Non-holonomic constraints in field theory

In summary, non-holonomic constraints in field theory refer to geometric restrictions on the possible motions of a system that cannot be expressed as equations of motion. These constraints can significantly affect the dynamics of a system by limiting its degrees of freedom and introducing additional forces. They cannot be derived from a Lagrangian and differ from holonomic constraints in their ability to be expressed as equations of motion and their impact on the equations of motion. Non-holonomic constraints cannot be eliminated from the equations of motion without fundamentally changing the dynamics of the system.
  • #1
Bobhawke
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My question is the following:

In field theory, if I have a constraint [tex] \chi(q_a, p_a, \partial_i q_a) [/tex] that depends on the generalised coordinates q_a, momenta p_a and spatial derivatives only of the q_a [tex] \partial_i q_a [/tex] does this count as a non-holonomic constraint? Or is it only non-holonomic if it depends on the time derivatives of q_a?

And related to this, if there is a non-holonomic constraint, can the dynamics of the system still be derived from a stationary action principle with Lagrange multipliers added into enforce the constraint? And does Dirac's procedure for constructing the Hamiltonian for constrained Hamiltonian systems produce the correct dynamics, or does it fail if there are non-holonomic constraints? If so, why?
 
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  • #2
In field theory, a constraint of the form \chi(q_a, p_a, \partial_i q_a) does not necessarily count as a non-holonomic constraint. It is possible for such a constraint to be holonomic depending on its exact form. A non-holonomic constraint must depend on the time derivatives of the q_a in order to be considered non-holonomic.It is still possible to derive the dynamics of a system with non-holonomic constraints using a stationary action principle with Lagrange multipliers, however, Dirac's procedure for constructing the Hamiltonian for constrained Hamiltonian systems may fail if there are non-holonomic constraints. This is because Dirac's procedure assumes that all the constraints are holonomic and thus the equations of motion derived by this method may not be correct if there are non-holonomic constraints.
 

1. What are non-holonomic constraints in field theory?

Non-holonomic constraints in field theory refer to restrictions on the possible motions of a system that are not expressible as equations of constraint. These constraints arise due to the geometry of the system and limit the degrees of freedom of the system.

2. How do non-holonomic constraints affect the dynamics of a system in field theory?

Non-holonomic constraints can significantly affect the dynamics of a system in field theory by restricting the possible motions and introducing additional forces into the equations of motion. This can lead to complex and non-intuitive behavior of the system.

3. Can non-holonomic constraints be derived from a Lagrangian in field theory?

No, non-holonomic constraints cannot be derived from a Lagrangian in field theory. They are geometric constraints that cannot be expressed as equations of motion and therefore cannot be derived from a Lagrangian.

4. How do non-holonomic constraints differ from holonomic constraints in field theory?

Holonomic constraints in field theory can be expressed as equations of motion, while non-holonomic constraints cannot. Additionally, holonomic constraints do not introduce additional forces into the equations of motion, while non-holonomic constraints do.

5. Can non-holonomic constraints be eliminated from the equations of motion in field theory?

No, non-holonomic constraints cannot be eliminated from the equations of motion in field theory. They are inherent to the geometry of the system and cannot be removed without fundamentally changing the dynamics of the system.

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