kuniga.me > NP-Incompleteness > Lagrangian Mechanics
Lagrangian Mechanics
08 Jun 2026
I’ve been reading the book The Theoretical Minimum [1] by Leonard Susskind and George Hrabovsky and the first topic I don’t recall learning in school is Lagrangian mechanics.
In this post I’d like to cover this formulation and connect it with the mathematical concepts we studied previously.
The Lagrangian
Suppose we have a function $q: \mathbb{R} \rightarrow \Omega$. The region $\Omega$ is called a configuration space. If we assume the domain $\mathbb{R}$ is time and $\Omega$ the set of valid positions of a particle in space, then one way to interpret the function $q(t)$ is an oracle that tells us the position of a particle at an instant $t$.
Note that $\Omega$ can encode the position of multiple particles. If talking about one particle in space, we’d have $\mathbb{R}^3$, but for a $N$ particle system, we would have $\mathbb{R}^{3N}$.
Let’s denote by $\dot{q}$ the derivative of $q$ with respect to $t$. Following our previous interpretation, $\dot {q}(t)$ corresponds to the velocity of the particle at an instant $t$.
The Lagrangian is a scalar function $L : \Omega \times \Omega \times \mathbb{R} \rightarrow \mathbb{R}$. In physics, it depends on $q, \dot{q}$ at an instant of time $t$, so we write
\[L(q(t), \dot{q}(t), t)\]but it’s common to write in the abbreviated form:
\[L(q, \dot{q}, t)\]The Action Functional
We define the action functional as:
\[S[q] = \int_{t_0}^{t_1} L(q(t), \dot{q}(t), t) dt\]Note that $S$ is a function of $q$, which is also a function, so $S$ is a functional which returns a scalar. One way to see $q$ is as a curve from $q(t_0)$ to $q(t_1)$. And then we can also interpret $S[q]$ as a “score” for the curve $q$. A more mathematical interpretation of $S$ is of a generalized length. That is, if $q$ is a curve, then
\[\int_{t_0}^{t_1} \norm{\dot{q}(t)} dt\]Is the length of this curve, so we could think of the Lagrangian as a more generalized version of length.
First Variation and the Fréchet Derivative
Now suppose we “wiggle” curve $q$ and obtain another $q_\epsilon$ that coincides with $q$ on $t_0$ and $t_1$. Such path can be represented by:
\[q_\epsilon(t) = q(t) + \epsilon\eta(t)\]Where we can think of $\eta$ as a function yielding some direction and $\epsilon$ an infinitesimal scalar. This is the infinite-dimensional analogue of translating a finite vector towards a direction $v$:
\[q_\epsilon = q + \epsilon v\]Now suppose we want to compute the change in $S$ by moving towards direction $\eta$. We can then define the (first) variation of $S$ as:
\[(1) \quad \delta S[q; \eta] = \lim_{\epsilon \rightarrow 0} \frac{S[q + \epsilon\eta] - S[q]}{\epsilon}\]We can connect it to the Fréchet derivative [2]. Recall that the Fréchet derivative at $q$ is the linear map $DS_q: \Omega \rightarrow \mathbb{R}$ such that
\[\lim_{\norm{h} \rightarrow 0} \frac{\norm{S[q + h] - S[q] - DS_q[h]}}{\norm{h}} = 0\]For every $h \in \Omega$, so it should hold for $h = \eta$ and thus $\delta S[q; \eta] = DS_q[\eta]$. This is the infinite dimension analogue of computing the differential for a multi-valued function $Df_q$ and then applying it to a value $x$: $Df_q(x)$.
Gradient and Riesz Representation
In [2], we discussed the Riesz Representation Theorem which states that if $f$ is a linear functional over a Hilbert space $H$, then there’s a unique vector $y$ in $H$ such that:
\[f(x) = \langle x, y \rangle, \quad \forall x \in H\]Now consider the Fréchet derivative $DS_q$. Since it’s a linear map and we’re assuming $q$ is over the Euclidean space which is a Hilbert one, we can use the theorem and conclude that there’s a unique element in $\Omega$ such that:
\[DS_q[\eta] = \langle g, \eta \rangle, \quad \forall \eta \in H\]Now we denote $g$ as $\nabla S(q) = g$, so we could write:
\[\delta S[q; \eta] = \langle \nabla S(q), \eta \rangle,\]For finite dimensions, if $Df_x$ is the derivative of $f$ at $x$, then:
\[Df_x(v) = \langle \nabla f(x), v \rangle = \nabla f(x) \cdot v\]Principle of Stationary Action
This is also known as the principle of least action, but this is mathematically inaccurate because it implies the minimization of some function but it could be maximization too.
