## Action integral

The problem statement, all variables and given/known data

Show the stationary value of,

$$J = \int_{a}^{b} dt~L(...;x_i;\dot{x}_i;...;t)$$

subject to the constraint,

$$\phi(...;x_i,\dot{x}_i;...;t) = 0$$

is given by the free variation of,

$$I = \int_{a}^{b} dt~F = \int_{a}^{b}dt~\left[L(...;x_i;\dot{x}_i;...;t)-\lambda(t)\phi(...;x_i,\dot{x}_i;...;t)\right]$$

The attempt at a solution

Not sure where to start here; or really what's wanted... Do I start with $J$ and $\phi$ and get to the variation of $I$?

Is the free variation of $I$ given by,

$$\delta I = \int_{a}^{b}dt~\left[\frac{\partial F}{\partial x} \delta x \frac{\partial F}{\partial \dot{x}} \delta{\dot{x}}\right]$$ ?
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 It's been a while since I did this, but you may want to take a look at "Lagrange multipliers", that might get you on track.