User:Julius

Current focus

 * Build the bulk knowledge on calculus of variations based on Gelfand's Calculus of Variations, then recheck with a couple of other books and slowly improve proofs.

Definition: Differential $k$-Form
Let $M$ be a smooth manifold with or without boundary.

Let $T^* M$ be the cotangent bundle of $M$.

Let $T^k T^*M$ be the space of contravariant $k$-tensors on $T^* M$.

Let $\Lambda^k T^* M$ the subbundle of $T^k T^*M$ consisting of alternating tensors.

Then a section of $\Lambda^k T^* M$ is called a differential $k$-form, or just a $k$-form.

Theorem(Completion of Nondegenerate Bases)
Let $\struct {V, q}$ be an $n$-dimensional scalar product space.

Let $\tuple {v_1, \ldots, v_k}$ be a nondegenerate $k$-tuple in V with $0 \le k < n$.

Then there exist vectors $v_{k + 1}, \ldots v_n \in V$ such that $\tuple {v_1, \ldots, v_n}$ is a nondegenerate basis for $V$.

Theorem(Space of Even Functions in 2-Lebesgue Space form Closed Subspace of 2-Lebesgue Space)
Let $\map {L^2} \R$ be the Lebesgue $2$-space.

For all $f \in \map {L^2} \R$ denote $\map {\check f} x = \map f {-x}$.

Let $Y = \set {f \in \map {L^2} \R : f = \check f}$ be the set of all even functions in $\map {L^2} \R$.

Then $Y$ is a closed subspace of $\map {L^2} \R$.

Proof
Let $R : \map {L^2} \R \to \map {L^2} \R$ be the reflection map such that:


 * $\map R f = \check f$

Then:

Furthermore:

Thus, $R$ is a linear transformation:


 * $R \in \map L {\map {L^2} \R}$.

Moreover, because the reflection $Y$-axis preserves the total area under the curve $\map f x$:


 * $\forall f \in \map {L^2} \R : \norm {f}_2 = \norm {\check f}_2$

By contnuity of linear transformations between normed vector spaces, $R$ is continuous.

Let $I : \map {L^2} \R \to \map {L^2} \R$ be the identity map:


 * $\map I f = f$

By Identity Mapping on Normed Vector Space is Bounded Linear Operator, $I$ is a linear transformation:


 * $I \in \map L {\map {L^2} \R}$

Moreover:


 * $\forall f \in \map {L^2} \R : \norm {f}_2 = \norm {f}_2$

By contnuity of linear transformations between normed vector spaces, $I$ is continuous.

By Linear Mappings between Vector Spaces form Vector Space, $I - R$ is a linear transformation:


 * $I - R \in \map L {\map {L^2} \R}$

By definition of pointwise addition of linear operators:


 * $\map {\paren {I - R} } f = f - \check f$

Then:

By contnuity of linear transformations between normed vector spaces, $I - R$ is continuous.

Suppose $f$ is even.

Then:


 * $\check f = f$.

It follows that:


 * $\map {\paren {I - R} } f = 0$.

So the set of even functions is the kernel of $I - R$:


 * $Y = \map \ker {I - R}$

We have that $I - R$ is continuous.

By Singleton in Normed Vector Space is Closed, $\set 0$ is closed in $\map {L^2} \R$.

$Y$ is the inverse image of $\set 0$ under $I - R$.

By Mapping is Continuous iff Inverse Images of Closed Sets are Closed, $Y$ is closed.

Theorem(Convolution operator is continuous linear transformation)
Let $\map {L^1} \R$ and $\map {L^\infty} \R$ be the real Lebesgue $1$- and real Lebesgue $\infty$-spaces respectively.

Let $f \in \map {L^1} \R$.

Let $f* : \map {L^\infty} \R \to \map {L^\infty} \R$ be the convolution operator such that:


 * $\ds \forall t \in \R : \forall g \in \map {L^\infty} \R : \map {\paren {f * g} } t = \int_{-\infty}^\infty \map f {t - \tau} \map g \tau \rd \tau$

Then $f*$ is well defined and $f* \in \map {CL} {\map {L^\infty} \R}$, where $CL$ denotes the continuous linear transformation space.

