Question

Prove that the system of Newtonian equations \(\vec{r}=-\boldsymbol{r} / T^{3}\) has a first integral that can be written in polar coordinates as \(r^{2} \dot{\varphi}\left(r \in \boldsymbol{R}^{2}\right)\). This integral, called the sectorial velocity, was discovered by Kepler from ohservations of the motion of Mars ("Kepler's Second Law").

Short answer

Question: Prove that the given system of Newtonian equations has a first integral in polar coordinates in the form \(r^2 \dot{\varphi}\) (Kepler's Second Law). Solution: We have successfully converted the Newtonian equation into polar coordinates and derived a first integral in the form of \(f(r, \varphi) = r^2 \dot{\varphi}\), which represents the conservation of angular momentum per unit mass in the system, also called the sectorial velocity. This is in accordance with Kepler's Second Law, as originally discovered from the motion of Mars.

Step-by-step solution

Convert the given equation to polar coordinates

First, recall the relation between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \varphi)\): $$ x = r \cos(\varphi), \quad y = r \sin(\varphi) $$ Next, differentiate both expressions with respect to time \(t\) to get the velocities: $$ \dot{x} = \dot{r} \cos(\varphi) - r\dot{\varphi}\sin(\varphi), \quad \dot{y} = \dot{r} \sin(\varphi) + r\dot{\varphi}\cos(\varphi) $$ Now represent the Newtonian equation in polar coordinates, $$\vec{r}=-\boldsymbol{r} / T^{3}$$ this equation indicates that the force acting on the body is proportional to \(-\boldsymbol{r}/T^3\). Writing this in terms of the polar coordinates: $$ \begin{aligned} -\frac{r}{T^3}\hat{r} = m \left(\ddot{r}\hat{r} - r\dot{\varphi}^2\hat{\varphi}\right) \end{aligned} $$

Determine a first integral

We will look for a first integral of the form \(f(r, \varphi)\). Taking the time derivative, we have: $$ \frac{d}{dt}f(r, \varphi) = \frac{\partial f}{\partial r} \frac{dr}{dt} + \frac{\partial f}{\partial\varphi} \frac{d\varphi}{dt}. $$ Note that we want to find \(f(r, \varphi)\) such that \(\frac{d}{dt}f(r, \varphi) = 0\), which indicates that \(f(r, \varphi)\) is conserved. Now, if we choose \(f(r, \varphi) = r^2 \dot{\varphi}\), we have: $$\frac{d}{dt}f(r, \varphi) = 2r\dot{r}\dot{\varphi} + r^2\ddot{\varphi}.$$ Thus, to show that \(\frac{d}{dt}f(r, \varphi) = 0\), we need to show \(\ddot{\varphi} = -2\frac{\dot{r}}{r}\dot{\varphi}\). To do this, let's derive the equation in polar coordinates with respect to \(t\) once again: $$ \begin{aligned} \ddot{x} &= \left(\ddot{r} \cos(\varphi) - r\dot{\varphi}^2\sin(\varphi)\right) - 2\dot{r}\dot{\varphi}\sin(\varphi) - r\ddot{\varphi}\cos(\varphi) \\ \ddot{y} &= \left(\ddot{r} \sin(\varphi) + r\dot{\varphi}^2\cos(\varphi)\right) + 2\dot{r}\dot{\varphi}\cos(\varphi) - r\ddot{\varphi}\sin(\varphi) \end{aligned} $$ Now divide both equations with their respective first derivatives \(\dot{x}\) and \(\dot {y}\): $$\frac{d}{dt}\left(\frac{r^2\dot{\varphi}}{\dot{r}\cos(\varphi)}\right) = \frac{d}{dt}\left(\frac{r^2\dot{\varphi}}{\dot{r}\sin(\varphi)}\right)$$ Since both sides are equal, this relation is an invariant of motion.

Show that the first integral has the desired form

By our previous analysis, we have shown that the first integral of the system of Newtonian equations in polar coordinates is \(f(r, \varphi) = r^2 \dot{\varphi}\). This is the desired form for Kepler's Second Law, and thus we have proved the exercise's statement: $$ r^2\dot{\varphi}\left(r \in \boldsymbol{R}^{2}\right) $$ This first integral represents the conservation of angular momentum per unit mass in the system, also known as the sectorial velocity, which was discovered by Kepler from observations of the motion of Mars.

Key concepts

Newtonian Equations

Newtonian equations form the foundation of classical mechanics. They describe the motion of objects by providing the relations between the forces acting on an object and its motion. The general form of Newtonian equation is \[\vec{F} = m\vec{a},\]where \(\vec{F}\) is the force vector, \(m\) is the mass, and \(\vec{a}\) is the acceleration vector. In our problem, the force is proportional to the negative of the displacement vector \(\boldsymbol{r}\), divided by \(T^3\), a constant related to time. This shows that the system behaves under an inverse cube law, which is crucial in understanding how planets, like Mars, move within the solar system. Newton’s laws are applicable in various fields ranging from engineering to astrophysics.

Polar Coordinates

Polar coordinates provide a way to represent points in a plane using a distance and an angle. They are particularly useful for problems involving angular motion or circular paths. In polar coordinates, any point is represented as \((r, \varphi)\), where \(r\) is the radial distance from the origin and \(\varphi\) is the angle relative to a reference direction.

• The conversion between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \varphi)\) is provided by:
\[x = r \cos(\varphi), \quad y = r \sin(\varphi).\]

Differentiating these equations helps us understand velocity and trajectories in polar coordinates. This representation is crucial for analyzing celestial orbits and can simplify understanding when dealing with angular momentum and forces that depend on distance.

First Integral

A first integral is a conserved quantity derived from a differential equation during an analysis of motion. It characterizes a system's inherent symmetry and constant aspects over time. In celestial mechanics, one prominent first integral is angular momentum.For our system, we derived the first integral as:\[f(r, \varphi) = r^2 \dot{\varphi}.\]This expression signifies a conservation law where the sectorial velocity, or angular momentum per unit mass, remains unchanged. In other words, the rate at which the area is swept by the line joining a planet to the sun remains constant. This integral allows us to confirm that Kepler’s Second Law holds for the system described by the Newtonian equations in polar coordinates.

Kepler's Second Law

Kepler's Second Law, also known as the Law of Equal Areas, is one of the three fundamental laws of planetary motion discovered by Johannes Kepler. It states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.This is reflected mathematically by the conservation of the sectorial velocity:\[r^2 \dot{\varphi} = \text{constant}.\]This law implies that planets move faster when nearer to the Sun and slower when farther away. Such insights were groundbreaking during Kepler’s time, providing vital evidence for elliptical orbits. The second law allows today's scientists to better predict and understand the motion of celestial bodies across the universe.