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I have a math problem coming from physics.

How do you push an object $1$ m in $1$ sec under the force $F=-v^2$, so the energy used is minimal?

We have to minimize

$$\int_0^1\left(\frac{1}{2}a^2+v'(t)*v(t)+v(t)^3 \right) dt$$

under the constraints

\begin{align} v(0) &= a \\ \int_{0}^{1} v(t) dt &= 1 \\ v'(t) + v(t)^{2} &\ge 0\qquad for\, all \,\,t:\quad 0<t<1 \end{align}

My problem is that I can't get the inequality constraint working with the Euler-Lagrange equation in a meaningful way.

I have tried to guess some functions for $v(t)$, and if the mass of the object is $1$ kg, my record for the energy used is $1.31$ Joule.

Is it possible to break this record using calculus of variations? - or using other methods?

Foldager
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  • There is no explicit $t$ dependence, so there is a first integral $v' \partial_{v'} L - L = C$. It yields the solution $v = a$ after the first constraint. The solution only matches the second constraint though if $a = 1$. The final condition is then satisfied. But I think there might be a problem with the question. – Matthew Cassell Dec 06 '19 at 16:12
  • If v(t)=1, the integral gives 1.5. Therefore the energi used is 1.5 Joule and this is not a record. There must exist an optimal v(t), that minimize the integral. – Foldager Dec 06 '19 at 16:35
  • Crossposted to https://physics.stackexchange.com/q/518060/2451 – Qmechanic Dec 06 '19 at 20:11

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