Constraints in Hamilton's equations

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SUMMARY

This discussion focuses on the treatment of constraints within Hamilton's equations, contrasting it with the more commonly addressed Lagrangian viewpoint. It highlights that constraints are inherently considered when constructing the Lagrangian (L) using generalized coordinates, which eliminates certain degrees of freedom. The conversation also notes that while non-holonomic constraints may require different handling, the fundamental approach remains consistent with the Hamiltonian framework. The user seeks techniques analogous to undetermined constants in the Lagrangian method for addressing constraint equations in Hamiltonian mechanics.

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  • Understanding of Hamiltonian mechanics and Hamilton's equations
  • Familiarity with Lagrangian mechanics and generalized coordinates
  • Knowledge of constraints in classical mechanics, including holonomic and non-holonomic constraints
  • Basic proficiency in variational principles and their applications in physics
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  • Research the application of constraints in Hamiltonian mechanics
  • Study the differences between holonomic and non-holonomic constraints
  • Explore techniques for incorporating constraints in Hamilton's equations
  • Learn about variational principles and their role in both Lagrangian and Hamiltonian frameworks
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This discussion is beneficial for physicists, particularly those specializing in classical mechanics, as well as students and researchers seeking to deepen their understanding of constraints in Hamiltonian systems.

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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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.
 
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.
 

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