User:Julius

Current focus

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


 * So I just noticed that vector notation is being used in Gelfand's for higher dimensional functionals. This implies rewriting all multivariable functionals. Implement this gradually.

Lemmas and theorems for Bernstein's Theorem on Unique Extrema (1978)
Raw material

Isoperimetric problem, no fixed points
Variation of area functional of length constraint gives us a circle.

However, now the endpoints are not fixed.

This requires differentiating the effective functions wrt $y'$ and setting the result at both ends to zero:


 * $\displaystyle \frac {y'}{\sqrt{1 + y'^2} } \paren {a}= 0$

which simply means $y' \paren {a} = 0$

But this does not make sense for a circle. There is something missing.

The thing is, naive approach does not work, because variable endpoints correspond to variable "y" at some fixed "x".

Our case is the opposive, fixed "y" and varied "x".

Most possibly this can be done by applying appropriate variational approach.

But for now, traditional vanishing derivative of Area function wrt "a" will suffice.

Length constraint: $l = 2 \lambda \map {\arctan} {\frac a {\sqrt{\lambda^2 - a^2} } }$

Solution for $\lambda$ is transcendental.

Solutions for $a$ are:


 * $a = \pm \frac {\lambda \map {\tan}{\frac l {2\lambda} } }{\sqrt{1 + \map {\tan^2}{\frac l {2\lambda} } } }$.

By assumption, $a > 0$. Square root in the denominator is always positive. $\lambda$ is also always positive.

$l$ is a fixed constant constraining our problem. $\lambda$ is a variable over which we variate. From before we now that $\lambda$ is somewhere between infinity and $\frac l {\pi}$, because in this case we can have at most a semicircle. Hence, $0 < \frac l {2\lambda} < \frac \pi 2$.

Substitute those into the area equation. Area:


 * $A = \int_{-a}^a \paren{\sqrt{\lambda^2 - x^2} - \sqrt{\lambda^2 - a^2} } \rd x = \paren {\frac 1 2 \paren {x \sqrt{\lambda^2 - x^2} + \lambda^2 \arctan \frac x {\sqrt{\lambda^2 - x^2} } } - x \sqrt{\lambda^2 - a^2} }\vert_{-a}^a = \frac 1 2 \paren {\lambda^2 \arctan \frac x {\sqrt{\lambda^2 - a^2} } - a \sqrt{\lambda^2 - a^2} }$

becomes :$A = \frac \lambda 2 \paren {l - \lambda \sin \frac l \lambda}$

The only variable left is $\lambda$. Calculate the derivative:


 * $\dfrac {\d A}{\d \lambda} = \frac 1 2 \paren {l + l \cos \frac l \lambda - 2 \lambda \sin \frac l \lambda} = \frac \lambda 2 \paren {\frac l \lambda + \frac l \lambda \cos \frac l \lambda - 2 \sin \frac l \lambda}$

This is solved by $\frac l \lambda = (2k + 1)\pi, k \in \Z$ and $\frac l \lambda = 0$

Substitute these into the formula for the area. In the first case we have:


 * $A = \frac {l^2} {\paren {1 + 2 k} 2\pi}$. From these the area is maximised with k = 0. Then $A = \frac {l^2} {2\pi} = \frac {\pi^2 R^2} {2 \pi} = \frac 1 2 \pi R^2$

For $\lambda = \infty$ apply H'ospital's rule.

$\lim_{\lambda \to \infty} \lambda \sin \frac l \lambda = \infty \dot 0 =\lim_{\lambda \to \infty} \frac{\sin \frac l \lambda}{\frac 1 \lambda} = \lim_{\lambda \to \infty} \frac {- \frac l {\lambda^2}\cos \frac l \lambda}{- \frac 1 {\lambda^2} } = l$

$\lim_{\lambda \to \infty} A = \lim_{\lambda \to \infty} \frac \lambda 2 \paren {l - \lambda \sin \frac l \lambda} = \infty \dot 0 = \lim_{\lambda \to \infty} \frac 1 2 \frac {l - \lambda \sin \frac l \lambda} {\frac 1 \lambda} = \frac 1 2 \frac{-\sin \frac 1 \lambda + \frac l \lambda \cos \frac l \lambda}{- \frac 1 {\lambda^2} } = \frac 1 2 \paren{\sin \frac l \lambda - \sin \frac l \lambda} = 0$

The second part with more than a semicircle: substitute again, $\lambda$ is bounded between finite values. Again find the extremum.

