Constraints in Hamilton's equations

In summary, the conversation discusses the role of constraints in the Hamiltonian viewpoint and how they are incorporated into the Hamilton's equations and the construction of L. It also mentions that for non-holonomic constraints, the treatment may differ. The speaker is looking for a technique similar to the use of undetermined constants in the Lagrangian viewpoint for handling constraint equations.
  • #1
somy
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My question is about the constraints in Hamiltonian viewpoint. I mean, where the constraints are put into the Hamilton's equations. Constraints are usually studied in Lagrange's equations in textbooks (such as Jose and Goldstein). However, I couldn't find anything about constraints in Hamiltonian viewpoint.
Thanks in advance.
 
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  • #2
When you are constructing L, you do it in terms of generalised co-ordinates, which are the least number of free co-ordinates. The constraints have already been taken into account when constructing L by eliminating some degrees of freedom. When you are using Hamilton’s principle, you are making integral Ldt stationary, and, so here too, the constraints have been taken into account.

For non-holonomic constraints, the treatment may be somewhat different, but the essence is the same.
 
  • #3
Dear Shooting star,
Thanks for writing. But I'm looking for some techniqe jist similar to one we use in Lagrangian viewpoint as undetermined constants and the way we treat with constraint equations there.
 

1. What are constraints in Hamilton's equations?

Constraints in Hamilton's equations refer to any restrictions or limitations on the motion of a physical system. These can include physical boundaries or conditions, as well as mathematical constraints that arise from the equations of motion themselves.

2. Why are constraints important in Hamilton's equations?

Constraints play a crucial role in Hamilton's equations because they help to simplify and reduce the complexity of the equations of motion. By accounting for the constraints, the number of degrees of freedom of the system can be reduced, making it easier to solve and analyze.

3. How are constraints incorporated into Hamilton's equations?

Constraints are typically incorporated into Hamilton's equations using Lagrange multipliers. These multipliers are added to the Hamiltonian function and serve as a way to enforce the constraints while still allowing the equations of motion to be solved.

4. What is the difference between holonomic and nonholonomic constraints in Hamilton's equations?

Holonomic constraints are those that can be expressed as equations involving only the generalized coordinates of the system. Nonholonomic constraints, on the other hand, cannot be expressed in this way and may involve time derivatives of the coordinates. Nonholonomic constraints are generally more difficult to incorporate into Hamilton's equations.

5. Can a system have both holonomic and nonholonomic constraints?

Yes, a system can have both holonomic and nonholonomic constraints. In fact, many physical systems have a combination of both types of constraints. In these cases, the constraints must be carefully considered and incorporated into the equations of motion for an accurate analysis of the system.

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