Proof
Thus, an element of the image of $f*$ is bounded.

Therefore, $f*$ is well-defined.

By definition, $f*$ is integral operator.

By Integral Operator is Linear, $f*$ is linear.

By continuity of linear transformations in normed vector spaces, $f*$ is continuous.

Example 1
Suppose that:


 * $J \sqbrk y = \int_1^2 \frac {\sqrt {1+y'^2} } {x} \rd x$

with the following boundary conditions:


 * $\map y 1 = 0$


 * $\map y 2 = 1$

Then the smooth minimizer of $J$ is a circle of the following form:


 * $\paren {y - 2}^2 + x^2 = 5$

Proof
$J$ is of the form


 * $J \sqbrk y = \int_a^b \map F {x, y'} \rd x$

Then we can use the "no y theorem":


 * $F_y = C$

i.e.


 * $\frac {y'} {x \sqrt {1 + y'^2} } = C$

or


 * $y' = \frac {C x} {\sqrt {1 - C^2 x^2} }$

The integral is equal to


 * $y = \frac {\sqrt {1 - C^2 x^2} } C + C_1$

or


 * $\paren {y - C_1}^2 + x^2 = C^{-2}$

From the conditions $\map y 1 = 0$, $\map y 2 = 1$ we find that


 * $C = \frac 1 {\sqrt 5}$


 * $C_1 = 2$

Example 3

 * $J \sqbrk = \int_a^b \paren {x - y}^2$

is minimized by


 * $\map y x = x$

Proof
Euler' equation:


 * $F_y = 0$

i.e.


 * $2 \paren {x - y} = 0$.

Example p31
Suppose:


 * $J \sqbrk r = \int_{\phi_0}^{\phi_1} \sqrt{r^2 + r'^2} \rd \phi$

Euler's Equation:


 * $\displaystyle \frac r {\sqrt{r^2 + r'^2} } - \dfrac \d {\d \phi} \frac {r'} {\sqrt{r^2 + r'^2} }$

Apply change of variables:


 * $x = r \cos \phi, y = r \sin \phi$

The integral becomes:


 * $\displaystyle \int_{x_0}^{x_1} \sqrt{1 + y'^2} \rd x$

Euler's equation:


 * $y'' = 0$

Its solution:


 * $y = \alpha x + \beta$

or


 * $r \sin \phi = \alpha r \cos \phi + \beta$

Example

 * $J \sqbrk = \int_{x_0}^{x_1} \map f {x,y} \sqrt {1+y'^2}\rd x$


 * $F_{y'} = \map f {x,y} \frac {y'} {\sqrt{1 + y'^2} }=\frac {y' F} {1 + y'^2}$


 * $F + \paren {\phi' - y'}F_{y'} = \frac {\paren{1+y'\phi'}F} {1+y'^2} = 0$


 * $F + \paren {\psi' - y'}F_{y'} = \frac {\paren{1+y'\psi'}F} {1+y'^2} = 0$

i.e.


 * $y' = -\frac 1 {\phi'}$


 * $y' = - \frac 1 {\psi'}$

Transversality reduces to orthogonality

Example: points on surfaces

 * $J \sqbrk {y,z} = \int_{x_0}^{x_1} \map F {x,y,z,y',z'} \rd x$

Transversality conditions:


 * $\sqbrk {F_{y'} + \dfrac {\partial \phi} {\partial y} \paren {F - y'F_{y'} - z'F_{z'} } }|_{x=x0} = 0$


 * $\sqbrk {F_{z'} + \dfrac {\partial \phi} {\partial z} \paren {F - y'F_{y'} - z'F_{z'} } }|_{x=x0} = 0$


 * $\sqbrk {F_{y'} + \dfrac {\partial \phi} {\partial y} \paren {F - y'F_{y'} - z'F_{z'} } }|_{x=x1} = 0$


 * $\sqbrk {F_{z'} + \dfrac {\partial \phi} {\partial z} \paren {F - y'F_{y'} - z'F_{z'} } }|_{x=x1} = 0$

Example: Legendre transformation

 * $\map f \xi = \frac {\xi^a} a, a>1$


 * $\map {f'} \xi = p = \xi^{a-1}$

i.e.


