| ∂A | δq1 + | ∂A | δq2 + ... = 0, | ∂B | δq1 + | ∂B | δq2 + ... = 0, &c. |
| ∂q1 | ∂q2 | ∂q1 | ∂q2 |
(29)
Introducing indeterminate multipliers λ, μ, ..., one for each of these equations, we obtain in the usual manner n equations of the type
| d | ∂T | − | ∂T | = Qr + λ | ∂A | + μ | ∂B | + ..., | |
| dt | ∂q˙r | ∂qr | ∂qr | ∂qr |
(30)
in place of § 2 (10). These equations, together with (28), serve to determine the n co-ordinates q1, q2, ... qn and the m multipliers λ, μ, ....
When t does not occur explicitly in the relations (28) the system is said to be holonomic. The term connotes the existence of integral (as opposed to differential) relations between the co-ordinates, independent of the time.
Again, it may happen that although there are no prescribed relations between the co-ordinates q1, q2, ... qn, yet from the circumstances of the problem certain geometrical conditions are imposed on their variations, thus
A1δq1 + A2δq2 + ... = 0, B1δq1 + B2δq2 + ... = 0, &c.,
(31)