Categories: Classical mechanics, Physics.

**Lagrangian mechanics** is a formulation of classical mechanics, which is equivalent to Newton’s laws, but offers some advantages. Its mathematical backbone is the calculus of variations, and hence it is built on the **principle of least action**, which states that the path taken by a system will be a minimum of the **action** (i.e. energy cost) of that path.

For a moving object with position \(x(t)\) and velocity \(\dot{x}(t)\), we define the Lagrangian \(L\) as the difference between its kinetic and potential energies:

\[\begin{aligned} \boxed{ L(x, \dot{x}, t) \equiv T - V = \frac{1}{2} m \dot{x}^2 - V(x) } \end{aligned}\]

From variational calculus we then get the Euler-Lagrange equation, which in this case turns out to just be Newton’s second law:

\[\begin{aligned} \dv{t} \Big( \pdv{L}{\dot{x}} \Big) = \pdv{L}{x} \qquad \implies \qquad m \ddot{x} = - \pdv{V}{x} = F \end{aligned}\]

But compared to Newtonian mechanics, Lagrangian mechanics scales better for large systems. For example, to describe the dynamics of \(N\) objects \(x_1(t), ..., x_N(t)\), we only need a single \(L\) from which the equations of motion can easily be derived. Getting these equations directly from Newton’s laws could get messy.

At no point have we assumed Cartesian coordinates: the Euler-Lagrange equations keep their form for any independent coordinates \(q_1(t), ..., q_N(t)\):

\[\begin{aligned} \dv{t} \Big( \pdv{L}{\dot{q_n}} \Big) = \pdv{L}{q_n} \end{aligned}\]

We define the **canonical momentum conjugate** \(p_n(t)\) and the **generalized force conjugate** \(F_n(t)\) as follows, such that we can always get Newton’s second law:

\[\begin{aligned} \boxed{ p_n \equiv \pdv{L}{\dot{q}_n} \qquad F_n \equiv \pdv{L}{q_n} } \qquad \implies \qquad \dv{p_n}{t} = F_n \end{aligned}\]

But this is actually a bit misleading, since \(p_n\) need not be a momentum, nor \(F_n\) a force, although often they are. For example, \(p_n\) could be angular momentum, and \(F_n\) torque.

Another advantage of Lagrangian mechanics is that the conserved quantities can be extracted from \(L\) using Noether’s theorem. In the simplest case, if \(L\) does not depend on \(q_n\) (then known as a **cyclic coordinate**), then we know that the “momentum” \(p_n\) is a conserved quantity:

\[\begin{aligned} F_n = \pdv{L}{q_n} = 0 \qquad \implies \qquad \dv{p_n}{t} = 0 \end{aligned}\]

Now, as the number of particles \(N\) increases to infinity, variational calculus will give infinitely many coupled equations, which is obviously impractical.

Such a system can be regarded as continuous, so the \(N\) functions \(q_n\) can be replaced by a single density function \(u(x,t)\). This approach can also be used for continuous fields, in which case the complex conjugate \(u^*\) is often included. The Lagrangian \(L\) then becomes:

\[\begin{aligned} L(u, u^*, u_x, u_x^*, u_t, u_t^*, x, t) = \int_{-\infty}^\infty \mathcal{L}(u, u^*, u_x, u_x^*, u_t, u_t^*, x, t) \dd{x} \end{aligned}\]

Where \(\mathcal{L}\) is known as the **Lagrangian density**. By inserting this into the functional \(J\) used for the derivation of the Euler-Lagrange equations, we get:

\[\begin{aligned} J[u] = \int_{t_0}^{t_1} L \dd{t} = \int_{t_0}^{t_1} \int_{-\infty}^\infty \mathcal{L} \dd{x} \dd{t} \end{aligned}\]

This is simply 2D variational problem, so the Euler-Lagrange equations will be two PDEs:

\[\begin{aligned} 0 &= \pdv{\mathcal{L}}{u} - \pdv{x} \Big( \pdv{\mathcal{L}}{u_x} \Big) - \pdv{t} \Big( \pdv{\mathcal{L}}{u_t} \Big) \\ 0 &= \pdv{\mathcal{L}}{u^*} - \pdv{x} \Big( \pdv{\mathcal{L}}{u_x^*} \Big) - \pdv{t} \Big( \pdv{\mathcal{L}}{u_t^*} \Big) \end{aligned}\]

If \(\mathcal{L}\) is real, then these two Euler-Lagrange equations will in fact be identical.

Finally, note that for abstract fields, the Lagrangian density \(\mathcal{L}\) rarely has a physical interpretation, and is not unique. Instead, it must be reverse-engineered from a relevant equation.

- R. Shankar,
*Principles of quantum mechanics*, 2nd edition, Springer.

© Marcus R.A. Newman, a.k.a. "Prefetch".
Available under CC BY-SA 4.0.