notes

materials/notes.md

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+# Principle of Least Action
+
+from Chapter 14 of Eriksson, Kenneth, Donald Estep, and Claes Johnson. 2004. *Applied Mathematics: Body and Soul*. Volume 2. Berlin, Heidelberg: Springer Berlin Heidelberg. <https://doi.org/10.1007/978-3-662-05798-8>.
+
+The principle of least action states that a system's trajectory $u(t)$ over an interval of time $[t_1, t_2]$ is such that the following integral, called the action integral, remains constant:
+$$
+I(u)=\int_{t_1}^{t_2} T(\dot{u}(t)) - V(u(t)) dt
+$$
+where $T(\dot{u}(t))$ is the kinetic energy and $V(u(t))$ is the potential energy.
+
+**Example 1.** Consider a ball that gets thrown into the air, under the influence of gravity. It's trajectory $u: [t_1, t_2] \to \mathcal{R}$ is a parabola between times $t_1$ and $t_2$, with $u(t_1) = u_1$ denoting the initial position, and $u(t_2) = u_2$ denoting the final position of the ball.
+The kinetic energy of the ball is  $T(\dot{u}(t)) = \frac{1}{2}m\dot{u}(t)^2$ and the potential energy is $V(u(t)) = mgu(t)$.
+
+We now explain what is meant with requiring that the integral $I(u)$ is stationary. Consider a possible trajectory  $u: [t_1, t_2] \to \mathcal{R}$ and consider its perturbed version $v: [t_1, t_2] \to \mathcal{R}$, given by $v(t) = u(t) + \epsilon w(t)$, or written equivalently, $v(t)=(u + \epsilon w)(t)$ where $w(t)$ is such that $w(t_1)=w(t_2)=0$, so that $v(t_1) = u(t_1)$ and $v(t_2) = u(t_2)$.
+For any such perturbation $w(t)$ the integral is stationary when $u(t)$ satisfies 
+$$
+\frac{d}{d\epsilon}I(u + \epsilon w) = 0 \text{ for } \epsilon = 0
+$$
+**Remark:** Why can't we simplify $u + \epsilon w$ to $u$ in the above equation (since $\epsilon = 0$...)? The reason becomes clear when we apply the limit definition of the derivative, taking into account that $\epsilon = 0$:
+$$
+\frac{d}{d\epsilon}I(u + \epsilon w) = \lim_{\Delta \epsilon \to 0} \frac{I(u + \Delta \epsilon w) - I(u)}{\Delta \epsilon}
+$$
+The above expansion makes it clear that  $\frac{d}{d\epsilon}I(u) \neq \frac{d}{d\epsilon}I(u + \epsilon w)$. 
+
+Intuitively, the above constraint resembles the constraint on a minimum or maximum point of a continuous function. Except that in this case, the minimum is a perturbed function (instead of a point).
+
+**Example 2.** Let's use the principle of least action to try to guess what the motion $u(t)$ of a ball that is thrown up at $\dot{u}_1 = 1$ (m/s) is, in the interval $[t_1=0, t_2=10]$. Let's take the following three possible trajectory functions:
+$a(t) = \dot{u}_1 + \frac{1}{2} g t^2  $
+