Action integral

CNX

The problem statement, all variables and given/known data

Show the stationary value of,

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

subject to the constraint,

[tex]\phi(...;x_i,\dot{x}_i;...;t) = 0[/tex]

is given by the free variation of,

[tex]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][/tex]

The attempt at a solution

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

Is the free variation of [itex]I[/itex] given by,

[tex]\delta I = \int_{a}^{b}dt~\left[\frac{\partial F}{\partial x} \delta x \frac{\partial F}{\partial \dot{x}} \delta{\dot{x}}\right][/tex] ?

Matterwave

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.