PlanetPhysics/Transformation From Rectangular to Generalized Coordinates

We take a [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|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 [[../CommutativeRingWithUnit/|generalized coordinates]] we denote by $q_{\alpha }$ $(l\le \alpha \le \nu )$. Since the generalized coordinates completely specify the [[../Position/|position]] of their system, $x_i$ are their unique [[../Bijective/|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/|velocity]]. Differentiating the function of many variables $x_i(\dots q_{\alpha })$ with respect to time, we have

$\frac{d{x}_i}{dt}=\sum _{\alpha =1}^{\nu }\frac{\partial x_i}{\partial q_{\alpha }}\frac{d{q}_{\alpha }}{dt}$

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 [[../AlbertEinstein/|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/|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 $\beta$.

References

[1] Kompaneyets, A. "[[../PhysicalMathematics2/|Theoretical physics]]." Foreign Languages Publishing House, Moscow, 1961.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1]