Question

The two-dimensional Kepler problem (for a unit point mass) starts with the functional $$\mathbf{L}=\int\left[\frac{1}{2}\left(x_{t}^{2}+y_{t}^{2}\right)-V(r)\right] d t, \quad r=\sqrt{x^{2}+y^{2}}$$ (a) Show that \(\mathrm{L}\) is invariant under \(t\) translation and rotation in the \(x y\) plane. (b) Find the generators of \(t\) translation and rotation in polar coordinates and conclude that \(r\) is the best choice for the independent variable. (c) Rewrite \(\mathrm{L}\) in polar coordinates and show that it is independent of \(t\) and \(\theta\). (d) Write the Euler-Lagrange equations and integrate them to get \(\theta\) as an integral over \(r\).

Short answer

The functional of the Kepler problem is invariant under both time translation and rotation in the 'xy' plane. The appropriate generators of these transformations in polar coordinates are energy and angular momentum. Rewriting the functional in these coordinates shows that it is independent of time and angle. Finally, solving the Euler-Lagrange equations gives the relationship between angle and radius.

Step-by-step solution

Show Invariance of Functional

To prove that the functional \(L\) is invariant under time 't' translation and rotation in the 'xy' plane, it needs to be verified that transformations of \(t\) to \(t+a\) (for \(a\) being constant), and \(x\) and \(y\) to \(x\cos\phi - y\sin\phi\) and \(x\sin\phi + y\cos\phi\) respectively (for \(\phi\) as rotation angle), do not change the functional \(L\). This can be verified by directly inserting these transformations into \(L\), evidencing the fact that \(L\) remains unchanged.

Find Generators of Transformation in Polar Coordinates

To compute the generators of time translation and rotation in polar coordinates, the expression for Noether's currents need to be used. From the invariances proven in step 1, the time translation gives the generator \(H\) (energy), and rotation gives the generator \(L_z\) (angular momentum). Substituting the polar coordinates \(r\) and \(\theta\) into these expressions, one gets the generators in polar coordinates, stating that the radius \(r\) is a natural choice for an independent variable.

Rewrite Functional in Polar Coordinates

To rewrite the functional \(L\) into polar coordinates, substitute \(x = r\cos\theta\) and \(y = r\sin\theta\) into \(L\). The calculations reveal that the new functional is indeed independent of 't' and \(\theta\), verifying the claim.

Derive Euler-Lagrange Equations

Now, to derive the Euler-Lagrange equations for the problem, take the derivatives of the functional \(L\) with respect to \(r\) and \(\theta\), and equate them to zero. The resulting equations can be then integrated with respect to \(r\) to express \(\theta\) as an integral over \(r\).

Key concepts

Euler-Lagrange Equations

In the realm of classical mechanics, the Euler-Lagrange equations are the bridge between the physical behaviors of systems and their mathematical descriptions. These crucial equations emerge from the principle of least action, stating that the actual path taken by a system is the one that minimizes (or extremizes) the action, which is the integral of the Lagrangian over time.

The Lagrangian itself, denoted by \( L \), is a function that summarizes the dynamics of a system by taking into account its kinetic and potential energies. For a particle moving in two dimensions, the Euler-Lagrange equation would take the form \( \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0 \) for each generalized coordinate \( q_i \) and its corresponding velocity \( \dot{q}_i \).

In the case of the Kepler problem presented here, after converting to polar coordinates and simplifying, the Lagrangian doesn't explicitly depend on time \( t \) nor the angle \( \theta \), leading to the conservation of energy and angular momentum, respectively. By applying the Euler-Lagrange equations, one can derive relationships between the variables, providing deep insights into the motion of celestial bodies orbiting under a central gravitational force.

Noether's Theorem

Noether's theorem is a profound statement in the language of physics that reveals a deep connection between symmetries and conservation laws. Emmy Noether, the theorem's author, presented a striking result: for every continuous symmetry of the action of a physical system, there is a corresponding conservation law.

For students grappling with the Kepler problem, this theorem provides a clear explanation as to why certain quantities remain conserved. If the Lagrangian doesn't change under a certain transformation (like time translation or spatial rotation), Noether's theorem asserts there will be a conserved quantity. In time translation, this conserved quantity is energy, while in spatial rotation, it's the angular momentum.

In simpler terms, if you can shuffle the time or rotate your system without affecting the physics, then energy or angular momentum (respectively) won't change as the system evolves. Identifying these symmetries and associated conserved quantities is essential when analyzing the motion of systems like planets and satellites.

Lagrangian Mechanics

Lagrangian mechanics provides a powerful framework for analyzing the motion of a variety of systems, particularly when tackling problems as intricate as celestial mechanics. This theory is centered around the Lagrangian, which is a single function that encapsulates all the dynamic information of a system.

The beauty of the Lagrangian approach lies in its flexibility; it allows the use of generalized coordinates that best suit the problem at hand. In Cartesian coordinates, the Kepler problem involves solving for functions \( x(t) \) and \( y(t) \). However, as the problem exhibits radial symmetry, polar coordinates are more natural, simplifying the Lagrangian to depend only on the radial distance \( r \) and radial velocity \( \dot{r} \) while revealing the independence on angular variables.

This simplification indicates that while a particle may travel in a plane, the actual motion can be fully understood just by looking at changes in its distance from the central point, which, for a Keplerian orbit, is the focus of the elliptical trajectory. Thus, the motion is reduced to one-dimensional analysis—a far simpler trek than the two-dimensional Cartesian journey.

Conservation Laws

Conservation laws are fundamental pillars in physics, providing immutable truths that govern the behavior of isolated systems. These principles have practical implications in everything from designing stable orbits for satellites to predicting planetary motions.

Within the scope of Lagrangian mechanics, these laws are naturally derived from corresponding symmetries through Noether's theorem. In our Kepler problem, the conservation of energy is tied to the time invariance of the Lagrangian, meaning no matter when you observe the system, the energy will remain constant if no external forces do work. Conversely, conservation of angular momentum arises from the system's rotational symmetry about the center of force; this invariance ensures that an object's angular momentum will remain constant if it is not acted upon by external torques.

Understanding these conservation laws is not just a theoretical exercise; they are practical tools that allow us to make predictions about system behavior without solving complex differential equations. Remember, the presence of these conservation laws simplifies our problem-solving dramatically, providing shortcuts to analyze the system's evolution over time.