Discussion
Suppose a system has holonomic and conservative part, and also some part of forces acting on the system are not conservative. Let the typical non-conservative force acting on jth particle is denoted as \(Q_{i}\). The Hamilton's equations of motion modify to :
Answer: B
The modified equations are \(\dot{q}_{i} = \frac{\partial H}{\partial p_{i}}\), \(\dot{p}_{i} = -\frac{\partial H}{\partial q_{i}} + Q_{i}\).
The conservative forces are contained in H. A non-conservative generalized force \(Q_{i}\) adds to the rate of change of momentum, as in Lagrange's equation, where \(\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_{i}}\) minus \(\frac{\partial L}{\partial q_{i}}\) equals \(Q_{i}\). Since \(p_{i} = \partial L/\partial \dot{q}_{i}\) and \(\partial L/\partial q_{i} = -\partial H/\partial q_{i}\), this gives \(\dot{p}_{i} = -\partial H/\partial q_{i} + Q_{i}\), while the first equation, which only defines the velocity, is unchanged.
Options A and C give \(\dot{p}_{i}\) the wrong sign of \(\partial H/\partial q_{i}\), and options C and D subtract the applied force instead of adding it.
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