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

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 minimzer 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.


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

Brachistochrone
Starting from the point $P = \paren {a, A}$, a heavy particle slides down a curve in the vertical plane. Find the curve such that the particle reaches the vertical line $x = b \paren {\ne a} in the shortest time.$

Proof
Velocity of motion along the curve:


 * $v = \frac {\d s} {\d t} = \sqrt{1 + y'^2} \frac {\d x} {\d t}$

Hence:


 * $\rd t = \frac {\sqrt {1 + y'^2} } v \rd x = \frac {\sqrt {1 + y'^2} } {\sqrt {2 g y}} \rd x$

Transit time:


 * $T = \int \frac {\sqrt {1 + y'^2} } {\sqrt {2 g y}} \rd x$

Solution - family of cycloids:


 * $x = r \paren {\theta - \sin \theta} + c, y = r \paren {1 - \cos \theta}$

Curve passes through the origin, so $c = 0$

To find $r$, use the second condition:


 * $F_{y'} = \frac {y'} {\sqrt {2 g y} \sqrt {1 + y'^2} } = 0, x = b$

This implies, that the tangent to the curve at its right end point must be horizontal, i.e. $r = \frac b \pi$.

All in all,


 * $x = \frac b \pi \paren {\theta - \sin \theta}, y = \frac b \pi \paren {1 - \cos \theta}$

Proof 2
Suppose that the curve passes through the point $\tuple {x, y}$ for some value of parammeter $t$.

Due to smoothness of the curve, one can define velocity $v$ at a point $\tuple {\map x t, \map y t}$:


 * $v = \dfrac {\d s} {\d t}$

where $\d s$ is an infinitesimal length element.

In Euclidean space we have:


 * $\d s = \sqrt{1 + y'^2} \d x$

Hence:


 * $v = \sqrt{1 + y'^2} \dfrac {\d x} {\d t}$

Due to the symmetries of Euclidean space, the following energy conservation identity holds:


 * $\frac {m v^2} 2 + m g y = E$

where $E$ is a constant of motion.

To determine $E$ use the following initial conditions:


 * $\tuple {\map x 0, \map y 0} = \mathbf 0$


 * $\tuple {\map {\dfrac {\d x} {\d t}} 0, \map {\dfrac {\d y} {\d t}} 0} = \mathbf 0$

Then it follows that:


 * $E = 0$

and:


 * $v = \sqrt{2 g y}$

Then the total travel time, integrated $x \in \closedint a b$ is:


 * $T = \int_a^b \frac {\sqrt{1 + y'^2}}{\sqrt{2 g y}} \rd x$

Application of Euler's Equation yields:


 * $\frac {\sqrt{1 + y'^2}} {\sqrt{2 g y}} - y' \frac {2 y'} {2 \sqrt {2 g y} \sqrt {1 + y'^2}} = c$

or


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

where


 * $ C = \frac 1 {2 c^2 g}$

and $c$ is a real constant.

This equation can be solved for $\map x y$:


 * $x = \pm \paren {\sqrt {C - y} \sqrt y - C \arctan {\sqrt{\frac{y}{C - y}}}+C_1}$

From the initial condition $\tuple {\map x 0, \map y 0} = \mathbf 0$ it follows that:


 * $C_1 = 0$

To bring the solution to parametric form introduce the following parametric dependence:


 * $\sqrt{\frac y {C - y}} = \tan \theta$

which can be solved for $y$:


 * $y = \frac {c \tan^2 \theta} {1 + \tan^2 \theta}$

Substitution into the expression for $x$ results in:


 * $\displaystyle x = \pm c\paren { \frac {\sin 2 \theta} 2 - \theta} = \pm c\paren { \frac {\sin \phi} 2 - \frac \phi 2} = \pm \frac c 2 \paren { {\sin \phi} - \phi}$

We can rewrite the expression for $y$ in the following way:


 * $y = \frac {c \tan^2 \theta} {1 + \tan^2 \theta} = c \tan^2 \theta \cos^2 \theta = c \sin^2 \theta = \frac c 2 \paren {1 - \cos 2 \theta} = \frac c 2 \paren {1 - \cos \phi}$

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:


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


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

Geodesics
Suppose we have a surface $\sigma$ specified by a vector equation:


 * $\mathbf r = \map{\mathbf r} {u, v}$

Shortest curve lying on $\sigma$ and connecting two points is called the geodesic.

A curve on the surface $\mathbf r$ can be specified as $u = \map u t$, $v = \map v t$

The arc length between the points corresponding to $t_1$ and $t_2$ equals


 * $\displaystyle J \sqbrk {u, v} = \int_{t_0}^{t_1} \sqrt {E u'^2 + 2 F u'v' + G v'^2} \rd t$

where $E, F, G$ are the coefficients of the first fundamental form:


 * $E = {\mathbf r}_u \cdot {\mathbf r}_u, F = {\mathbf r}_u \cdot {\mathbf r}_v, G = {\mathbf r}_v \cdot {\mathbf r}_v$

Euler's Equations:


 * $\displaystyle \frac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t}\frac {E u' + F v'} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$


 * $\displaystyle \frac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t}\frac {F u' + G v'} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$

Special case: cylinder

 * $\mathbf r = \paren {a \cos \phi, a \sin \phi, z}$

Then:


 * $E = a^2, F = 0, G = 1$

Geodesics are:


 * $\dfrac \d {\d t} \frac {a^2 \phi'} {\sqrt{a^2 \phi'^2 + z'^2} } = 0, \dfrac \d {\d t} \frac {z'} {\sqrt{a^2 \phi'^2 + z'^2} } = 0$

Integrate both equations, divide one by the other:

$\dfrac {\d z} {\d \phi} = c_1$

Solution is:


 * $z = c_1 \phi + c_2$

Helical lines.

Isoperimetric example

 * $J \sqbrk y = \int_{-a}^a y \rd x$

subject to conditions


 * $\map y {-a} = \map y a = 0, K \sqbrk y = \int_{-a}^a \sqbrk{1 + y'^2} \rd x = l$

Form the functional:


 * $J \sqbrk y + \lambda K \sqbrk y = \int_{-a}^a \paren {y + \lambda \sqrt {1 + y'^2} }\rd x$

Euler's Equation:


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

Implies:


 * $x + \lambda \frac {y'} {\sqrt{1 + y'^2} } = C_1$

Integration yields:


 * $\paren {x - C_1}^2 + \paren {y - C_2}^2 = \lambda^2$

Constants are determined from boundary conditions and the constraint.

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)

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