Area of more than a semidisk:


 * $A = \pi \frac {\lambda^2}{2} + 2\lambda\sqrt{\lambda^2 - a^2}-\int_{-\lambda}^{-a}\paren{\sqrt{\lambda^2 - a^2}-\sqrt{\lambda^2-x^2}}\rd x-\int_{a}^{\lambda}\paren{\sqrt{\lambda^2 - a^2}-\sqrt{\lambda^2-x^2}}\rd x =$

$\pi \frac {\lambda^2}{2} + 2\lambda\sqrt{\lambda^2 - a^2} - 2 \paren{\lambda - a} \sqrt{\lambda^2-a^2} + \pi \frac{\lambda^2} 2 - a \sqrt{\lambda^2 - a^2} - \lambda^2 \arctan{\frac a {\sqrt{\lambda^2-a^2}}}=$ $=\pi \lambda^2 + a \sqrt{\lambda^2 - a^2} - \lambda^2 \arctan \paren{\frac a {\sqrt{\lambda^2 - a^2} } }$

Length constraint in this case is:


 * $l = 2 \lambda \paren{\pi - \arctan\paren {\frac a {\sqrt{\lambda^2 - a^2} } } }$

It can be solved for $a$ in the following form:


 * $a = \pm \frac {\tan{\frac{2\pi\lambda - l}{2 \lambda}}}{\sqrt{1+\tan^2{\frac{2\pi\lambda - l}{2 \lambda}}}}$

Substitution to the area formula leads to


 * $A = \frac {\lambda}{2} \paren{l - \lambda \sin \frac l \lambda}$

Derivative reads as


 * $\frac 1 2 \paren{l + l \cos \frac l \lambda -2 \lambda \sin \frac l \lambda}$

Within interval $\frac l {2\pi} =< \lambda =< \frac l \pi$ this is extremised at $\lambda = l \lambda$.

Together with the previous result this means that area is maximised for $\lambda = \frac l \pi$

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


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

Propagation of light in inhomogeneous medium
Suppose 3d space is filled with an optically inhomogeneous medium such that at each point speed of light $v = \map v {x, y, z}$

If the curve joining two points $A$ and $B$ is specified by $y = \map y x$ and $z = \map z x$

then the time it takes to traverse the curve equals:


 * $\displaystyle \int_a^b \frac {\sqrt{1 + y'^2 + z'^2} } {\map v {x, y, z} }$

Euler's Equations:


 * $\displaystyle \dfrac {\partial v} {\partial y} \frac {\sqrt{1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \frac {y'} {v\sqrt {1 + y'^2 + z'^2} } = 0$


 * $\displaystyle \dfrac {\partial v} {\partial z} \frac {\sqrt{1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \frac {z'} {v \sqrt {1 + y'^2 + z'^2} } = 0$

Shortest path on a sphere
Sphere:


 * $x^2 + y^2 + z^2 = a^2$

Curve passes through $\paren {x_0, y_0, z_0}, \paren {x_1, y_1, z_1}$

Length of the curve:


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

Auxiliary functional:


 * $\int_{x_0}^{x_1} \sqbrk {\sqrt{1 + y'^2 + z'^2} + \map {\lambda} x \paren{x^2 + y^2 + z^2} } \rd x$

Euler's Equations


 * $2 y \map \lambda x - \dfrac \d {\d x} \frac {y'} {\sqrt{1 + y'^2 + z'^2} } = 0$


 * $2 z \map \lambda x - \dfrac \d {\d x} \frac {z'} {\sqrt{1 + y'^2 + z'^2} } = 0$

Minimize a functional when endpoints lie on curves
Suppose end points lie on curves $y = \map \phi x$, $y = \map \psi x$


 * $\displaystyle \delta J = F_{y'}|_{x=x_1}\delta y_1 + \paren {F-F_{y'}y'}|_{x=x_1}\delta x_1-F_{y'}|_{x=x_0}\delta y_0 - \paren {F - F_{y'}y'}|_{x=x0}\delta x_0$


 * $\displaystyle \delta J = \paren {F_{y'}\psi' + F - y' F_{y'} }|_{x=x_1} \delta x_1 - \paren {F_{y'}\phi' + F - y' F_{y'} }|_{x=x_0}\delta x_0 = 0$


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


 * $\sqbrk {F + \paren {\psi' - y'}F_{y'} }|_{x=x_1}=0$

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 $H = \const$

Kinetic energy

 * $T = \frac 1 2 \sum_{i = 1}^n m_i \paren {\dot {x_i}^2 + \dot {y_i}^2 + \dot {z_i}^2}$

Potential energy

 * $U = \map U {t, x_1, y_1, \ldots x_n, y_n, z_n}$

Force:


 * $X-i = - \dfrac {\partial U} {\partial x_i}$


 * $Y_i = - \dfrac {\partial U} {\partial y_i}$


 * $Z-i = - \dfrac {\partial U} {\partial z_i}$

Lagrangian Function of the system of particles

 * $L = T - U$

Principle of least action
The motion of a system of $n$ particles during the time interval $\sqbrk {t_0, t_1}$ is described by those functions $\map {x_i} t$, $\map {y_i} t$, $\map {z_i} t$, $1 \le i \le n$ for which the integral


 * $\int_{t_0}^{t_1} L \rd t$

called the action, is a minimum.

