Discussion
A dynamical system consists of three particles \(m_{1}, m_{2}, m_{3}\) in motion in space. The distance between \(m_{1}\) and \(m_{2}\), \(m_{2}\) and \(m_{3}\), \(m_{1}\) and \(m_{3}\) remain constant throughout the motion. The motion will be governed by \(n\) Euler Lagrange equations. Then :
Answer: B
The number of equations is \(n = 6\).
Three free particles in space have \(3 \times 3 = 9\) coordinates. The three fixed distances \(m_{1}m_{2}\), \(m_{2}m_{3}\) and \(m_{1}m_{3}\) are three holonomic constraints, leaving
\[ 9 - 3 = 6 \]
degrees of freedom. The system is a rigid body (three non-collinear points fix it), which indeed has 3 translational and 3 rotational degrees of freedom. There is one Euler-Lagrange equation per generalised coordinate, so 6 equations. 9 ignores the constraints, 3 counts only the constraints, and 7 has no basis.
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