Express the hamiltonian function in elliptic coordinates. Solution. The lines \(\breve{\mathrm{c}}=\) const are ellipses with fociat \(\mathrm{O}_{1}\) and \(\mathrm{O}_{2}\); the lines \(\eta=\) const are hyperbolas with the same foci (Figure 205). They are mutually orthogonal; therefore, $$ d s^{2}=a^{2} d \xi^{2}+b^{2} d \eta^{2} . $$

Elliptic coordinates are a two-dimensional orthogonal coordinate system that generalize polar coordinates. They are based on two fixed points, known as the foci, and can be extremely useful for solving problems that possess a certain elliptical symmetry.

In elliptic coordinates, any point in the plane is represented by two numbers \(\xi\) and \(\eta\). The parameter \(\xi\) is associated with ellipses, while the parameter \(\eta\) corresponds to hyperbolas. Every combination of \(\xi\) and \(\eta\) will intersect at exactly one point in the plane, and the two parameters determine the shape of the ellipses and hyperbolas that are centered on the foci of the coordinate system.

Mathematically, the relationship between Cartesian and elliptic coordinates can be described by the following formulas for a point (x, y):
\[x = a\cosh(\eta)\cos(\xi)\]
\[y = b\sinh(\eta)\sin(\xi)\]
Where \(a\) and \(b\) are parameters that scale the coordinates appropriately. These relations are crucial when transforming expressions of energy from Cartesian to elliptic coordinates.

Kinetic energy, which represents the energy of motion, can be expressed in various coordinate systems. In Cartesian coordinates, it is written as \(T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2)\), where \(m\) is the mass, and \(\dot{x}\) and \(\dot{y}\) denote the velocities in the x and y directions respectively.

However, when one switches to elliptic coordinates, representing the kinetic energy requires changing the velocity components to be expressed in terms of \(\xi\) and \(\eta\). This transformation necessitates the use of the chain rule to relate the derivatives of x and y with respect to time to the derivatives with respect to \(\xi\) and \(\eta\), ultimately leading to a new expression for the kinetic energy in terms of \(\dot{\xi}\) and \(\dot{\eta}\), which are the time derivatives of the elliptic coordinates.

Doing so acknowledges the inherent relation between the velocity components and the shape of the coordinate grid lines—ellipses and hyperbolas governed by \(\xi\) and \(\eta\), respectively. The resulting equation takes into account the possibility that the mass distribution and motion may not be uniform, which can introduce additional complexity into the kinetic energy expression, depending on the specific scenario or potential being analyzed.

Potential energy is the energy associated with the configuration of a system of objects that exert forces on one another, typically arising from the objects' positions within a force field, like gravity or electromagnetism.

In classical mechanics problems, we frequently encounter potential energy in the form of \(V(x, y)\), which depends on the Cartesian coordinates. To translate this into elliptic coordinates, one must substitute the expressions that relate the Cartesian coordinates to elliptic coordinates. The equations \(x = a\cosh(\eta)\cos(\xi)\) and \(y = b\sinh(\eta)\sin(\xi)\) serve as the transformational bridge. By replacing the Cartesian coordinates in the potential energy function with these relations, \(V(x, y)\) is rewritten as \(V(\xi, \eta)\).

This new expression makes it possible to evaluate the potential energy for systems where elliptic coordinates offer a more natural and simplified description, such as when dealing with the gravitational field around two massive bodies - a classic two-body problem in astrophysics. The potential energy in elliptic coordinates takes on a form that neatly encapsulates the geometry of the situation.

Classical mechanics is a branch of physics that deals with the motion of bodies under the action of forces, including kinematic descriptions and dynamic explanations. That is, it looks at how objects move (kinematics) and why they move (dynamics).

The principal equation of motion in classical mechanics is Newton's second law, which states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. However, numerous problems in classical mechanics are often more conveniently solved by using energy methods and principles of conservation. One such concept is the Hamiltonian function, which is central to this discussion.

The Hamiltonian function represents the total energy of a system, which is the sum of its kinetic and potential energies, and is used for formulating the equations of motion. By expressing the Hamiltonian in different coordinate systems, such as elliptic coordinates, it becomes possible to leverage the symmetries and special structures of the problem, leading to more elegant and oftentimes simpler solutions. For students and researchers dealing with scenarios that feature elliptical or hyperbolic symmetries, understanding the Hamiltonian in elliptic coordinates is crucial for mastering classical mechanics.