If \(q_{1}, q_{2}, q_{3}\) are orthogonal curvilinear co-ordinates, and the element of length in the \(q_{i}\) direction is \(h_{i} \mathrm{~d} q_{i}\), write down (a) the kinetic energy \(T\) in terms of the generalized velocities \(\dot{q}_{i}\), (b) the generalized momentum \(p_{i}\) and (c) the component \(\boldsymbol{e}_{i} \cdot \boldsymbol{p}\) of the momentum vector \(\boldsymbol{p}\) in the \(q_{i}\) direction. (Here \(\boldsymbol{e}_{i}\) is a unit vector in the direction of increasing \(\left.q_{i}\right)\)

Kinetic energy represents the energy that an object possesses due to its motion. In classical mechanics, we're accustomed to expressing kinetic energy in terms of Cartesian coordinates. However, many physical problems lend themselves more naturally to descriptions in different coordinate systems, such as cylindrical, spherical, or any set of orthogonal curvilinear coordinates.

In these alternative coordinate systems, the expression for kinetic energy involves 'generalized velocities' which are the time derivatives of the generalized coordinates. Instead of using plain \(x, y, z\) coordinates, we use \(q_1, q_2, q_3\) and their respective velocities \(\dot{q}_1, \dot{q}_2, \dot{q}_3\). These generalized velocities describe how quickly the position of the object is changing within the curvilinear framework.

To find the kinetic energy in terms of these generalized velocities, we consider each component's contribution. The mass \(m\) of the particle and the scale factors \(h_i\) for each coordinate direction, which account for the non-uniformity and curvature of the coordinates, are critical. The total kinetic energy \(T\) is then given by the sum of the kinetic energies in each of the generalized coordinates:
\[T = \frac{1}{2} m (\dot{q}_1^2 h_1^2 + \dot{q}_2^2 h_2^2 + \dot{q}_3^2 h_3^2).\]
This formula allows us to calculate the kinetic energy of a particle moving in a space defined by any set of orthogonal curvilinear coordinates.

Momentum, fundamentally, is a quantity of motion. In Cartesian coordinates, we think of it as mass times velocity. But when dealing with curvilinear coordinates, the concept of generalized momentum becomes essential, especially in the field of analytical mechanics.

Generalized momentum is derived from the Lagrangian formulation of mechanics and is represented by the symbol \(p_i\). It is defined as the derivative of the kinetic energy \(T\) with respect to the corresponding generalized velocity \(\dot{q}_i\), for each generalized coordinate \(q_i\).
\[p_i = \frac{\partial T}{\partial \dot{q}_i} = m h_i^2 \dot{q}_i\]
Here, \(h_i\) are the scale factors that adjust for the dimensions and distortions introduced by different coordinate systems. Generalized momentum is a powerful concept as it blends the geometric properties of motion (mediated through the scale factors) with the physical properties, encapsulating the motion of a particle in a comprehensive mathematical term.

In an orthogonal curvilinear coordinate system, the momentum vector of a particle is intrinsically linked with the nature of the coordinate system itself. This vector can be decomposed into components that align with the coordinate axes represented by the unit vectors \(\boldsymbol{e}_i\).

The component of the momentum vector \(\boldsymbol{p}\) in the direction of a specific coordinate \(q_i\) is expressed as \(\boldsymbol{e}_{i} \cdot \boldsymbol{p}\), which is the dot product of the unit vector in that direction with the total momentum vector.
\[\boldsymbol{p} = p_1 \boldsymbol{e}_1 + p_2 \boldsymbol{e}_2 + p_3 \boldsymbol{e}_3\]
However, due to the orthogonality of the coordinate system, the dot product of different unit vectors is zero, leaving us with:
\[\boldsymbol{e}_i \cdot \boldsymbol{p} = p_i \]
Thus, in orthogonal curvilinear coordinates, the component of the momentum vector along a coordinate direction is exactly the generalized momentum \(p_i\) for that coordinate. Physically, this means that for determining the motion or predicting the future state of the particle along a specific curvilinear axis, we only need to consider the component of momentum parallel to that axis. This simplifies calculations in complex systems and is a cornerstone in the study of dynamics in non-Cartesian geometries.