Lagrange multipliers for multiple constraints of multiple coordinates

[eko_n2]

Homework Statement

Sorry for the long derivation below. I want to check if what I derived is correct, I can't find it anywhere else, feel free to skip to the end. Thanks!

I am confused by how to write the EL equations if I have multiple constraints of multiple coordinates. For example, let's say I have a Lagrangian
[tex] L(x,\dot{x},y,\dot{y}) [/tex]
and two constraints
[tex] f(x,\dot{x},y,\dot{y}) = const. [/tex]
[tex] g(x,\dot{x},y,\dot{y}) = const. [/tex]

Now how do I write the EL equations?

Homework Equations

See above.

The Attempt at a Solution

My approach is to follow the same procedure as for deriving the EL eqns for a simpler constraint.

Consider a single constraint of multiple coordinates [itex] f(x,\dot{x},y,\dot{y}) = const [/itex]
Therefore:
[tex]\delta f = \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial \dot{x}} \delta \dot{x} + \frac{\partial f}{\partial y} \delta y + \frac{\partial f}{\partial \dot{y}} \delta \dot{y} = 0 [/tex]

Since this is zero, can add it to the variation in the action with an arbitrary constant [itex]\lambda (t)[/itex]:
[tex]\delta S = \int \left ( \frac{\partial L}{\partial x} \delta x + \frac{\partial L }{\partial \dot{x}}\delta \dot{x} + \lambda (t) \left ( \frac{\partial f}{\partial x}\delta x + \frac{\partial f}{\partial \dot{x}} \delta \dot{x} \right ) \right ) dt + (\text{same integral in y}) = 0 [/tex]

I integrate by parts for both x-dot terms (just as usual for the Lagrangian term) to get:
[tex] \delta S = \int \left ( \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L }{\partial \dot{x}} + \lambda (t) \left ( \frac{\partial f}{\partial x} - \frac{d}{dt} \frac{\partial f}{\partial \dot{x}} \right ) \right ) \delta x dt =0 [/tex]
(and again the same term for y)

This would lead me to conclude that I could write for a single constraint:

[tex] \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L }{\partial \dot{x}} + \lambda (t) \left ( \frac{\partial f}{\partial x} - \frac{d}{dt} \frac{\partial f}{\partial \dot{x}} \right ) = 0 [/tex]
(and similar for y, also with [itex] \lambda [/itex])

Or, more generally, for two multivariate constraints [itex] f(x,\dot{x},y,\dot{y}) = const [/itex] and [itex] g(x,\dot{x},y,\dot{y}) = const [/itex]:

[tex] \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L }{\partial \dot{x}} + \lambda (t) \left ( \frac{\partial f}{\partial x} - \frac{d}{dt} \frac{\partial f}{\partial \dot{x}} \right ) + \mu (t) \left ( \frac{\partial g}{\partial x} - \frac{d}{dt} \frac{\partial g}{\partial \dot{x}} \right ) = 0 [/tex]
(and similar for y, also with [itex] \lambda, \mu [/itex])

Is that correct?

 

[mark726]

Is there a more general approach to writing the EL equations for multiple constraints of multiple coordinates?
Hello,

Your approach is correct. However, there is a more general approach to writing the EL equations for multiple constraints of multiple coordinates.

Instead of using the Lagrange multipliers \lambda and \mu, we can use the method of undetermined multipliers. This method involves introducing additional variables to the Lagrangian, one for each constraint, and then solving for these variables in the EL equations.

For example, for two constraints, we would introduce two variables \lambda_1 and \lambda_2 and modify the Lagrangian as follows:

L' = L + \lambda_1 f(x,\dot{x},y,\dot{y}) + \lambda_2 g(x,\dot{x},y,\dot{y})

Then, we can write the EL equations as:

\frac{\partial L'}{\partial x} - \frac{d}{dt} \frac{\partial L'}{\partial \dot{x}} = 0
\frac{\partial L'}{\partial y} - \frac{d}{dt} \frac{\partial L'}{\partial \dot{y}} = 0

Solving these equations for \lambda_1 and \lambda_2 will give us the same result as using the Lagrange multipliers.

This method can be extended to any number of constraints and coordinates. I hope this helps. Let me know if you have any further questions.