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Physics LibreTexts

2.7: Undetermined Multipliers

  • Page ID
    8566
  • Let ψ(x, y, z) be some function of x, y and z. Then if x, y and z are independent variables, one would ordinarily understand that, where ψ is a maximum, the derivatives are zero:

    \[ \frac{ \delta \psi}{ \delta x} = \frac{ \delta \psi}{ \delta y} = \frac{ \delta \psi}{ \delta z} = 0.\]

    However, if x, y and z are not completely independent, but are related by some constraining equation such as f(x, y, z) = 0, the situation is slightly less simple. (In a thermodynamical context, the three variables may be, for example, three “intensive state variables”, P, V and T, and ψ might be the entropy, which is a function of state. However the intensive state variables may not be completely independent, since they are related by an “equation of state”, such as PV = RT.)

    If we move by infinitesimal displacements dx, dy, dz from a point where ψ is a maximum, the corresponding changes in ψ and f will both be zero, and therefore both of the following equations must be satisfied.

    \[ d \psi = \frac{ \delta \psi}{ \delta x} dx + \frac{ \delta \psi}{ \delta y} dy + \frac{ \delta \psi}{ \delta z} dx = 0,\]

    \[ df = \frac{ \delta f}{ \delta x} dx + \frac{ \delta f}{ \delta y} dy + \frac{ \delta f}{ \delta z} dx = 0.\]

    Consequently any linear combination of ψ and f, such as Φ = ψ + λf, where λ is an arbitrary constant, also satisfies a similar equation. The constant λ is sometimes called an “undetermined multiplier” or a “Lagrangian multiplier”, although often some additional information in an actual problem enables the constant to be identified.

    In summary, the conditions that ψ is a maximum (or minimum or saddle point), if x, y and z are related by a functional constraint f (x, y, z) = 0, are

    \[ \begin{matrix} \frac{ \delta \Phi}{ \delta x} = 0 & \frac{ \delta \Phi}{ \delta y} = 0, & \frac{ \delta \Phi}{ \delta z} = 0, \end{matrix}\]

    where

    \[ \Phi = \psi + \lambda f.\]

    Of course, if ψ is a function of many variables x1 , x2 , x3..., and the variables are subjected to several constraints, such as f = 0, g = 0, h = 0, etc., where f, g, h, etc., are functions connecting all or some of the variables, the conditions for ψ to be a maximum (etc.) are

    \[ \frac{ \delta \psi}{ \delta x_i} + \lambda \frac{ \delta \psi}{ \delta x_i} + \mu \frac{ \delta \psi}{ \delta x_i} + \nu \frac{ \delta \psi}{ \delta x_i} + ... = 0,~ i = 1,~2,~3\]