Dot cancellation (holonomic constraints)

In summary, dot cancellation is a process used to eliminate dot products in the equations of motion for systems with holonomic constraints. It can significantly simplify the dynamics of a system and reveal important relationships between the generalized coordinates and velocities. However, it is not applicable to systems with non-holonomic constraints and may become difficult to manage for complex systems. Dot cancellation is closely related to the principle of virtual work and can be applied to systems with external forces, but may make the analysis more complex.
  • #1
amiras
65
0
I started to read Analytical Mechanics. It said that if holonomic constraints are defined as:

r = r(q1, q2, ... qn, t) (or without time)

This equation holds (dot cancellation):

∂r'/∂q_k' = ∂r/∂q_k

where ' specified derivatives.

And the question was given to check if it works for two simple examples:

1) a mass sliding without friction down a stationary inclined plane
2) the rotating bead on a wire

So if we talk about 1) I should express the position r with some generalized coordinate?
 
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  • #2
The position r is the generalised coordinate
 

1. What is dot cancellation in the context of holonomic constraints?

Dot cancellation refers to the process of eliminating dot products in the equations of motion for a system with holonomic constraints. This is done by using the constraints to express the velocities in terms of the generalized coordinates, which allows for a simplified and more efficient solution to the equations of motion.

2. How does dot cancellation affect the dynamics of a system?

Dot cancellation can significantly simplify the equations of motion for a system with holonomic constraints, leading to a more efficient and accurate analysis of the system's dynamics. It reduces the number of equations that need to be solved and can reveal important relationships between the generalized coordinates and velocities.

3. Are there any limitations to using dot cancellation?

While dot cancellation can be a powerful tool in analyzing systems with holonomic constraints, it is not applicable to systems with non-holonomic constraints. Additionally, it may not be feasible for complex systems with a large number of constraints, as the process can become cumbersome and difficult to manage.

4. How is dot cancellation related to the principle of virtual work?

The principle of virtual work states that the virtual work done by the forces on a system in equilibrium is equal to zero. Dot cancellation is closely related to this principle, as it allows for the simplification of the equations of motion to the point where the virtual work done by the forces can be easily calculated and equated to zero.

5. Can dot cancellation be applied to systems with external forces?

Yes, dot cancellation can be applied to systems with external forces. However, in these cases, the equations of motion will also include terms for the virtual work done by the external forces, in addition to the virtual work done by the constraint forces. This can make the analysis more complex, but dot cancellation can still be used to simplify the equations and make them more manageable.

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