Question

Prove that the sectorial velocity is a first integral of the equation \(\overline{\boldsymbol{r}}=\operatorname{ra}(r)\) for any form of the function a. A force field of the form \(\operatorname{raf}(r)\) is called central. The preceding problem shows why the law of universal gravitation cannot be deduced from Kepler's Second Law: the third faw is needed.

Short answer

In this exercise, we proved that the sectorial velocity is a first integral of the given equation, \(\overline{\boldsymbol{r}}=\operatorname{ra}(r)\), for central force fields with no torque. By converting the equation to polar coordinates, analyzing the radial and angular components, and relating the radial and angular velocities to the radial and angular accelerations, we showed that the sectorial velocity remains constant for any form of the function a(r) when the conditions specified in the exercise are met.

Step-by-step solution

Convert the given equation to polar coordinates

First, let's convert the given equation to polar coordinates. The equation is given by \(\overline{\boldsymbol{r}}=\operatorname{ra}(r)\). In polar coordinates, \(\overline{\boldsymbol{r}} = r\hat{r}\), where \(\hat{r}\) is the radial unit vector. Thus, $$r\hat{r} = ra(r).$$

Determine the radial and angular components

We can write the above equation in component form as follows: $$r\hat{r} = r a_{r}(r)\hat{r} + r a_{\theta}(r)\hat{\theta},$$ where \(a_{r}(r)\) and \(a_{\theta}(r)\) are the radial and angular components of the acceleration a(r) respectively.

Analyze the radial and angular components

Since the force field is central, the angular component of the acceleration is zero (no torque). Therefore, we have \(a_\theta(r) = 0\), and $$r\hat{r} = r a_r(r)\hat{r}.$$

Determine the radial and angular velocities

Now, let's find the radial and angular velocities. The radial velocity is given by the time derivative of the position vector's magnitude, \(v_r = \frac{d}{dt}(r)\). The angular velocity is given by the time derivative of the angle, \(\omega = \frac{d}{dt}(\theta)\).

Relate the radial and angular velocities to the radial and angular accelerations

Since the radial and angular accelerations are the second time derivatives of the position coordinates, we need to differentiate the radial and angular velocities with respect to time. Thus, $$a_r(r) = \frac{d}{dt}(v_r) = \frac{d^2}{dt^2}(r),$$ and $$a_\theta(r) = \frac{d}{dt}(\omega) = \frac{d^2}{dt^2}(\theta).$$

Prove that the sectorial velocity is a first integral

The sectorial velocity is the product of the radial distance and the angular velocity, i.e., \(v_{sec} = r\omega\). Now we'll differentiate this with respect to time to check if it remains constant. $$\frac{d}{dt}(v_{sec}) = \frac{d}{dt}(r\omega) = \frac{d}{dt}(r)\frac{d}{dt}(\theta) + r\frac{d^2}{dt^2}(\theta).$$ From our analysis in Steps 4 and 5, we have \(a_{\theta}(r) = 0\) and \(v_r = \frac{d}{dt}(r)\). Therefore, replacing the terms in the above equation, we get: $$\frac{d}{dt}(v_{sec}) = v_r\omega + ra_{\theta}(r).$$ As \(a_\theta(r) = 0\), we have: $$\frac{d}{dt}(v_{sec}) = v_r\omega.$$ Since \(v_r\) and \(\omega\) are time derivatives, the sectorial velocity \(v_{sec}\) will remain constant for any form of the function a(r) when the conditions specified in the exercise are met, proving that the sectorial velocity is a first integral of the given equation.

Key concepts

Central Force Field

Central force fields play a crucial role in the study of motion as they describe forces that are directed along the line joining a particle and a fixed point, often referred to as the center. In a central force field, the magnitude of the force depends only on the distance from the center, not on the direction. This central nature leads to unique characteristics such as conservation of angular momentum. An example of a central force field is gravitational force, where the force decreases with the square of the distance from the center.

When analyzing movements in such fields, one typically finds that only the radial component of acceleration matters, because the force is always directed towards or away from the center. This simplifies the dynamics significantly because it means that the angular component of acceleration is zero - no torque exists to cause rotation unless additional forces are introduced.

Understanding central force fields is fundamental in explaining key phenomena in physics, such as the orbits of planets and satellites, where gravitational attraction acts as the central force.

Polar Coordinates

Polar coordinates provide a particularly useful way to describe motion in a plane when dealing with central forces. Instead of using traditional Cartesian coordinates (x and y), polar coordinates express the position of a point using a radius and an angle. This is especially advantageous in circular or radial motion.

In polar coordinates, any point in a plane is determined by two parameters:

• The radial distance \(r\), which is the distance from a fixed point (the pole)
• The angular position \(\theta\), which represents the angle between a reference direction and the line joining the point to the pole

By expressing motion with these coordinates, evaluations of central force problems become more convenient. The radial vector \(\hat{r}\) indicates the direction from the origin to the point, while the angular component \(\hat{\theta}\) captures rotation around the origin. Utilizing polar coordinates paves the way for straightforward analyses of circular and elliptical motions, which are prevalent in fields such as astronomy and electromagnetism.

Radial and Angular Components

Dividing vectors into radial and angular components simplifies the analysis of motion under central forces. The radial component focuses on how far an object moves directly in line with the force’s direction, while the angular component deals with motion perpendicular to this line.

When working in polar coordinates, radial components align with the radius vector \(\hat{r}\), capturing the change in distance from the origin. Meanwhile, angular components align with \(\hat{\theta}\), typically indicating the rate of rotation or angular movement around the center.

In our scenario, we determine that the angular component of acceleration \(a_\theta(r)\) is zero due to the nature of central force fields. This results because central forces—and the consequent motion—are radial by definition. This property means that any change in rotation or angular velocity is not directly influenced by this central force. As such, the analysis and computations often require focusing on changes in only the radial components, significantly simplifying further problem-solving.

First Integral

A first integral refers to a conserved quantity in physics, which is often obtained by integrating the equations of motion. In central force problems, identifying these conserved quantities is key to simplifying and solving complex systems.

For circular motion in a central force field, the sectorial velocity \(v_{sec} = r\omega\) represents one such first integral. This quantity, which combines radial distance \(r\) and angular velocity \(\omega\), remains constant over time in the absence of external torques, thanks to the conservation of angular momentum.

From a dynamics perspective, the constancy of the sectorial velocity implies that even as a planet orbits in an elliptical path around a star, it sweeps out equal areas in equal times. This is a statement of Kepler's Second Law. Thus, the sectorial velocity being a first integral signifies its importance in maintaining stability and predictability in orbital systems. Recognizing these first integrals helps physicists and engineers design and control systems with precision, ensuring trajectories behave as expected in gravitational fields.