The first variation is a generalized directional derivative of a functional . Their properties are relevant in applied mathematics and theoretical physics . The first variation plays a central role in the calculus of variations and is used in analytical mechanics . A related concept is functional derivation .
definition
Be a function space; a functional with or ; Functions and . Then the first variation of the functional after is defined as
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![J: X \ rightarrow {\ mathbb {K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/209d001872ebd08d473c285bf5af5e0a1409f3ec)
![{\ mathbb {K}} = {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa0cac96c9853f0cba61605679cf8963983b5d9)
![{\ mathbb {K}} = {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1a8ee77fce00b8ddbd288a297dccae8eab7f8f)
![y, h \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff6e016bf5fe9681e72405d3ed2af472e975307)
![\ varepsilon \ in {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd62e0a7627b820957f2c201ab6e4f9d8186135)
![J](https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
-
.
This corresponds to the Gâteaux differential of the functional at the point in direction .
![J](https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
properties
- The first variation is a linear map :
![{\ displaystyle \ delta (F (y) + \ alpha G (y)) (h) = \ delta F (y) (h) + \ alpha \ delta G (y) (h) \ quad \ forall \ alpha \ in \ mathbb {K}, \, \ forall F, G \ in {\ mathcal {D}} ^ {\ prime} (\ Omega)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b81dbac745166db0e1477e7ab9fedb8a881dad22)
- The product rule applies to a functional product:
![F (y) = G (y) H (y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3038f6ba477ea2721583fd2e786def7e5d6947ac)
![\ delta F (y) (h) = \ delta G (y) (h) \ H (y) + G (y) \ \ delta H (y) (h)](https://wikimedia.org/api/rest_v1/media/math/render/svg/328ebcadf396e9156d2cc5494732f0c8f1736b46)
example
The first variation of
![{\ displaystyle J (y) = \ int _ {a} ^ {b} yy '\, \ mathrm {d} x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b28bc63b014e48fa9c4e642743900fcf5bda61d9)
is according to the above definition
![{\ displaystyle {\ begin {aligned} \ delta J (y) (h) & = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} \ varepsilon}} J (y + \ varepsilon h) \ right | _ {\ varepsilon = 0} \\ & = {\ frac {\ mathrm {d}} {\ mathrm {d} \ varepsilon}} \ int _ {a} ^ {b} (y + \ varepsilon h) (y ^ {\ prime} + \ varepsilon h ^ {\ prime}) \, \ mathrm {d} x {\ Bigg |} _ {\ varepsilon = 0} \\ & = {\ frac {\ mathrm {d} } {\ mathrm {d} \ varepsilon}} \ int _ {a} ^ {b} (yy ^ {\ prime} + y \ varepsilon h ^ {\ prime} + y ^ {\ prime} \ varepsilon h + \ varepsilon ^ {2} hh ^ {\ prime}) \, \ mathrm {d} x {\ Bigg |} _ {\ varepsilon = 0} \\ & = \ int _ {a} ^ {b} \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} \ varepsilon}} (yy ^ {\ prime} + y \ varepsilon h ^ {\ prime} + y ^ {\ prime} \ varepsilon h + \ varepsilon ^ {2 } hh ^ {\ prime}) \ right | _ {\ varepsilon = 0} \, \ mathrm {d} x \\ & = \ int _ {a} ^ {b} (yh ^ {\ prime} + y ^ {\ prime} h + 2 \ varepsilon hh ^ {\ prime}) {\ bigg |} _ {\ varepsilon = 0} \, \ mathrm {d} x \\ & = \ int _ {a} ^ {b} (yh ^ {\ prime} + y ^ {\ prime} h) \, \ mathrm {d} x \,. \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23ec73b78b8a1a3016837e563e49b2df3e4d97d)
See also
Web links