*The motion of a particle in a plane may be described in terms of elliptic co-ordinates \(\lambda, \theta\) defined by $$ x=c \cosh \lambda \cos \theta, \quad y=c \sinh \lambda \sin \theta, \quad(\lambda \geq 0,0 \leq \theta \leq 2 \pi) $$ where \(c\) is a positive constant. Show that the kinetic energy function may be written $$ T=\frac{1}{2} m c^{2}\left(\cosh ^{2} \lambda-\cos ^{2} \theta\right)\left(\dot{\lambda}^{2}+\dot{\theta}^{2}\right) $$ Hence write down the equations of motion.

When dealing with the motion of a particle, one often uses curvilinear coordinates rather than the standard Cartesian coordinates. This is especially helpful in systems where symmetries or constraints make certain coordinate systems more natural, like the elliptic coordinates used for a particle's motion in a plane.

In curvilinear coordinates, expressing kinetic energy can seem more complex at first glance. Kinetic energy, which is the energy due to motion, is generally given by the equation \( T = \frac{1}{2}mv^2 \), where \( m \) is the mass of the particle and \( v \) is its velocity. However, in curvilinear coordinates, velocities are not always directly related to derivatives with respect to time because the basis vectors of the coordinate system can change direction and magnitude.

To properly express kinetic energy in these coordinates, as shown in the original exercise, we derive the velocities in terms of the curvilinear coordinates and their time derivatives. We then plug these velocities into our typical kinetic energy expression, resulting in a formula that accounts for the nature of the coordinate system being used. In the given elliptical coordinate system, one obtains \( T=\frac{1}{2} m c^{2}(\cosh ^{2} \lambda -\cos ^{2} \theta)(\dot{\lambda}^{2}+\dot{\theta}^{2}) \), where \( c \) is a constant characteristic of the system, \( \lambda \) and \( \theta \) are the elliptic coordinates, and dots represent time derivatives.

The Euler-Lagrange equations are foundational in the field of classical mechanics. These equations emerge from the principle of least action, which states that the path taken by a system between two states is the one for which the action integral is minimized.

In the scenario where the kinetic energy function is known, the Lagrangian \( L \) can typically be defined as the difference between kinetic and potential energy, \( L = T - V \). With no potential energy explicitly mentioned in the exercise, the Lagrangian is reduced to the kinetic energy function. The Euler-Lagrange equations then provide a powerful method to derive the equations of motion from the Lagrangian.

The general form of the Euler-Lagrange equation for a coordinate \( q \) is \( \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0 \). By calculating the partial derivatives of the Lagrangian with respect to each coordinate and its time derivative, and then plugging these into the Euler-Lagrange equation, one can derive the equations governing the dynamics of the system.

The equations of motion describe how a system evolves over time following the laws of physics. They are the final goal when analyzing a physical system as they tell us everything about the system's future behavior, provided we know the initial conditions.

Using the kinetic energy function and following the procedure laid out by the Euler-Lagrange equations, the equations of motion for a particle in elliptic coordinates can be derived. By calculating the derivatives as per the Euler-Lagrange equations, we're led to the equations that govern the evolution of \( \lambda \) and \( \theta \) over time.

These equations are vital as they enable prediction of future states of the system. For the given problem, the equations of motion obtained tell us how the coordinates \( \lambda \) and \( \theta \) change with time, which can be used to compute the trajectory of the particle in the plane. In classical mechanics, solving the equations of motion often gives complete information about the physical state of a system at any given time.