## Sunday, 14 September 2008

### Lagrangian Mechanics: From the Principle of Least Action to the Euler-Lagrange Equation

A quantity known as the Action, A is defined as
A = \int L \,dt

where L is known as the Lagrangian. The simplest Lagrangian is given as the kinetic energy T of a system minus the potential energy V of the system:
L(x,t) = T(x,t) - V(x,t)

By extremising the Action (minimising it, in this case) we can obtain the Euler-Lagrange Equation, a key equation for Lagrangian and Hamiltonian mechanics. The concept is also used extensively in quantum mechanics and particle physics, particularly when dealing with gauge theories.

Consider an extremised path x(t) from a point x(t0) to a point x(t1), and some excursion from this path, as shown in the figure below. The excursion is given by some small function a(t), and the velocity is changed accordingly:
\begin{align*}x(t) \rightarrow x(t) + a(t) \\v(t) \rightarrow v(t) + \dot{a}(t)\end{align*}

If we take x(t) to be the extremal path from x(t0) to x(t1), with the end-points fixed, and a(t) to be some small but general excursion from that path which must pass through the end-points, we can assert that:
a(t_0) = a(t_1) = 0

The Lagrangian will be changed as a result of these excursions. To first-order in small a(t), the Lagrangian transforms as:
\begin{align*}L(x,v) \rightarrow L(x+a, v+\dot{a}) \\= L(x, v) + a(t)\frac{\partial L}{\partial x} + \dot{a}(t)\frac{\partial L}{\partial v}\end{align*}

The Action therefore transforms according to
A \rightarrow A + \delta A
where:
\delta A = \int_{t_0}^{t_1}\,dt \left( a(t)\frac{\partial L}{\partial x} + \frac{da}{dt}\frac{\partial L}{\partial v} \right)

The second term in the brackets above can be integrated by parts:
\int_{t_0}^{t_1}\,dt \frac{da}{dt}\frac{\partial L}{\partial v} = \left[ a(t)\frac{\partial L}{\partial v} \right]_{t_0}^{t_1} - \int_{t_0}^{t_1}dt\,a(t)\frac{d}{dt}\frac{\partial L}{\partial v}

Since a(t0) = a(t1) = 0, the integrated part (in square brackets) vanishes, leaving the following form for delta A:
\delta A = \int dt\,a(t)\left(\frac{\partial L}{\partial x} - \frac{d}{dt}\frac{\partial L}{\partial v}\right)

For arbitrary a(t), to minimise the action (by having delta A zero), the part in brackets must be zero. This is the Euler-Lagrange Equation, rewritten for a generalised coordinate q (in place of x) and using the time-derivative of q in place of the velocity v, above:
\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0

As I mentioned above, the Euler-Lagrange Equation is critical for the understanding of several aspects of quantum mechanics and gauge theories. Since I plan on making posts about some of these topics in the future, I felt I should begin by explaining Lagrangian mechanics, rather than jumping in at the deep end.