Explaining the General Brachistochrone Problem

Consider a problem about the curve of fastest descent in the following generalized statement. Suppose that we have a Lagrangian system $$L(x,\dot x)=\frac{1}{2}g_{ij}(x)\dot x^i\dot x^j-V(x),\quad x=(x^1,\ldots, x^m)\in M.\qquad (*)$$ Here $M$ is a smooth configuration manifold, $\mathrm{dim}\,M=m$. The functions $g_{ij}$ form a Riemann metric in $M$. Assume also that our system is constrained with the following (possibly nonholonomic) constraints $$a^k_l(x)\dot x^l=0,\quad k=1,\ldots, n<m.\qquad (**)$$

Fix two points $x_1,x_2\in M$.  There is a continuous family  of smooth curves $y(s)$ on $M$ that connect the points $x_i$ i.e. $y(s_i)=x_i,\quad s\in [s_1,s_2],\quad i=1,2$.  Consider such a curve as an additional ideal constraint of our system so that  system (*)-(**) can move only along the curve $y(s)$.

We are looking for a curve $y(s)$ such that it takes minimal time for the system (*)-(**) to slide from $x_1$ to $x_2$ along this curve.

 

The parameter  $s$ is a generalized coordinate in a new system with a single degree of freedom and the Lagrangian

$$L^*(s,\dot s)=\frac{1}{2}g(s)\dot s^2-v(s),\quad v(s)=V(y(s)),\quad g(s)=g_{ij}(y(s))\frac{d y^i(s)}{ds}\frac{d y^j(s)}{ds}.$$

 

The constraints (**) imply the following conditions on the curve $y(s)$: $$a^k_l(y(s))\frac{d y^l(s)}{ds}=0.\qquad (!!).$$ The geometrical sense of this condition is clear: the tangent vector to the curve $y$ must belong to the distribution (**).

In other words let the curve $y$ be fixed and satisfies (!!). We can solve system with Lagrangian $L^*$  and obtain $s=s(t)$. So that the function $x(t)=y(s(t))$ satisfies (**).

The system with Lagrangian $L^*$ has the “energy” integral $H(s,\dot s)=\frac{1}{2}g(s)\dot s^2+v(s).$

Integrating the corresponding equation $\frac{1}{2}g(s)\dot s^2+v(s)=h=const$ we get

$$T=t_2-t_1=\int_{s_1}^{s_2}\sqrt{\frac{g(s)}{2(h-v(s))}}ds.$$ This is the time which takes our system to move from the point $y(s_1)$ to the point $y(s_2)$.

Rewrite this formula in the detailed form:

$$T(y(\cdot))=\int_{s_1}^{s_2}\sqrt{\frac{1}{2(h-v(s))}g_{ij}(y(s))\frac{d y^i(s)}{ds}\frac{d y^j(s)}{ds}}ds$$

Thus the brachistochrone that connects the points $x_1,x_2$ on the energy level $h$ is an extremal of the functional $T$ in the class of curves $y(s)$ such that $y(s_i)=x_i$ and (!!) is fulfilled.

In the absence of constraints (!!) the problem about the brachistochrone  turns to a problem about the geodesics of the metric

$$g_{ij}^*=\frac{1}{2(h-v(s))}g_{ij}.$$ In the general case it is a vaconomic type variational problem.