Lagrangian with constraint forces

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Discussion Overview

The discussion revolves around the formulation of the Lagrangian in the context of constraint forces within classical mechanics, specifically addressing the potential energy term in the Lagrangian equation L = T - U. Participants explore whether the potential energy U should include contributions from constraint forces, particularly when some of these forces are conservative.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question why the potential energy U in the Lagrangian is defined as corresponding only to nonconstraint forces, suggesting that conservative constraint forces might also contribute.
  • Others argue that constraint forces do no work and are always perpendicular to the direction of motion, thus asserting that they do not contribute to the potential energy in the Lagrangian formulation.
  • It is noted that the Lagrangian formulation L = T - V is applicable only when energy is conserved, implying that only conservative forces are considered in this context.
  • A participant mentions the need for Lagrange multipliers when dealing with nonconservative forces, indicating that additional terms must be included in the Lagrangian in such cases.
  • There is a recognition of the difficulty in defining appropriate generalized coordinates for kinetic energy T and potential energy V.

Areas of Agreement / Disagreement

Participants express differing views on the role of constraint forces in the Lagrangian formulation. While some maintain that constraint forces do not contribute to potential energy, others challenge this notion, leading to an unresolved discussion regarding the inclusion of conservative constraint forces.

Contextual Notes

The discussion highlights assumptions about the nature of constraint forces and their work, as well as the conditions under which the Lagrangian is applied. There is an acknowledgment of the complexities involved in defining generalized coordinates.

C. Lee
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I am now reading Lagrange's equations part in Taylor's Classical Mechanics text.

It says:

When a system of interest involves constraint forces, F_cstr, and all the nonconstraint forces are derivable from a potential energy(U), then the Lagrangian for the system L is L = T - U, where U is the potential energy for the nonconstraint forces only, and thus this definition of L excludes the constraint forces.

Here's the question: How do we know that U in L = T - U is the potential energy for the nonconstraint forces only? Shouldn't it have contribution from constraint forces if some of constraint forces are conservative?
 
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C. Lee said:
Here's the question: How do we know that U in L = T - U is the potential energy for the nonconstraint forces only? Shouldn't it have contribution from constraint forces if some of constraint forces are conservative?

The reason is that constraint forces do no work.They just maintain the constrains of the system and their direction is always perpendicular to the direction of motion.
Scientists used work formulas to derive the Lagrangian equation,so the potential energy "U" in the Lagrangian corresponds to non-constraint conservative forces only and constraints forces have no contribution.
 
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amjad-sh said:
The reason is that constraint forces do no work.They just maintain the constrains of the system and their direction is always perpendicular to the direction of motion.
Scientists used work formulas to derive the Lagrangian equation,so the potential energy "U" in the Lagrangian corresponds to non-constraint conservative forces only and constraints forces have no contribution.
I would like to make more explicit this statement. The Lagrangian of L = T - V is used only when energy is conserved. Thus, going back to the statement that only conservative forces are being considered.
 
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bluejay27 said:
I would like to make more explicit this statement. The Lagrangian of L = T - V is used only when energy is conserved. Thus, going back to the statement that only conservative forces are being considered.
When you want to deal with nonconservative forces, you have to use Lagrange multipliers, where you are including an additional term to the L
 
The difficulty in using L is in defining the right generalized coordinates for T and V
 

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