Question

Prove that each component of the cross product \([r, \vec{r}]\) is a first integral for motion in any central force field in three-dimensional space ("the law of conservation of angular momentum").

Short answer

Question: Prove that each component of the cross product \([\vec{r}, \dot{\vec{r}}]\) is a first integral for motion in any central force field in three-dimensional space. Answer: To prove this, we need to show that the time derivative of each component of the angular momentum vector \(\vec{L} = \vec{r} \times \dot{\vec{r}}\) is equal to zero. Using the product rule, we find the time derivative of angular momentum as \(\frac{d\vec{L}}{dt} = \vec{r} \times \ddot{\vec{r}}\). Substituting the expression for the force vector and considering central force field is radial, we find that the cross product of the position vector and the force vector is zero. This results in \(\frac{d\vec{L}}{dt} = 0\), proving that each component of the angular momentum vector is a first integral for motion in any central force field in three-dimensional space.

Step-by-step solution

Write the equation for angular momentum in vector form

The angular momentum \(\vec{L}\) is given by the cross product of the position vector \(\vec{r}\) and its time derivative, which represents the linear momentum per unit mass \(\dot{\vec{r}}\). This can be written in vector form as: $$ \vec{L} = \vec{r} \times \dot{\vec{r}} $$

Differentiate each component of angular momentum with respect to time

Find the time derivative of each component of the angular momentum vector. Since \(\vec{L} = (L_x, L_y, L_z)\), we need to calculate \(\frac{dL_x}{dt}\), \(\frac{dL_y}{dt}\), and \(\frac{dL_z}{dt}\). To calculate the time derivative of a cross product, we can use the product rule as follows: $$ \frac{d}{dt}(\vec{r} \times \dot{\vec{r}}) = \frac{d\vec{r}}{dt} \times \dot{\vec{r}} + \vec{r} \times \frac{d\dot{\vec{r}}}{dt} $$ Note that \(\frac{d\vec{r}}{dt} = \dot{\vec{r}}\) and \(\frac{d\dot{\vec{r}}}{dt} = \ddot{\vec{r}}\). The previous equation can be rewritten as: $$ \frac{d\vec{L}}{dt} = \dot{\vec{r}} \times \dot{\vec{r}} + \vec{r} \times \ddot{\vec{r}} $$

Evaluate the cross product of the velocity vector

Since the velocity vector is parallel to itself, the cross product of the velocity vector with itself is zero. Therefore, \(\dot{\vec{r}} \times \dot{\vec{r}} = 0\). Now, rewrite the equation for the time derivative of the angular momentum as: $$ \frac{d\vec{L}}{dt} = \vec{r} \times \ddot{\vec{r}} $$

Substitute the expression for the acceleration vector

According to Newton's Second Law, the acceleration vector can be substituted by the force acting on the particle per unit mass: $$ \ddot{\vec{r}} = \frac{\vec{F}}{m} $$ Where \(m\) is the particle mass and \(\vec{F}\) is the central force field. Since the central force field is radial, it can be written in the form \(\vec{F} = F(r)\hat{r}\). Therefore, we can substitute \(\ddot{\vec{r}}\) with the force per unit mass expression: $$ \frac{d\vec{L}}{dt} = \vec{r} \times \frac{F(r)\hat{r}}{m} $$

Prove that the time derivative of angular momentum is zero

Since the central force field is radial, the force vector is always parallel to the position vector. Therefore, the cross product of the position vector and the force vector is zero as they are parallel: $$ \vec{r} \times \frac{F(r)\hat{r}}{m} = 0 $$ Now, we can conclude that each component of the angular momentum vector is a first integral for motion in any central force field in three-dimensional space, as the time derivative of the angular momentum is zero: $$ \frac{d\vec{L}}{dt} = 0 $$ This property is known as the "law of conservation of angular momentum".

Key concepts

Angular Momentum

Angular momentum is a fundamental concept in physics that describes the rotational motion of objects. It is a vector quantity, meaning it has both magnitude and direction. Angular momentum \( \vec{L} \) for a particle is calculated using the cross product of the particle's position vector \( \vec{r} \) and its linear momentum, which is mass times velocity. This can be expressed mathematically as:

• \( \vec{L} = \vec{r} \times \dot{\vec{r}} \) where \( \dot{\vec{r}} \) is the velocity of the particle.

The concept of angular momentum is essential for understanding how objects behave under rotational motion. It helps explain phenomena such as why planets orbit stars or why a spinning ice skater can change speed by pulling their arms inwards or outwards.
This quantity is conserved in isolated systems, meaning the total angular momentum remains constant if no external torques are acting on it.

Central Force Field

A central force field is a force field where the force acting on a particle is always directed towards a fixed point, usually called the center, and its magnitude only depends on the distance from the center. This means that the force has radial symmetry.
Examples of central force fields include gravitational force, which acts between any two masses, and electrostatic force, acting between charged particles.

• The force \( \vec{F} \) in a central force field can be expressed as: \( \vec{F} = F(r)\hat{r} \)
• Here, \( F(r) \) is a function of the distance \( r \), and \( \hat{r} \) is the unit vector pointing radially outward from the center.

Central force fields are key to understanding planetary motions and satellite orbits, as they provide a simplification where forces depend only on radius and not on directions perpendicular to the radius.

Cross Product

The cross product is a mathematical operation used with vectors in three-dimensional space. It results in another vector that is perpendicular to the plane formed by the original vectors and has a magnitude proportional to the sine of the angle between them.
To compute the cross product of two vectors \( \vec{a} \) and \( \vec{b} \), you can use the formula:

• \( \vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin(\theta)\hat{n} \)
• Where \( |\vec{a}| \) and \( |\vec{b}| \) are the magnitudes of the vectors, \( \theta \) is the angle between them, and \( \hat{n} \) is a unit vector perpendicular to both.

This tool is crucial in physics for calculating angular momentum, torque, and other vector operations. The direction of the resulting vector follows the right-hand rule: if you point your fingers along the first vector and curl them towards the second, your thumb points in the direction of the cross product.

Newton's Second Law

Newton's Second Law is one of the core laws of motion, formulated by Sir Isaac Newton, that describes how the velocity of an object changes when it is subjected to an external force. It states that the force acting on an object is equal to the mass of that object times its acceleration, and is usually expressed as:

• \( \vec{F} = m\ddot{\vec{r}} \)
• Here, \( \vec{F} \) is the force vector, \( m \) is the mass of the object, and \( \ddot{\vec{r}} \) is the acceleration.

In the context of the conservation of angular momentum, Newton's Second Law helps relate the gravitational forces experienced by celestial bodies to their motions. This connection is vital for solving problems involving central force fields, as it allows us to substitute acceleration with the force per unit mass.
By using Newton's Second Law in such problems, we can often determine the behavior of a system over time, providing a foundation for understanding the dynamics of moving objects in a large array of physical contexts.