrow \ddot{x} = - \sin x\) (4) \(F = -m x \dot{x} /(1+\dot{x})\) \(\Rightarrow m \ddot{x} = -m x \dot{x} /(1+\dot{x})\) \(\Rightarrow \ddot{x} = - x \dot{x} /(1+\dot{x})\) Now, we have the equations of motion for each case.

Investigating Phase Portraits and Trajectories for each case

To analyze the phase portraits and trajectories, we should look at possible fixed points (where \(\dot{x} = 0\) and \(\ddot{x} = 0\)) and the behavior of the trajectories as \(x\) changes. Each given force will be analyzed separately: (1) Equilibrium Points: \(x=0\) and \(x=2/3\). For \(x<0\), \(m\ddot x>0\) so motion is oscillatory, so \(x=2/3\) is a stable equilibrium point, and \(x=0\) is an unstable equilibrium point due to the negative slope. (2) Equilibrium Points: \(x=-1, x=0, x=1\). For \(x<-1\), \(m\ddot x>0\), for \(x \in(-1, 0)\), \(m\ddot x<0\), and for \(1>x>0\), \(m\ddot x>0\) and for \(x>1\), \(m\ddot x>0\). Thus, we have two stable equilibrium points at \(x=-1\) and \(x=1\) and one unstable equilibrium point at \(x=0\). (3) Equilibrium Points: multiple equilibria at \(x = n\pi\) where \(n\) is any integer. For \(x \in (2n\pi, (2n+1)\pi)\), we have \(m\ddot x<0\), and for \(x \in ((2n-1)\pi, 2n\pi)\), we have \(m\ddot x>0\). Therefore, odd multiples of \(\pi\) are stable equilibrium points while even multiples are unstable. (4) Equilibrium Points: The only equilibrium point is at \(x=0\). Note that for \(x > 0\), we have a positive motion, while for \(x < 0\), we have a negative motion. Behavior changes near the line \(v=-1\) in the phase plane which is the next step.

Analyzing the Fourth Case as Trajectories Approach the Line \(v=-1\) in the Phase Plane

According to the given force, \(F=-m x \dot{x} /(1+\dot{x})\), when \(\dot{x}=-1\): \(F=-m x \dot{x} / (1 - 1)\) which leads to a division by zero situation. As the trajectories approach the line \(v=-1\) in the phase plane, we will have an infinite force, indicating that motion in the line \(v=-1\) is not physically possible. In practice, the motion will be prevented, and the trajectories will never reach the line \(v=-1\) in the phase plane.

Key concepts

Phase Portrait

A phase portrait is a visual representation of the trajectories of a dynamical system in the phase plane. The phase plane is a graphical space in which each point, often denoted as (x, v), represents the state of the system at a particular time. Here, **x** is the position, and **v** is the velocity of the system, which can also be considered as the first derivative of the position.

Drawing a phase portrait allows us to visualize how the system evolves over time without needing to know the specific time evolution of each particle.

• **Flow Patterns**: Arrows or lines in the phase plane indicate the direction in which the motion progresses as time advances.
• **Equilibrium Points**: These are typically labeled in the phase portrait and mark stable or unstable states of the system (where velocity (\(\dot{x}\)) is zero).
• The system's dynamics can be clearly seen, helping us to predict future motion or behavior.

Phase portraits provide essential insights into the behavior of complex systems and serve as a crucial tool in analytical dynamics.

Equilibrium Points

Equilibrium points in a dynamical system are states where the system can be at rest if undisturbed. At these points, the net force acting on the particle is zero, resulting in no acceleration.

In our exercise, identifying equilibrium points helps us understand when and where a particle will cease to accelerate, essentially marking the static points of the system. Furthermore, the stability of these points indicates how the system will respond to small deviations:

• **Stable Equilibrium**: If perturbed, the system will return to this point. These are attractive, like a marble in a bowl.
• **Unstable Equilibrium**: A small disturbance will lead the system to move away, analogous to a marble atop a hill.

The analysis of equilibrium points provides an understanding of both static and dynamic aspects of particle behavior in various force fields, as seen in the solutions.

Newton's Second Law

Newton's Second Law is a fundamental principle in classical mechanics stating that a force acting on a mass (\(m\)) causes it to accelerate (\(a\)) according to the equation \(F = ma\). This principle allows us to determine how a system evolves over time by relating force with the resultant acceleration.

In practice, Newton's second law helps us derive the equations of motion for different force scenarios by considering each specific force expression:

• It gives a mathematical framework to analyze and predict the motion of objects subject to various forces.
• By substituting the different force expressions into \(F = m \ddot{x}\), we obtain second-order differential equations describing the system's dynamics.

Understanding Newton's Second Law within this context helps in constructing a clearer picture of how forces influence motion and can be analyzed through phase portraits and equilibrium points.

Equations of Motion

Equations of motion describe the mathematical relationships detailing how the position of a particle changes over time under the influence of forces. These relations are derived using Newton's laws and provide insights into future positions and velocities of particles.

In the exercise, we have several force expressions, which when applied to Newton's Second Law, provide us with several equations:

• The equations help in predicting how the particle will continue to move by involving \(x\) (position), \(\dot{x}\) (velocity), and \(\ddot{x}\) (acceleration).
• Through these equations, one can understand the impact of different forces, oscillations, stability, and other motion aspects.
• By solving these equations, we derive phase portraits and equilibrium points, giving complete dynamical information regarding the particle's behavior.

Thus, equations of motion are central to understanding and predicting the mechanics of any system studied in analytical dynamics.