Problem 3 - Classical Mechanics - An Undergraduate Text

Question

A particle of mass \(m\) moves under the attractive inverse square field \(\boldsymbol{F}=-\left(m \gamma / r^{2}\right) \widehat{\boldsymbol{r}}\) Show that the equation satisfied by the apsidal distances is $$ 2 E r^{2}+2 \gamma r-L^{2}=0 $$ where \(E\) and \(L\) are the specific total energy and angular momentum of the particle. When \(E<0\), the orbit is known to be an ellipse with \(O\) as a focus. By considering the sum and product of the roots of the above equation, establish the elliptic orbit formulae $$ L^{2}=\gamma b^{2} / a, \quad E=-\gamma / 2 a $$

Step-by-Step Solution

Answer

From the force field and subsequent conservation laws, the equation for the apsidal distances is derived as \(2E r^{2}+2\gamma r-L^{2}=0\). With \(-E\) being the constant term here and corresponding to the semi-major and semi-minor axes of the ellipse, the elliptic orbit formulae \(L^{2}=\gamma b^{2} / a\) and \(E=-\gamma / 2a\) are established by considering their sum and product.

Step 1: Obtain the equation of motion

Start by writing the equation of motion for a particle under a central force: \(\dot{r}^{2}=2 E-\left(\frac{2 \gamma}{r}\right)-\frac{L^{2}}{r^{2}}\) using the conservation of mechanical energy and angular momentum.

Step 2: Derive required equation

Next, differentiate this equation once to have it in terms of \(\dot{r}\) and then solve it for the apsidal distances (\(d\dot{r} / dt=0\)). Simplifying results in \(2Er^2 + 2γr - L^2 = 0\)

Step 3: Calculate roots

With \(E<0\), the above equation is quadratic and its roots provide the apsidal distances, which are the values of the semi-major, a, and semi-minor, b, axes of the ellipse. The roots are given by \(r=\frac{-\gamma \pm \sqrt{\gamma^{2}+2 E L^{2}}}{2 E}\).

Step 4: Establish elliptic orbit formulae

The characteristics of these roots (sum and product) can be used to relate a and b to E, L and γ. By comparing these characteristics, we find the relations \(L^{2}=\gamma b^{2} / a\) and \(E=-\gamma / 2a\), which are the elliptic orbit formulae.

Key Concepts

Inverse Square Law

In central force motion, where a particle is influenced by a force that obeys the inverse square law, the force experienced by the particle is inversely proportional to the square of the distance from the source of the force. This means that if the distance doubles, the force becomes one fourth as strong. In our exercise, the force acting on the particle is described by the equation \(-\left(\frac{m \gamma}{r^2}\right) \widehat{\mathbf{r}}\), showcasing that the force diminishes rapidly as the distance \(r\) increases.

Understanding this concept is crucial because many naturally occurring forces, such as gravitational and electrostatic forces, follow this law. Here, \(\gamma\) represents the force constant associated with the specific central force field, revealing details about its intensity and influence radius.

• Force is directly proportional to an object's mass \(m\) and force constant \(\gamma\).
• The direction of the force is radial, pointing toward the mass creating the field.
• This concept is foundational in deriving essential equations for motion under central force fields.

Energy Conservation

In the realm of central force motion, the principle of energy conservation plays a pivotal role. The total mechanical energy \(E\) of the particle is conserved, meaning it remains constant over time. This energy comprises kinetic energy and potential energy.

The equation of motion, derived from energy conservation, is given by \(\dot{r}^{2}=2 E-\left(\frac{2 \gamma}{r}\right)-\frac{L^{2}}{r^{2}}\), which describes how energy is distributed between kinetic and potential forms. In the context of our problem, different configurations of \(r\) demonstrate how energy shifts between potential energy (dependent on \(r\)) and kinetic energy (dependent on the change of \(r\)).

• Conserving energy allows the setup of the crucial equation for apsidal distances and further understanding orbital mechanics.
• Specific energy \(E\) could be negative, indicating a bound system, like an ellipse.

Angular Momentum

Angular momentum \(L\) is a key concept in describing rotational dynamics for central force motion. It is conserved both in magnitude and direction when no external torque acts on the system. Angular momentum conservation implies that the particle maintains a constant motion pattern, making its path predictable.

Mathematically, this is expressed through \(L = mr^2\dot{\theta}\), where \(r\) is the radial distance and \(\dot{\theta}\) is the rate of angular displacement. In our problem, angular momentum affects the form of the motion equation, translating radial movement into a comprehensive orbital path.

• Conserved angular momentum results in the predictable shape of orbits, often leading to elliptical orbits.
• It simplifies problem-solving by reducing the degrees of freedom in rotational systems.

Elliptic Orbits

Elliptic orbits are a specific result of negative specific energy \(E\) in central force motion. When the particle's energy is less than zero, the resulting trajectory is an ellipse, as given by the relationships \(L^{2}=\gamma b^{2} / a\) and \(E=-\gamma / 2a\). These are derived from the solutions to the quadratic equation equating apsidal distances.

In an elliptic orbit, the central attracting body occupies one of the foci of the ellipse, not the center, emphasizing the non-circular nature of such paths. Parameters \(a\) and \(b\) represent the semi-major and semi-minor axes of the ellipse, respectively, crucial for describing the shape and size of the orbit.

• The semi-major axis \(a\) correlates with the total energy of the system, affecting the orbit's size and period.
• The semi-minor axis \(b\) influences the orbit's width, connected to the conservation of angular momentum.

Elliptic orbits help explain the natural paths of planets, satellites, and other space-bound bodies, governed by gravitational forces.