General Brachistochrone Problem

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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 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 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.

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2 replies
1. Greg Bernhardt says:

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information,  come to any new conclusions or is it possible to reword the post?

2. stephen8686 says:

It's always fun tackling famous problems. Thanks for posting!