(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Optimize the following cost integral

[tex] x(1)^2 + \displaystyle \int_0^1 (x^2 + \dot{x}^2) dx [/tex]

subject to x(0) =1, x(1) is free

2. Relevant equations

Now our prof showed us a method of doing this. In general, if we want to minimize

[tex] f(b,x(b)) + \displaystyle \int_a^b L(t,x,\dot{x}) dx [/tex]

where x(b) is free, then we can change the problem to minimizing

[tex] \displaystyle \int_a^b L(t,x,\dot{x}) + \frac{\partial f}{\partial t} + \sum_i \frac{\partial f}{\partial x_i} \dot{x}_i dx [/tex]

3. The attempt at a solution

Now we he goes through the example above, he changes the Lagrangian to

[tex] \displaystyle \int_0^1 \left[ 2x\dot{x} + (x^2 + \dot{x}^2) \right] dx [/tex]

My problem is that I don't see where [itex] 2x\dot{x} [/itex] comes from. The only way this conforms to the above equation is if f has the form of the original Lagrangian. At least in this case, I figure that [itex] f(t,x(t)) = x(t) [/itex] in which case

[tex] \displaystyle \frac{\partial f}{\partial t} + \sum_i \frac{\partial f}{\partial x_i} \dot{x}_i = \dot{x} + \dot{x} = 2\dot{x} [/tex]

which varies from what he got by the factor of x

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# Homework Help: Calculus of Variations

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