# Lagrangian mechanics

The Lagrangian $$\mathcal L$$ is a function which summarizes the entire state of a classical mechanical system. In simple systems of $$N$$ particles $$m_k$$ with position $$\vec r_k$$, we have the following non-relativisitc Lagrangian where $$\mathcal U$$ is the potential energy.

$\mathcal L = \mathcal T - \mathcal U \hspace{6mm} \text{ where } \hspace{6mm} \mathcal T = \frac 1 2 \sum_{k=1}^ N m_k\dot r_k^2$

From the Lagrangian arises the notion of Action $$\mathcal S$$, which is defined as the time integral of the Lagrangian between two points in time, representing an integral over some trajectory or path between two configurations of the system.

$\mathcal S = \int_{t_1}^{t_2} L\,dt$

The Principle of Least Action states that out of all possible paths, a system will take the one for which this quantity is stationary to the first order, in other words, $$\delta \mathcal S = 0$$.

#### Lagrange's equations of the first kind

Out of the principle of least action arises the Lagrange equations, which fully describe a system of $$N$$ particles and $$C$$ constraint functions, which are basically restrictions to the configuration of the system.

$\frac{\partial \mathcal L}{\partial \vec r_k} - \frac{d}{dt} \frac{\partial \mathcal L}{\partial \dot{\vec r}_k} + \sum_{i=1}^C \lambda_i \frac{\partial f_i}{\partial \vec r_k} = 0$

For example, a constraint for a pendulum of length $$\ell$$ is $$f(x,y) = x^2 + y^2 = \ell^2$$.

#### Euler-Lagrange equations (second kind)

A much more useful method for Lagrangian mechanics comes from the Euler-Lagrange equations, whereby, if you can describe a system with generalized coordinates $$\vec q_j$$, which describe a configuration space embodying the constraints on the system, we no longer need to worry about Lagrange multipliers, and have a simpler relation:

$\frac d {dt} \left(\frac{\partial\mathcal L}{\partial \dot q_j}\right) = \frac {\partial \mathcal L}{\partial q_j}$

For example, the pendulum above can instead be described by a generalized coordinate $$\varphi$$ where $$x = \ell\sin\varphi,\, y=\ell\cos\varphi$$. This completely contains the information from the restriction of $$f(x,y)$$ to $$\ell^2$$.

#### Example: 2 dim'l pendulum

Continuing on with our pendulum, let's use Lagrangian mechanics to find the equations of motion. First, we need our Lagrangian $$\mathcal L(\vec q, \dot {\vec q}, t)$$ in terms of our generalized coordinate $$\varphi(t)$$:

$\vec r = \ell\hat\varphi, \,\, \dot r = \ell\dot\varphi$ $\mathcal L = \mathcal T - \mathcal U = \frac 1 2 m\dot r^2 + mgy = \frac 1 2 m \ell^2 \dot \varphi^2 + mg\ell\cos\varphi$

Now, we can apply the Euler-Lagrange equations for $$\varphi$$,

$\left[ \frac d {dt} \left(\frac{\partial\mathcal L}{\partial \dot \varphi}\right) = \frac d {dt} \left( m\ell^2\dot\varphi \right) = m\ell^2\ddot\varphi \right] = \left[ \frac {\partial \mathcal L}{\partial \varphi} = -mg\ell \sin\varphi \right]$ $\implies \ddot \varphi + \frac g \ell \sin\varphi = 0$

Which, of course, is the equation of motion for a simple pendulum, and we can even apply our small angle approximation:

$\varphi \ll 1 \implies \sin\varphi\approx\varphi \implies \ddot\varphi + \frac g \ell \varphi = 0$