PX285 - C6a - multi-coordinate problems

for problems with more than one coordinate, $d > 1$ , $T$ and $V$ can be written down in terms of the positions and velocities of these coordinates
hence, the lagrangian can be determined and trajectories that minimize $A$ can be identified
denote the coordinates using $q$ - eg: $q_{1}, q_{2}, \dots q_{d} ⟹ v_{i} = {\dot{q}}_{i}$
the lagrangian:

L (q_{1}, q_{2}, \dots q_{d}, {\dot{q}}_{1}, {\dot{q}}_{2}, \dots {\dot{q}}_{d})

the action:

A = \int_{t_{0}}^{t_{1}} L (q_{1}, q_{2}, \dots q_{d}, {\dot{q}}_{1}, {\dot{q}}_{2}, \dots {\dot{q}}_{d}) d t

identify a path that is extremal, ie: the action is a minimum for small variations about that path
vary about this path:
$q_{i} (t) \to q_{i} (t) + a_{i} (t)$
${\dot{q}}_{i} (t) \to {\dot{q}}_{i} (t) + {\dot{a}}_{i} (t)$
the lagrangian will be a function of $d$ positions and $d$ velocities:

\begin{matrix} L (q_{1} + a_{1}, \dots q_{d} + a_{d}, {\dot{q}}_{1} + {\dot{a}}_{1}, \dots {\dot{q}}_{d} + {\dot{a}}_{d}) \\ \approx L (q_{1}, \dots q_{d}, {\dot{q}}_{1}, \dots {\dot{q}}_{d}) + s m a l l \\ \approx L (q_{1}, \dots q_{d}, {\dot{q}}_{1}, \dots {\dot{q}}_{d}) + \sum_{i = 1}^{d} \frac{\partial L}{\partial q_{i}} a_{i} + \sum_{i = 1}^{d} \frac{\partial L}{\partial {\dot{q}}_{i}} {\dot{a}}_{i} + \dots \end{matrix}

the action:

A \to A + δ A = A + \sum_{i = 1}^{d} \int_{t_{0}}^{t_{1}} [\frac{\partial L}{\partial q_{i}} a_{i} + \frac{\partial L}{\partial {\dot{q}}_{i}} {\dot{a}}_{i}] d t

using integration parts on all terms:

\int_{t_{0}}^{t_{1}} {\dot{a}}_{i} \frac{\partial L}{\partial {\dot{q}}_{i}} d t = {[\frac{\partial L}{\partial q_{i} a_{i}}]}_{t_{0}}^{t_{1}} - \int_{t_{0}}^{t_{1}} a_{i} \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}} d t

- $a_{i} (t_{0}) = a_{i} (t_{1}) = 0 \forall i ⟹$ first term will be zero:
$$ \int_{t_{0}}^{t_{1}} \dot a_{i}\frac{\partial L}{\partial \dot q_{i}},dt = - \int_{t_{0}}^{t_{1}} a_{i} \frac{d}{dt} \frac{\partial L}{\partial \dot q_{i}},dt$$

substituting back to $δ A$ , which must vanish for any excursion, $a_{i} (t)$ , from the classical trajectory:

δ A = \sum_{i = 1}^{d} \int_{t_{0}}^{t_{1}} a_{i} (t) [\frac{\partial L}{\partial q_{i}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}}] d t = 0

this is only satisfied for any excursion, $a_{i} (t)$ , if the following euler-lagrange equation is true:

\frac{\partial L}{\partial q_{i}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}} = 0 \forall i

this gives $d$ euler-lagrange equations, one for each coordinate-velocity pair