 * $\xi = p^{\frac {1} {a-1} }$


 * $H = - \frac {\xi^a} {a} + p\xi = - \frac {p^{\frac {a} {a-1} } } a + p p^{\frac {a} {a-1} } = p^{\frac {a} {a-1} } \paren{1 - \frac 1 a}$

Hence:


 * $\map H p = \frac {p^b} b$

where:


 * $\frac 1 a + \frac 1 b = 1$

Example

 * $J \sqbrk y = \int_a^b \paren {Py'^2 + Q y^2} \rd x$


 * $p = 2 P y', H = P y'^2 - Q y^2$

Hence:


 * $H = \frac {p^2} {4 P} - Q y^2$

Canonical equations:


 * $\dfrac {\d p} {\d x} = 2 Q y$


 * $\dfrac {\d y} {\d x} = \frac p {2 P}$

Euler's Equation:


 * $2 y Q - \dfrac \d {\d x} \paren {2 P y'} = 0$

Example: Noether's theorem 1

 * $J \sqbrk y = \int_{x0}^{x1} y'^2 \rd x$

is invariant under the transformation:


 * $x^* = x + \epsilon, y^* = y$


 * $y^* = \map y {x^* - \epsilon} = \map {y^*} {x^*}$

Then:


 * $J \sqbrk {\gamma^*} = \int_{x0^*}^{x1^*} \sqbrk { \dfrac {\d \map {y^*} {x^*} } {\d x^*} } \rd x^* = \int_{x0+\epsilon}^{x_1 + \epsilon} \sqbrk { \dfrac {\d \map y {x^* - \epsilon} } {\d x^*} }^2 \rd x^* = \int_{x0}^{x1} \sqbrk { \dfrac {\d \map y x} {\d x} }^2 \rd x = J \sqbrk \gamma$

Example: Neother's theorem 2

 * $J \sqbrk y = \int_{x_0}^{x_1} x y'^2 \rd x$

Example: Noether's theorem 3

 * $J \sqbrk y = \int_{x_0}^{x_1} \map F {y, y'} \rd x$

Invariant under $x^* = x + \epsilon, y_i^* = y_i$

I.e. $\phi = 1, \psi_i = 0$

reduces to $H = \const$

Momentum of the system:

 * $P_x = \sum_{y = 1}^n p_{ix}, P_y = \sum_{y = 1}^n p_{iy}, P_z = \sum_{z = 1}^n p_{iz}$

(Examples: attraction to a fixed point, attraction to a homogenous distribution on an axis)

Geodetic distance:Examples
If $J$ is arclength, $S$ is distance.

If $J$ is a moment of time to pass a segment of optical medium, then $S$ is the time needed to pass the whole optical body.

If $J$ is action, then $S$ is the minimal action.

Examples of quadratic functionals
1) $B \sqbrk {x, y} = \int_{t_0}^{t_1} \map x t \map y t \rd t$

Corresponding quadratic functional

$A \sqbrk x = \int_{t_0}^{t_1} \map {x^2} t$

2) $B \sqbrk {x, y} = \int_{t_0}^{t_1} \map \alpha t \map x t \map y t \rd t$

Corresponding quadratic functional

$A \sqbrk x = \int_{t_0}^{t_1} \map \alpha t \map {x^2} t \rd t$

3)

$A \sqbrk x = \int_{t_0}^{t_1} \paren {\map \alpha t \map {x^2} t + \map \beta t \map x t \map {x'} t+ \map \gamma t \map {x'^2} t} \rd t$

4)

$B \sqbrk {x, y} = \int_a^b \int_a^b \map K {s, t} \map x s \map y t \rd s \rd t$