It says that:
\[(2) \quad \delta S[q; \eta] = 0 \quad \forall \eta\]with $\eta(t_0) = \eta(t_1) = 0$, which encodes the fact that the wiggled curve coincides with the original curve’s endpoints. One way to interpret this is that small wiggling of $q$ doesn’t change the score $S$.
For finite dimensions, this is analogous to the expression:
\[\nabla f(x) = 0\]since that implies
\[D f_x(v) = 0\]for all $v$.
Euler-Lagrange Equations
Now let’s plug the Lagrangian into $S$ and see what we get. We start by computing $S[q_\epsilon]$:
\[S[q_\epsilon] = \int_{t_0}^{t_1} L(q_\epsilon(t), \dot{q_\epsilon}(t), t) dt\]By some algebra and calculus we obtain that:
\[\delta S[q; \eta] = \int_{t_0}^{t_1} \left( \frac{\partial L}{\partial q} + \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \eta dt\]Let’s define:
\[E(L) = \frac{\partial L}{\partial q} + \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}\]By $(2)$ we have:
\[(3) \qquad \int_{t_0}^{t_1} E(L) \eta dt = 0 \qquad \forall \eta\]The Fundamental Lemma of the Calculus of Variations states that:
Lemma 1. Let $f: [a, b] \rightarrow \mathbb{R}$ be a continuous function. If
\[\int_{a}^b f(t) \eta(t) dt = 0\]for every smooth function satisfying $\eta(a) = \eta(b) = 0$, then
\[f(t) = 0 \qquad \forall t \in [a, b]\]Applied to our case, this means $E(L) = 0$ and hence:
\[(4) \quad \frac{\partial L}{\partial q} + \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = 0\]which is the Euler-Lagrange equations.
Sobolev Spaces
We mentioned at the beginning that $q$ is a function $q: \mathbb{R} \rightarrow \Omega$ but we haven’t really specified what kind of function this is. We didn’t need it to understand the derivations all the way up to the Euler-Lagrange equations but we shall do it now to keep things rigorous.
In Sobolev Spaces we learned about weak derivatives and different function spaces and their notations such as the space of smooth functions $C^\infty(D)$ for some domain $D$. The function space implicitly assumed in mechanics is $C^2([t_0, t_1])$, the space of twice differentiable functions over the interval $[t_0, t_1]$.
The need for twice-differentiability is due to this term in the Euler-Lagrange equation $(4)$:
\[\frac{d}{dt} \frac{\partial L}{\partial \dot{q}}\]Which expands to:
\[= \frac{\partial^2 L}{\partial q \partial \dot{q}} \dot{q} + \frac{\partial^2 L}{\partial \dot{q}^2} \ddot{q} + \frac{\partial^2 L}{\partial t \partial \dot{q}}\]So the term $\ddot{q}$ requires $q$ to be twice-differentiable. This condition can be too restrictive in the real world, since $q$ might not be differentiable everywhere along $t_0$ to $t_1$.
A more relaxed function space is used, $H^1$, which as we’ve seen in [3] is the Sobolev space $W^{1, 2}$. This means that if a function $f$ belongs to this space:
- It also belongs to $L^2$, i.e. it’s a square integrable function;
- It has the first weak derivative, and it also belongs to $L^2$
- $H^1$ is a Hilbert space, so $f$ has an inner product;
The Lagrangian is often defined as the kinetic energy minus potential energy:
\[L(q, \dot{q}, t) = T(q, \dot{q}, t) - V(q, t)\]and the kinetic energy is:
\[\frac{1}{2} m \dot{q}^2\]So the derivative of $q$ squared appears inside the integral of $S[q]$. So we just require that $q$ has a weak derivative and that it is square integrable, which is one of the properties of $H^1$.
Now how about $\ddot{q}$ in the Euler-Lagrange? Turns out that when we start from a weaker assumption such as $q$ being from a Sobolev space $H^1$, we don’t actually solve the Euler-Lagrange directly. This equation is only available when we can make stronger assumptions about $q$, but otherwise we need to work with $(3)$.
Manifolds
There’s one more piece we didn’t discuss yet: what $\Omega$ is. We mentioned that for a single particle in 3D it could be $\mathbb{R}^3$, but a more general object than the Euclidean space is a smooth manifold.
A smooth manifold is a geometric object that locally behaves like $\mathbb{R}^n$. The sphere is such an example, where in an infinitesimal patch it behaves like a plane.
Since I haven’t studied manifolds yet, I’ll leave this discussion for another time.
Conclusion
In this post we covered Lagrangian mechanics and made connections with math concepts such as functionals and Sobolev Spaces. I’ve seen these terms thrown around before and it’s nice to finally get a sense of what they are.
Even for manifolds, which I haven’t studied yet, I now have a better understanding where it comes into picture.
The book The Theoretical Minimum doesn’t go as deep on the mathematics, but I enjoyed the process of digging into it.
