I Lagrangian with constraint forces

[C. Lee]

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?

 

[amjad-sh]

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.

 

[bluejay27]

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.

 

[bluejay27]

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

 

[bluejay27]

The difficulty in using L is in defining the right generalized coordinates for T and V