The Hamilton--Jacobi Equation

Find the field function $p(x,y)$ and the field of extremals itself, passing through the origin, of the integral

It is given that a family of extremals passes through the point $(x_{0},y_{0})$. Denoting by $(x_{1},y_{1})$ the point of intersection of these extremals with some transversal, show that the integral

taken along the extremal has a constant value.

A one-parameter family of extremals is intersected by two transversals. Prove that the integral $\int f(x,y,y') \, dx$, taken along any extremal between the points of intersection with the transversals, has a constant value.

Among the lines joining the line $x=0$ with the point $A(x_{1},y_{1})$, where $x_{1} > 0$, $y_{1} > 0$, find the one that gives the minimum of the integral

Also form the Hamilton--Jacobi equation and find the equidistant curves for the extremals transversally intersecting the line $x=0$.

Find the Hamilton--Jacobi equation and the transversals for the field of extremals $y=ax$ of the integral $\int y'^{2} \, dx$.

Find the transversals for the field of extremals $y=ax$ of the integral

Find the transversals for the field of extremals $y=x+a$ of the integral

Find the transversals for the field of extremals $y = x - \frac{x^{2}}{a}$ of the integral

Find the extremals of the integral

by solving the Hamilton--Jacobi equation, and conversely --- find the solution of the Hamilton--Jacobi equation, starting from the first integral of Euler's equation.

Find the extremals of the integral

by means of solving the Hamilton--Jacobi equation.

Knowing that the paths of light rays are curves along which light propagates in the shortest time, and knowing that the speed of light is inversely proportional to the refractive index of the medium $n$, show that the brachistochrone (problem 5989) is the path of light in a medium in which the refractive index $n$ is expressed by the formula

In connection with this, find the paths of light rays in such a medium and the equation of the wave surfaces. For the brachistochrones, find the synchrones, i.e., the curves reached simultaneously by points moving from the point $(0,0)$ along different brachistochrones.

Find the trajectory of a material point moving under the influence of gravity, starting from the principle of least action, according to which for the trajectory of the point the first variation of the "action" integral vanishes:

Here $U$ is the potential; in this case $U = -gy$, if the axis $OY$ is directed upward.

Form the Hamilton--Jacobi equation and, having solved it, find the extremals, in particular those passing through the origin. Investigate the character of the minimum of the integral.

Find the trajectory for the motion of a point in a plane under the action of a repulsive force from the axis $OX$, proportional to the distance of the point from this line and directed parallel to the axis $OY$, under the condition that the energy integral has the form $\frac{v^{2}}{2} - \frac{y^{2}}{2} = 0$, starting from the "action" integral

A material point describes the circle $r = 2R\cos \theta$ (in polar coordinates $r$ and $\theta$) of radius $R$ under the action of a central force $\frac{k}{r^{5}}$, inversely proportional to the fifth power of the distance from the center of force, lying at the origin. Show that the "action" integral, taken along any arc of this circle: $-\frac{\pi }{2} < \theta_{1} \leq \theta \leq \theta_{2} < \frac{\pi }{2}$, is a strong minimum.

Study the motion of a material point under the action of a central force proportional to the distance from the center $O$ of the force, starting from the principle of least action and applying the Jacobi--Hamilton method.

Show that at the conjugate critical points of any trajectory of the point, regarded as an extremal of the "action" integral, the tangents are mutually perpendicular.

Solving the Hamilton--Jacobi equation, find the geodesic lines on a Liouville surface, for which the line element has the form

Prove that for surfaces whose line element is determined by the equality

the Jacobi--Hamilton equation for the problem of the geodesic line has the form

Find the shortest distance between a given point and the parabola $y^{2} = 2px$.

Find the minimum of the integral

if the values at the endpoints of the interval are not given.

Find the shortest distance between two circles $C_{1}$ and $C_{2}$, lying in one plane.

For the problem of finding the minimum of the integral

the Hamilton--Jacobi equation is given: $w_{x} = H(x,y,w_{y})$. Find the function $f(x,y,y')$.

Example. $\left(\frac{\partial w}{\partial x}\right)^{2} + \left(\frac{\partial w}{\partial y}\right)^{2} = 1$.

The Hamilton--Jacobi equation for the problem of the minimum of the integral $\int f(x,y,y') \, dx$ has the form

Find the function $f(x,y,y')$.