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Homework Statement
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] ?
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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