Lamé coefficients

of an orthogonal curvilinear coordinate system $ u , v , w $ in space

The quantities

$$ L _ {u} = \ \sqrt { \left ( \frac{\partial x }{\partial u } \right ) ^ {2} + \left ( \frac{\partial y }{\partial u } \right ) ^ {2} + \left ( \frac{\partial z }{\partial u } \right ) ^ {2} } , $$

$$ L _ {v} = \sqrt {\left ( \frac{\partial x }{\partial v } \right ) ^ {2} + \left ( \frac{\partial y }{\partial v } \right ) ^ {2} + \left ( \frac{\partial z }{\partial v } \right ) ^ {2} } , $$

$$ L _ {w} = \sqrt {\left ( \frac{\partial x }{\partial w } \right ) ^ {2} + \left ( \frac{\partial y }{\partial w } \right ) ^ {2} + \left ( \frac{\partial z }{\partial w } \right ) ^ {2} } . $$

The Lamé coefficients in the plane are defined similarly. In terms of the Lamé coefficients in the coordinates $ u , v , w $ one can express the element of arc length:

$$ d l = \ \sqrt { L _ {u} ^ {2} d u ^ {2} + L _ {v} ^ {2} d v ^ {2} + L _ {w} ^ {2} d w ^ {2} } ; $$

the element of surface area:

$$ d \sigma = \ \sqrt { ( L _ {u} L _ {v} d u d v ) ^ {2} + ( L _ {u} L _ {w} d u d w ) ^ {2} + ( L _ {v} L _ {w} d v d w ) ^ {2} } ; $$

and the volume element:

$$ d V = L _ {u} L _ {v} L _ {w} d u d v d w . $$

The Lamé coefficients occur in the expressions for the operations of vector analysis in the coordinates $ u , v , w $:

$$ \mathop{\rm grad} _ {u} \psi = \ \frac{1}{L _ {u} } \frac{\partial \psi }{\partial u } ,\ \ \mathop{\rm grad} _ {v} \psi = \ \frac{1}{L _ {v} } \frac{\partial \psi }{\partial v } ,\ \ \mathop{\rm grad} _ {w} \psi = \ \frac{1}{L _ {w} } \frac{\partial \psi }{\partial w } ; $$

$$ \mathop{\rm div} a = \ \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial {\partial u } ( a _ {u} L _ {v} L _ {w} ) + \frac \partial {\partial v } ( a _ {v} L _ {u} L _ {w} ) \right . + $$

$$ + \left . \frac \partial {\partial w } ( a _ {w} L _ {u} L _ {v} ) \right ] ; $$

$$ \mathop{\rm rot} _ {u} \mathbf a = \ \frac{1}{L _ {v} L _ {w} } \left [ \frac \partial {\partial v } ( a _ {w} L _ {w} ) - \frac \partial {\partial w } ( a _ {v} L _ {v} ) \right ] , $$

$$ \mathop{\rm rot} _ {v} \mathbf a = \frac{1}{L _ {u} L _ {w} } \left [ \frac \partial {\partial w } ( a _ {u} L _ {u} ) - \frac \partial {\partial u } ( a _ {w} L _ {w} ) \right ] , $$

$$ \mathop{\rm rot} _ {w} \mathbf a = \frac{1}{L _ {u} L _ {v} } \left [ \frac \partial {\partial u } ( a _ {v} L _ {v} ) - \frac \partial {\partial v } ( a _ {u} L _ {u} ) \right ] ; $$

$$ \Delta \psi = \mathop{\rm div} ( \mathop{\rm grad} \psi ) = \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial {\partial u } \left ( \frac{L _ {v} L _ {w} }{L _ {u} } \frac{\partial \psi }{\partial u } \right ) \right . + $$

$$ + \left . \frac \partial {\partial v } \left ( \frac{L _ {u} L _ {w} }{L _ {v} } \frac{\partial \psi }{\partial v } \right ) + \frac \partial {\partial w } \left ( \frac{L _ {u} L _ {v} }{L _ {w} } \frac{\partial \psi }{\partial w } \right ) \right ] . $$

For the Lamé coefficients of various orthogonal curvilinear coordinates see the corresponding articles on those coordinates.

Lamé coefficients were introduced by G. Lamé [1].

References

[1]	G. Lamé, "Leçons sur les coordonnées curvilignes et leurs diverses applications" , Paris (1859)
[2]	G.F. Laptev, "Elements of vector calculus" , Moscow (1975) (In Russian)
[3]	P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1 , McGraw-Hill (1953)

Comments

The name "Lamé coefficients" for the quantities $ L _ {u} $, $ L _ {v} $, $ L _ {w} $ is not often used in the Western literature. Instead one finds "scale factors of an orthogonal curvilinear coordinate systemscale factors" [3] or "metric coefficients of an orthogonal curvilinear coordinate systemmetric coefficients" [a1]. The latter terminology of course derives from the fact that the original Riemannian metric $ d s ^ {2} $ with respect to the new orthogonal curvilinear system of coordinates $ u , v , w $ takes the form

$$ d s ^ {2} = L _ {u} ^ {2} \ d u ^ {2} + L _ {v} ^ {2} \ d v ^ {2} + L _ {w} ^ {2} \ d w ^ {2} . $$

Thus the squares of the $ L _ {u} $, $ L _ {v} $, $ L _ {w} $ are the diagonal components of the standard metric tensor on $ \mathbf R ^ {3} $ expressed in terms of $ u , v, w $. The other components are zero because $ u , v , w $ is an orthogonal curvilinear coordinate system.

References

[a1]	I.S. [I.S. Sokolnikov] Sokolnikoff, R.M. Redheffer, "Mathematics of physics and engineering" , McGraw-Hill (1958)
[a2]	H.F. Davis, A.D. Snider, "Introduction to vector analysis" , Allyn & Bacon (1979)