We take a system with a total of
Cartesian coordinates of
which
are independent. We denote Cartesian coordinates
by the same letter
, understanding by this symbol all the coordinates
; this means that
varies from
to
, that is, from
to
. The generalized coordinates we denote by
. Since the generalized coordinates completely specify the position
of their system,
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
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 , 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 to avoid confusion with the symbol
, which denotes the summation in the expression for velocity (2). Thus we obtain the desired expression for
:
![]() |
(3) |
The first term on the right-hand side contains a double summation with respect to and
.
[1] Kompaneyets, A. "Theoretical Physics." Foreign Languages Publishing House, Moscow, 1961.
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