We take a system with a total of 3N ≡ n Cartesian coordinates of which ν 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_α (l ≤ α ≤ ν). Since the generalized coordinates completely specify the position of their system, x_i are their unique functions:

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

(1)

In the subsequent derivation we shall often have to perform summations with respect to all the generalized coordinates q_α, 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:

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

(2)

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

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

The Greek symbol over which the summation is performed is deonted by the letter β to avoid confusion with the symbol α, which denotes the summation in the expression for velocity (2). Thus we obtain the desired expression for x_i:

(3)

The first term on the right-hand side contains a double summation with respect to α and β.

0.1 References

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

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