- #1
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- Homework Statement
- How can we express a Lagrangian for a system with a constraint that is not expressed as F(x) = 0?
- Relevant Equations
- N/A
When a constraint is expressed as F(x)=0, I am quite comfortable in putting such constraints into the Lagrangian. Just add the function with an undetermined multiplier, then treat the multiplier as an additional coordinate, and proceed as before.
##L = T - V + \lambda F ##
For example, you can do motion on a surface by taking the F to be a function that defines the surface. A sphere, for example, could have this.
## F(x,y) = x^2 + y^2 - R^2 = 0 ##
This is a very usual thing for studying Lagrangian dynamics, and most people study some such problem when they study Lagrangians. Not bothered about that.
I have been trying to figure out how to do something along these lines for constraints that are not expressed as some function. For example, suppose I am considering a simple ballistics problem, launching a projectile and calculating its path under gravity. The Lagrangian is just the following.
##L = \frac{1}{2} m V_x^2 + \frac{1}{2} m V_y^2 - g y ##
And the Lagrange equations are just the usual Newtonian things. ##V_x## is constant, and ##\frac{d V_y}{dt} = -g##.
Now suppose the constraint that I have is that the particle should reach a particular location. Say (Xp,Yp) as the spot the particle should pass through. I am having a very hard time expressing this in a way I could add to the Lagrangian with a multiplier. Or suppose that the constraint is that, at distance Xp, the particle must pass through a gate extending from YL to YH. This also is resisting my attempts to express it as a function that could be added to the Lagrangian with a multiplier.
I can certainly solve the equations in terms of general initial values, then solve for the required initial values to get the required end point. Or the required range of end-point values. But that is not what I am trying to do. I am trying to set things up so that there is a modified Lagrangian which has Lagrange equations that include a constraint. And the resultant set of equations have as their solution that the particle goes where it is supposed to.
I have a book: "Constrained Optimization and Lagrange Multiplier Methods" by Dimitri P. Bertsekas. It's good for numerically solving constrained systems, including inequality type problems But it does not set things up the way I would hope. Rather it casts things in a way they can be solved numerically.
Any tiny clues appreciated. Is this even possible?
##L = T - V + \lambda F ##
For example, you can do motion on a surface by taking the F to be a function that defines the surface. A sphere, for example, could have this.
## F(x,y) = x^2 + y^2 - R^2 = 0 ##
This is a very usual thing for studying Lagrangian dynamics, and most people study some such problem when they study Lagrangians. Not bothered about that.
I have been trying to figure out how to do something along these lines for constraints that are not expressed as some function. For example, suppose I am considering a simple ballistics problem, launching a projectile and calculating its path under gravity. The Lagrangian is just the following.
##L = \frac{1}{2} m V_x^2 + \frac{1}{2} m V_y^2 - g y ##
And the Lagrange equations are just the usual Newtonian things. ##V_x## is constant, and ##\frac{d V_y}{dt} = -g##.
Now suppose the constraint that I have is that the particle should reach a particular location. Say (Xp,Yp) as the spot the particle should pass through. I am having a very hard time expressing this in a way I could add to the Lagrangian with a multiplier. Or suppose that the constraint is that, at distance Xp, the particle must pass through a gate extending from YL to YH. This also is resisting my attempts to express it as a function that could be added to the Lagrangian with a multiplier.
I can certainly solve the equations in terms of general initial values, then solve for the required initial values to get the required end point. Or the required range of end-point values. But that is not what I am trying to do. I am trying to set things up so that there is a modified Lagrangian which has Lagrange equations that include a constraint. And the resultant set of equations have as their solution that the particle goes where it is supposed to.
I have a book: "Constrained Optimization and Lagrange Multiplier Methods" by Dimitri P. Bertsekas. It's good for numerically solving constrained systems, including inequality type problems But it does not set things up the way I would hope. Rather it casts things in a way they can be solved numerically.
Any tiny clues appreciated. Is this even possible?