Proof
Euler's equations


 * $\dfrac L {x_i} - \dfrac \d {\d t} \dfrac {\partial L} {\partial \dot{x_i}}$


 * $\dfrac L {y_i} - \dfrac \d {\d t} \dfrac {\partial L} {\partial \dot{y_i}}$


 * $\dfrac L {z_i} - \dfrac \d {\d t} \dfrac {\partial L} {\partial \dot{z_i}}$

These can be rewritten as:


 * $- \dfrac {\partial U} {\partial x_i} - \dfrac \d {\d t} m_i \dot {x_i} = 0$


 * $- \dfrac {\partial U} {y_i} - \dfrac \d {\d t} m_i \dot {y_i} = 0$


 * $- \dfrac {\partial U} {z_i} - \dfrac \d {\d t} m_i \dot {z_i} = 0$

Since the derivatives are components of the force acting on the $i$th particle, the system reduces to


 * $m_i \ddot {x_i} = X_i$


 * $m_i \ddot {y_i} = Y_i$


 * $m_i \ddot {z_i} = Z_i$

Hamiltonian

 * $S = \int_{t_0}^{t_1} L \rd t = \int_{t_0}^{t_1} \paren {T - U} \rd t$


 * $p_{ix} = \dfrac L {\dot {x_i} } = m_i \dot {x_i}$


 * $p_{iy} = \dfrac L {\dot {y_i} } = m_i \dot {y_i}$


 * $p_{iz} = \dfrac L {\dot {z_i} } = m_i \dot {z_i}$


 * $H = \sum_{i = 1}^n \paren {\dot {x_i} p_{ix} + \dot {y_i} p_{iy} + \dot {z_i} p_{iz} } - L = 2 T - \paren {T - U} = T + U$

Conservation of momentum

 * $x^* = \map \Phi {x, y, y'; \epsilon} = x$


 * $y_i^* = \map {\Psi_i} {x, y, y'; \epsilon}$

implies the first integral


 * $\sum_{i = 1}^n$ F_{y_i} \psi_i = \const

where


 * $\map {\psi_i} {x, y, y'} = \dfrac {\partial \map {\Psi_i} {x, y, y'; \epsilon} } {\partial \epsilon} \vert_{\epsilon = 0}$

in this case:


 * $\map \phi {x, y, y'} = \dfrac {\partial \Phi {x, y, y'; \epsilon} } {\partial \epsilon} \vert_{\epsilon = 0} = 0$

The invariance of the functional under


 * $x_i^* = x_i + \epsilon, y_i^* = y_i, z_i^* = z_i$

implies that


 * $\sum_{i = 1}^n \dfrac {\partial L} {\partial \dot {x_i} } = \const$

or


 * $\sum_{i = 1}^n p_{i x} = \const$


 * $\sum_{i = 1}^n p_{i y} = \const$


 * $\sum_{i = 1}^n p_{i z} = \const$

Momentum of the system:


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

Conservation of angular momentum

 * $x_i^* = x_i \cos \epsilon + y_i \sin \epsilon$


 * $y_i^* = -x_i \sin \epsilon + y_i \cos \epsilon$


 * $z_i^* = z_i$

In this case:


 * $\psi_{ix} = \dfrac {\partial {x_i^*} } {\partial \epsilon} \vert_{\epsilon = 0} = y_i$


 * $\psi_{iy} = \dfrac {\partial {y_i^*} } {\partial \epsilon} \vert_{\epsilon = 0} = -x_i$


 * $\psi_{iz} = \dfrac {\partial {z_i^*} } {\partial \epsilon} \vert_{\epsilon = 0} = 0$

Noether's theorem implies


 * $\sum_{i = 1}^n \paren {\dfrac {\partial L} {\partial \dot {x_i} }y_i - \dfrac {\partial L} {\partial \dot {y_i} }x_i} = \const$

i.e.


 * $\sum_{i = 1}^n \paren {p_{ix}y_i - p_{iy}x_i} = \const$

(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)

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