Brachistochrone

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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