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Lagrange multipliers for multiple constraints of multiple coordinates

  1. Sep 23, 2014 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations

    See above.

    3. 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?
     
  2. jcsd
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