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transformation from rectangular to generalized coordinates (Topic)

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:

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

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

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

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:

$\displaystyle \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:

$\displaystyle \dot{x_i} = \frac{\partial x_i}{\partial q_\alpha} \dot{q_\alpha}$ (2)

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

$\displaystyle \ddot{x_i}= \frac{d}{dt}\left( \frac{\partial x_i}{\partial q_\al... ...ight ) \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha} $

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

$\displaystyle \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 $\ddot{x_i}$:

$\displaystyle \ddot{x_i} = \frac{\partial^2 x_i}{\partial q_\beta \partial q_\a... ..._\beta} \dot{q_\alpha} + \frac{\partial x_i}{\partial q_\alpha} \ddot{q_\alpha}$ (3)

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

References

[1] Kompaneyets, A. "Theoretical physics." Foreign Languages Publishing House, Moscow, 1961.

This entry is a derivative of the Public domain work [1]



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Cross-references: work, domain, theoretical physics, acceleration, Einstein, velocity, functions, position, generalized coordinates, system

This is version 2 of transformation from rectangular to generalized coordinates, born on 2008-07-15, modified 2008-07-15.
Object id is 283, canonical name is TransformationFromRectangularToGeneralizedCoordinates.
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Physics Classification: 45.20.Jj (Lagrangian and Hamiltonian mechanics)
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