Lagrangian mechanics

The Lagrangian, \(L: (\mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}) \rightarrow \mathbb{R}\) is simply a functional:

\begin{align*} L = L(t, f(t), f'(t)) \end{align*}

Where the Lagrangian represents some metric by which we calculate how optimized \(f(x)\) is. The action:

\begin{align*} J[f] = \int_{a}^{b}L(t, f(t), f'(t))dt \\ \end{align*}

Defines the actual relationship between \(f(t)\) and its level of optimization, where \(a\) and \(b\) represent the start and end points for a certain curve. For example, if you wanted to minimize the surface area of something, \(a\) and \(b\) would be the starting and end points of the surface.

We first define some function:

\begin{align*} g(t) := f(t) + \epsilon \nu(t) \end{align*}

Where \(f(t)\) is our optimized function and \(\nu(t)\) represents some function we add to \(f(t)\) such that we perturb it by some small amount. Now \(\epsilon\) is a small number such that the perturbation is small. Note that when \(\epsilon = 0\), \(g(t) = f(t)\), our optimized function.

\begin{align*} J[g] = \int_{a}^{b}L(t, g(t), g'(t))dt \end{align*}

Now \(J[g]\) is optimized when \(g(t)\) is a maximum or minimum with respect to the Lagrangian. \(\frac{dJ}{d\epsilon}\) represents the extent to which the action changes when the perturbation changes. When \(\epsilon = 0\), \(g(t) = f(t)\), which means \(\frac{dJ}{d\epsilon}\) evaluated at \(\epsilon = 0\) should be zero, by definition of maxima and minima.

\begin{align*} \frac{dJ[g]}{d\epsilon} = \int_{a}^{b}\frac{dL}{d\epsilon}dt \end{align*}

By the multivariable chain rule:

\begin{align*} \frac{dL}{d\epsilon} = \frac{\partial L}{\partial t}\frac{dt}{d\epsilon} + \frac{\partial L}{\partial g}\frac{dg}{d\epsilon} + \frac{\partial L}{\partial g'}\frac{dg'}{d\epsilon} \end{align*}

because \(t\) does not depend on \(\epsilon\), \(g = f + \epsilon\nu\), and \(g' = f' + \epsilon\nu'\):

\begin{align*} \frac{dL}{d\epsilon} = \frac{\partial L}{\partial g}\nu(t) + \frac{\partial L}{\partial g'}\nu'(t) \end{align*}

now substituting back into the integral:

\begin{align*} \frac{dJ}{d\epsilon} = \int_{a}^{b}(\frac{\partial L}{\partial g}\nu(t) + \frac{\partial L}{\partial g'}\nu'(t))dt \end{align*}

applying integration by parts to the right side:

\begin{align*} \frac{dJ}{d\epsilon} = \int_{a}^{b}\frac{\partial L}{\partial g}\nu(t)dt + \nu(t)\frac{\partial L}{\partial g'}\bigg|_{a}^{b} - \int_{a}^{b}\nu(t)\frac{d}{dt}\frac{\partial L}{\partial g'}dt \end{align*}

now \(\nu(t)\) can be any perturbation of \(f(t)\) but the boundary conditions must stay the same (every function that we are considering for optimization must have the same start and end points); therefore, \(\nu(a) = \nu(b) = 0\). We can evaluate the bar to be 0 as a result. Doing this, combining the integral, then factoring out \(\nu(t)\):

\begin{align*} \frac{dJ}{d\epsilon} = \int_{a}^{b}\nu(t)(\frac{\partial L}{\partial g} - \frac{d}{dt}\frac{\partial L}{\partial g'})dt \end{align*}

Now we finally set \(\epsilon = 0\). This means \(g(t) = f(t)\), \(g'(t) = f'(t)\), and \(\frac{dJ}{d\epsilon} = 0\):

\begin{align*} 0 = \int_{a}^{b}\nu(t)(\frac{\partial L}{\partial f} - \frac{d}{dt}\frac{\partial L}{\partial f'})dt \end{align*}

And now because \(\nu(t)\) can be an arbitrarily large or small valued function as long as the boundary conditions remain the same and the left hand side must be zero, we get the Euler-Lagrange equation:

\begin{align*} \frac{\partial L}{\partial f} - \frac{d}{dt}\frac{\partial L}{\partial f'} = 0 \end{align*}

This is because the integral implies that for all selections for this function \(\nu(t)\), \(\nu(t)(\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'}) = 0\). Because \(\nu(t)\) can be any function assuming it satisfies the boundary conditions, this can only be the case if \(\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'} = 0\). In physics, we re-cast \(f\) as \(q\) and \(f'\) as \(\dot{q}\), where \(q\) and \(\dot{q}\) are the generalized coordinates and generalized velocities respectively.

The Hamiltonian represents the total energy in the system; it is the Legendre Transformation of the Lagrangian. Applying the Legendre Transformation to the Lagrangian for coordinate \(\dot{q}\):

\begin{align*} L = \frac{1}{2}m\dot{q}^{2} - V(q) \\ H = \frac{\partial L}{\partial \dot{q}}\dot{q} - L \end{align*}

the Hamiltonian is defined as:

\begin{align*} H(q, p) = \sum _{i}p_{i}\dot{q_{i}} - L(q, \dot{q}) \end{align*}

Or:

\begin{align*} H(q, p) = \frac{p^{2}}{2m} + V(q) \end{align*}

where \(p\) is the generalized momentum, and \(q\) is a generalized coordinate. This results in two differential equations, the first of which is:

\begin{align*} \frac{\partial H}{\partial p_{i}} = \dot{q_{i}} \end{align*}

which follows directly from the Hamiltonian definition. Then, from the Euler-Lagrange equation:

\begin{align*} L = \sum_{i}p_{i}\dot{q_{i}} - H \\ \frac{\partial(\sum_{i}p_{i}\dot{q_{i}} - H)}{\partial q_{i}} - \frac{d}{dt}\frac{\partial(\sum_{i}p_{i}\dot{q_{i}} - H)}{\partial \dot{q_{i}}} = 0 \\ - \frac{\partial H}{\partial q_{i}} = \frac{dp_{i}}{dt} \\ \frac{\partial H}{\partial q_{i}} = - \frac{dp_{i}}{dt} \end{align*}

Although the generalized coordinate system in question does not have to be linear, we can encode all the differential equations for all the coordinates at once with the del operator:

\begin{align*} \vec{\nabla}_{p}H = \frac{d\vec{q}}{dt} \\ \vec{\nabla}_{q}H = -\frac{d\vec{p}}{dt} \end{align*}

this notation isn't standard and I kind of made it up, but I think it works, as long as you don't take the divergence or the curl of this system to really mean anything. Note that in both the Hamiltonian formulation and Lagrangian formulation, the differential equations reduce to Newtonian mechanics if we are working in a linear coordinate system with energy conservation.