PlanetPhysics/Transformation From Rectangular to Generalized Coordinates

We take a system with a total of $$3N \equiv n$$ Cartesian coordinates of which $$\nu$$ are independent. We denote Cartesian coordinates by the same letter $$x_i$$, understanding by this symbol all the coordinates $$x, y, z$$; this means that $$i$$ varies from $$1$$ to $$3N$$, that is, from $$1$$ to $$n$$. The generalized coordinates we denote by $$q_\alpha$$ $$(l \le \alpha \le \nu )$$. Since the generalized coordinates completely specify the position of their system, $$x_i$$ are their unique functions:

$$x_i = x_i (q_1, q_2,\dots q_\alpha,\dots,q_v)$$

From this it is easy to obtain an expression for the Cartesian components of velocity. Differentiating the function of many variables $$x_i(\dots q_\alpha)$$ with respect to time, we have

$$ \frac{ dx_i}{dt} = \sum_{\alpha=1}^{\nu} \frac{\partial x_i}{\partial q_\alpha} \frac{d q_\alpha}{dt} $$

In the subsequent derivation we shall often have to perform summations with respect to all the generalized coordinates $$q_\alpha$$, and double and triple sums will be encountered. In order to save space we will use Einstein summation.

The total derivative with respect to time is usually denoted by a dot over the corresponding variable:

$$ \frac{d x_i}{dt} = \dot{x_i}; \,\,\, \frac{d q_\alpha}{dt} = \dot{q_\alpha} $$

In this notation, the velocity (1) in abbreviated form becomes:

$$ \dot{x_i} = \frac{\partial x_i}{\partial q_\alpha} \dot{q_\alpha} $$

Differentiating this with respect to time again, we obtain an expression for the Cartesian components of acceleration:

$$\ddot{x_i}= \frac{d}{dt}\left( \frac{\partial x_i}{\partial q_\alpha} \right ) \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha} $$

The total derivative in the first term is written as usual:

$$ \frac{d}{dt}\left( \frac{\partial x_i}{\partial q_\alpha} \right ) = \frac{\partial^2 x_i}{\partial q_\beta \partial q_\alpha} \dot{q_\beta} $$

The Greek symbol over which the summation is performed is deonted by the letter $$\beta$$ to avoid confusion with the symbol $$\alpha$$, which denotes the summation in the expression for velocity (2). Thus we obtain the desired expression for $$\ddot{x_i}$$:

$$ \ddot{x_i} = \frac{\partial^2 x_i}{\partial q_\beta \partial q_\alpha} \dot{q_\beta} \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha} $$

The first term on the right-hand side contains a double summation with respect to $$\alpha$$ and $$\beta$$.