Categories: Classical mechanics, Physics.

Lagrangian mechanics

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.

References

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