I've seen a few other threads on here inquiring about what is the point of Lagrange Multipliers, or the like. My main question though is, how can I tell by looking at a system in a problem that Lagrange Multipliers would be preferred compared to generalized coordinates. I'm in a theoretical mechanics course, and we are just doing very basic systems (pendulums, points constrained to some shape).
The book I have just outlines Lagrange Multipliers incorporated into the Lagrangian Equation.
∂L∂qj−ddt∂L∂˙qj+∑kλk(t)∂fk∂qj=0.
The book gives about 2 examples of using these, but I wouldn't know whether or not to use them over just using the regular generalized coordinate example.
References:
- Thornton & Marion, Classical Dynamics of Particles and Systems, Fifth Ed.; p.221.
Answer
In the context of Lagrange equations
ddt∂(T−U)∂˙qj−∂(T−U)∂qj = Qj−∂F∂˙qj+m∑ℓ=1λℓaℓj,j ∈{1,…,n},
in classical mechanics, the Lagrange multipliers are used to impose semi-holonomic constraints
n∑j=1aℓj(q,t)˙qj+aℓt(q,t) = 0,ℓ ∈{1,…,m}.
See my Phys.SE answer here for notation.
If a semi-holonomic constraint is holonomic, it is not necessary to implement it via a Lagrange multiplier, except for the case where one is interested in calculating the corresponding constraint force.
References:
- H. Goldstein, Classical Mechanics; Chapter 1 & 2.
No comments:
Post a Comment