Globally-determined motion

Motion as described from the Lagrange equation is said to be local, as it evaluates motion point-wise in time. The opposite is global motion, which is determined from global properties of the entire system instead of local ones. It is easiest to explain by example.

In $\mathbb{R}^{2}$ , we can say that an object, like a point mass, begins its motion in $x_{\text{start}}$ and ends it in $x_{\text{end}}$ . Its coordinates $x$ and $y$ are bound by some function $u(x)=y$ , which describes a locus of points in $\mathbb{R}^{2}$ that says where the particle can be found. This space can be described as a Curve $\gamma$ , its trajectory. The length of $u$ is its measure and can be found through a Functional:

F[u]=\int_{x_{\text{start}}}^{x_{\text{end}}}\sqrt{ 1+u'(x)^{2} }\ dx

The length is a global property: it affects the whole motion, not just one piece at one time. As such, we can use it to set a condition for the motion to satisfy. For example, we can state the we want only the motion that leads to the shortest trajectory between $x_{\text{start}}$ and $x_{\text{end}}$ [^1]. In other words, we want whatever motion minimizes $F[u]$ . For it to be minimum, $u'(x)^{2}$ needs to be $0$ , which means that the shortest trajectory must be one for which $u'(x)=0$ everywhere. But this is just a constant, $u(x)=y_{0}$ , which is a straight line. This shouldn't be surprising: the shortest way from point A to point is B is a straight line after all.

Another useful property to minimize is the travel time, the time it takes for the particle to go from one end to the other. In other words, we're now looking not for the shortest path, but the fastest path, which may not be the same. To do so, we minimize the following functional:

T[u]=\int_{\gamma} \frac{ds}{v}=\int_{x_{\text{start}}}^{x_{\text{end}}} \frac{\sqrt{ 1+u'(x)^{2} }}{v(x,u(x))}dx

where the velocity is dependent on the position of the particle. This comes up a few times in physics.

> Say the [[potential energy]] is given by [[Gravity|gravity]], we get velocity $v=\sqrt{ 2gy }$. Our functional becomes > $$T[u]=\int_{x_{\text{start}}}^{x_{\text{end}}}\sqrt{ \frac{1+u'(x)^{2}}{2gu(x)} }dx

Finding the minimum gives us the correct trajectory.

These are nice example, but the meat of the discussion is in the minimization of a different quantity: the [[Lagrangian]]. Consider some Lagrangian $L:\mathbb{R}^{3}\to \mathbb{R}$, $L\equiv L(u,u',x)$. Our functional here is

F[u]=\int_{x_{\text{start}}}^{x_{\text{end}}}L(u(x),u'(x),x)dx

The variation of $F$ is

\begin{align} \delta F[u,\delta u]&=\left.{\frac{d}{d\alpha}\int_{x_{\text{start}}}^{x_{\text{end}}}L(u+\alpha\delta u,u'+\alpha \delta u',x)dx}\right|{\alpha=0} \ &=\int{x_{\text{start}}}^{x_{\text{end}}}\left( \frac{ \partial L }{ \partial u } \delta u+\underbrace{ \frac{ \partial L }{ \partial u' } \delta u' }{ fg' } \right)dx \ (\text{integration by parts})&=\int{x_{\text{start}}}^{x_{\text{end}}}\left( \frac{ \partial L }{ \partial u } \delta u- \frac{d}{dx}\frac{ \partial L }{ \partial u' } \delta u' \right)dx+\int_{x_{\text{start}}}^{x_{\text{end}}} \frac{d}{dx}\left( \frac{ \partial L }{ \partial u' } \delta u \right)dx \ &=\left.{\frac{ \partial L }{ \partial u' } \delta u}\right|{x{\text{start}}}^{x_{\text{end}}}-\int_{x_{\text{start}}}^{x_{\text{end}}}\left( \frac{d}{dx}\frac{ \partial L }{ \partial u' } - \frac{ \partial L }{ \partial u } \right)\delta udx \ &=\ldots \end{align}

The [[Lagrange equation]] spontaneously came up in the integral there. Before we move on, let's better define what $u$ even is by giving a solid definition of its space $U$:

U={ \text{functions }[x_{\text{start}},x_{\text{end}}]\to \mathbb{R}\text{ such that }u(x_\text{start})=y_\text{start},\ u(x_{\text{end}})=y_\text{end} }

$\delta u$ is defined such that $u$ and $u+\delta u$ are both in $U$ and it vanishes at the boundaries:

\delta u(x_\text{start})=0, \quad \delta u(x_\text{end})=0

We can use these boundary conditions to state that the first term in our calculation is *zero*, which leaves us with:

\ldots=-\int_{x_\text{start}}^{x_\text{end}}\left( \frac{d}{dx}\frac{ \partial L }{ \partial u' } -\frac{ \partial L }{ \partial u } \right)\delta udx

(TODO: Finish this, end of lesson 15/04/25) We know from the theory of functionals that a function $u_{0}$ for which the argument of this integral is zero is a stationary point. Thus, we can state that if a function $u$ is a stationary point for the functional

S[u]=\int_{x_\text{start}}^{x_\text{end}}L(u(x),u'(x),x)\ dx

it describes motion, and it is defined from the global properties of motion. We should first reflavor our variables to be more familiar. Firstly, let's definitively call $L$ the Lagrangian of our system. Then, in one dimensions, we can just rename $u(x)$ to be the more familiar $q(t)$ and get

\boxed{S[q]=\int_{t_{1}}^{t_{2}}L(q(t),q'(t),t)\ dt}

This functional is called **[[action]]** and the functions $q(t)$ that minimize it fully determine motion and solve the [[Lagrange equation]], a fact known as the [[least action principle]]. This can be trivially extended to multiple dimensions by adding more variables: if $L:\mathbb{R}^{2n+1}\to \mathbb{R}$, the action is

\boxed{S[q_{1},\ldots,q_{n}]=\int_{t_{1}}^{t_{2}}L(q_{1}(t),\ldots,q_{n}(t),\dot{q}{1}(t),\ldots,\dot{q}{n}(t))\ dt}

which is sometimes called the **Hamiltonian action**. [^1]: Such as trajectory is called a [[geodetic curve]] and is very important in general relativity.