Question

Two particles, each of mass \(m\), are moving under their mutual gravitational attraction, which is given by the potential \(U=-\gamma m / 2 r\), where \(2 r\) is their separation and \(\gamma\) is a constant. Find the equations of motion in terms of the coordinates \(X, Y, Z, r, \theta, \varphi\), where \(X, Y\), and \(Z\) are the Cartesian coordinates of the centre of mass and \(r, \theta\), and \(\varphi\) are the polar coordinates of one particle relative to the centre of mass.

Short answer

Question: Find the equations of motion for two particles with mass m which are gravitationally attracted to each other, given the potential energy function U = -γm/(2r), in terms of the Cartesian coordinates of the center of mass (X, Y, Z), and polar coordinates for one particle relative to the center of mass (r, θ, φ). Answer: The equations of motion for the two particles with mass m can be written in terms of the center of mass (X, Y, Z) and polar coordinates (r, θ, φ) as follows: For the individual particles: 1. \(\ddot{r_1} = -\frac{\gamma m}{r^2}\) 2. \(\ddot{r_2} = \frac{\gamma m}{r^2}\) For the center of mass: \(\ddot{R} = \ddot{r_1} + \ddot{r_2} = 0\) These equations of motion describe the movement of the two particles under gravitational attraction in terms of their position relative to the center of mass and their polar coordinates.

Step-by-step solution

Rewrite the potential energy in terms of relative coordinates

We have the potential energy function: \(U=-\frac{\gamma m}{2r}\). To write this potential in terms of relative coordinates, we introduce \(r_{1}\) and \(r_{2}\) as the position vectors of the two particles, and the center of mass position vector R. We can write the position vector of one particle relative to the center of mass as: \(r=\frac{r_1-r_2}{2}\). Substitute this in the potential: \(U = -\frac{\gamma m}{2\left[\frac{r_1-r_2}{2}\right]} = -\frac{\gamma m}{r_1 - r_2}\).

Find the forces acting on the particles

To find the forces acting on each particle, we need to take the gradient of the potential energy function with respect to the position vector of each particle: \(F_1 = -\nabla_{r_1}U = -\frac{\gamma m^2}{(r_1 - r_2)^2}\), and \(F_2 = -\nabla_{r_2}U = \frac{\gamma m^2}{(r_1 - r_2)^2}\).

Write down the equations of motion for the particles

Now, we can write down the equations of motion for each of the particles using Newton's second law, \(F = ma\), where \(F\) is the acting force and \(a\) is acceleration. For one of the particles, we have: \(m\ddot{r_1} = -\frac{\gamma m^2}{(r_1 - r_2)^2}\) Similarly for the other particle: \(m\ddot{r_2} = \frac{\gamma m^2}{(r_1 - r_2)^2}\)

Convert individual equations of motion to coordinates relative to the center of mass

Now, it is time to express these equations in terms of the coordinates relative to the center of mass. Recall that we defined \(r=\frac{r_1-r_2}{2}\). Rewrite the equations of motion in terms of \(r\): \(\ddot{r_1} = -\frac{\gamma m}{r^2}\) \(\ddot{r_2} = \frac{\gamma m}{r^2}\) Now express the Cartesian position of the centre of mass, R: \(R = \frac{r_1 + r_2}{2}\) Derive the equation of motion for the center of mass: \(\ddot{R} = \ddot{r_1} + \ddot{r_2} = -\frac{\gamma m}{r^2} + \frac{\gamma m}{r^2} = 0\) This means that the center of mass moves uniformly without acceleration. From here, we can obtain the equations of motion for particles in terms of \(X, Y, Z, r, \theta, \varphi\) coordinates.

Key concepts

Gravitational Attraction

Gravitational attraction is a fundamental force in the universe, governing the motion of objects with mass. According to Newton's Law of Universal Gravitation, every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

In our exercise, the potential energy function given, \( U = -\frac{\gamma m}{2r} \), is derived from the gravitational attraction between two particles of mass \( m \) separated by a distance \( 2r \). This negative sign indicates that gravitational force is attractive and the potential energy decreases as particles come closer to each other. The constant \( \gamma \) is related to the universal gravitational constant.

To analyze the motion of the particles under gravitational attraction, we considered their mutual forces and used the concept of potential energy to derive the equations of motion. The gravitational force is responsible for changing the velocity of the particles as they move towards or away from each other.

Potential Energy

Potential energy, in the context of the gravitational force, is the energy stored by an object due to its position in a gravitational field. In our exercise, the potential energy between two particles is given by \( U=-\frac{\gamma m}{2r} \), where \( \gamma \) is a constant encapsulating the strength of the gravitational pull and \( r \) is the separation between the particles.

This equation shows that potential energy is inversely related to the distance \( r \)—as the distance increases, the potential energy becomes less negative, indicating that the attractive gravitational force becomes weaker. In essence, the less negative the potential energy, the weaker the binding gravitational forces between the particles, and the more energy would be needed for one particle to escape the gravitational pull of the other.

Relative Coordinates

Relative coordinates refer to the positional description of an object concerning another object, instead of an absolute frame of reference. In our problem, the relative coordinate \( r \) denotes the separation between the two particles. Calculations become easier when we reframe the problem in terms of the displacements of each particle relative to their center of mass.

The substitution \( r=\frac{r_1-r_2}{2} \) allows us to simplify the potential energy in terms of this relative distance. By focusing on the relative positions, we can directly analyze the interaction between the two particles without worrying about their individual locations in space. This is particularly useful when dealing with forces, like gravity, that depend on the distance between two objects.

Center of Mass

The center of mass is a point that behaves as if all the mass of a system was concentrated there for the purpose of considering translational motion. In other words, it is the weighted average position of all the masses in a system.

In the given problem, we use the center of mass, \( R \), to express the equations of motion in a way that simplifies the interaction between the two particles. The equation \( R = \frac{r_1 + r_2}{2} \) shows that the center of mass lies exactly at the midpoint between the two particles. The notable result \( \ddot{R} = 0 \) implies that the center of mass of our two-particle system does not accelerate due to their mutual gravitational forces. It moves with a constant velocity, in accordance with the conservation of momentum.

Polar Coordinates

Polar coordinates (\( r, \theta, \varphi \)) are a two-dimensional coordinate system that specifies the position of a point on a plane by the distance from a reference point (the origin) and the angle from a reference direction. In three dimensions, polar coordinates are extended with an additional angle, \( \varphi \) often called the azimuthal angle, to describe an object's position in space.

In this exercise, the exercise improvement advice suggests transforming the problem into polar coordinates to obtain a clearer understanding of the symmetry of the system. We use polar coordinates to describe the position of one particle relative to the center of mass, highlighting the radial and angular components of motion. This coordinate system is especially convenient when dealing with problems where the force is directed towards a point or along a circular path.