Lagrangian mechanics ed
An elegant way of describing dynamics.
Table of Contents
Starting point ed
A system's configuration is constrained to a n dimensional manifold \(Q\). Each point \(q \in Q\) describes a configuration. Also we use coordinates \(q^i\) on \(Q\) (yes, same symbol...). For now, \(Q\) does not change with time.
Over time a configuration will change and form a parametrices curve \(q(t)\) in \(Q\). The tangential vectors are \(\dot q(t)\) and live in the tangential bundle \(TQ\).
The system's dynamics will be encoded in the Lagrangian \( L : TQ \times \mathbb R \rightarrow \mathbb R \). And we define a curve's action as the functional\[ S[q(t)] = \int_{t_0}^{t_1} L(q(t), \dot q(t), t) dt \]
Euler-Lagrange ed
The correct time evolution \(q(t)\) is found by minimizing the action:\[ S[q] \stackrel{!}{=} \operatorname{min} \]
We can find the solution by taking the derivative\[ 0 \stackrel{!}{=} \partial_\epsilon \, S[q(t) + \epsilon \, \phi(t)]_{|\epsilon=0} \qquad \forall \phi \]
That is\[ 0 = \partial_\epsilon \int_{t_0}^{t_1} L(q(t) + \epsilon \, \phi(t), \dot q(t) + \epsilon \, \dot \phi(t), t) \, dt_{|\epsilon=0} \]\[ 0 = \int_{t_0}^{t_1} \left( \frac{\partial L}{\partial q^i}(q(t),\dot q(t), t) \, \phi^i(t) + \frac{\partial L}{\partial \dot q^i}(q(t),\dot q(t), t) \, \dot \phi^i(t) \right) dt \]
The second term can be seen as\[ \frac{\partial L}{\partial \dot q^i}(\dots) \, \dot \phi^i(t) = \frac{d}{dt} \left(\frac{\partial L}{\partial \dot q^i}(\dots) \, \phi^i(t)\right) - \left(\frac{d}{dt}\frac{\partial L}{\partial \dot q^i}(\dots) \right) \phi^i(t) \]
Since the new middle term is a time derivative, the integral is just evaluation of \(\frac{\partial L}{\partial \dot q}(\dots) \, \phi(t)\) on the interval boundary. ....bla bla bla....
\[ 0 = \int_{t_0}^{t_1} \left( \frac{\partial L}{\partial q^i}(\dots) - \frac{d}{dt}\frac{\partial L}{\partial \dot q^i}(\dots) \right) \phi(t) dt \]
Since the integral vanishes for all \(\phi\), the rest of the integrand has to vanish. In short (Euler Lagrange Equations):\[ \frac{\partial L}{\partial q^i}(\dots) - \frac{d}{dt}\frac{\partial L}{\partial \dot q^i}(\dots) = 0 \]
Discussion ed
The Euler-Lagrange-Equations are an algebraic equation for \(\ddot q\).
The Lagrangian can be separated into (assuming differentiability around \(\dot q = 0\))\[L = L^0(q,t) + L^1_{i}(q,t) \dot q^i + \frac{1}{2} L^2_{ij}(q,t) \dot q^i \dot q^j + L^3 \dots \]Since we're looking for the minimum, we need positivity and the odd \(L^{2k+1}\) to vanish (um....nope....). The even terms can be symmetric.