Problem 1
In the examples below, identify whether \(f\) is a function or a functional. \(\bullet\) A parabola is described by \(f(x)=x^{2}\). \(\bullet\) Given two forms of energy and a path \(y(t), f\) is the Lagrangian of the system \(\mathcal{L}\left(t, y, \frac{d y}{d t}\right)\) \(\bullet\) Given the magnitude of the velocity \(|\vec{v}(t)|\) of an object, \(f\) represents the distance that the object travels from time 0 to time 3600 seconds. \(\bullet\) Given the position \((x, y, z)\) in space, \(f(x, y, z)\) represents the distance from that point to the origin.
Problem 2
A system has the Lagrangian \(\mathcal{L}\left(t, y, \frac{d y}{d t}\right)=\left(\frac{d y}{d t}\right)^{3}+e^{3 y}\). Find an equation for the path \(y(t)\) that minimizes the action \(\int_{t_{1}}^{t_{2}} \mathcal{L}\left(t, y, \frac{d y}{d t}\right) d t\).
Problem 3
A system has Lagrangian \(\mathcal{L}\left(t, y, \frac{d y}{d t}\right)=\frac{1}{2}\left(\frac{d y}{d t}\right)^{2}+\frac{1}{2} \cdot y^{-2}\). Find the corresponding equation of motion.
Problem 6
The purpose of this problem is to derive the shortest path \(y(x)\) between the points \(\left(x_{0}, y_{0}\right)\) and \(\left(x_{1}, y_{1}\right) .\) Consider an arbitrary path between these points as shown in the figure. We can break the path into differential elements \(d \vec{l}=d x \hat{a}_{x}+d y \hat{a}_{y} .\) The magnitude of each differential element is $$|d \vec{l}|=\sqrt{(d x)^{2}+(d y)^{2}}=d x \sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$$ The distance between the points can be described by the action $$\mathbb{S}=\int_{x_{0}}^{x_{1}} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x .$$ To find the path \(y(x)\) that minimizes the action, we can solve the Euler- Lagrange equation, with \(\mathcal{L}=\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}\) as the Lagrangian, for this shortest path \(y(x)\). This approach can be used because we want to minimize the integral of some functional \(\mathcal{L}\) even though this functional does not represent an energy difference \([163,\) p. 33\(]\). Set up the Euler-Lagrange equation, and solve it for the shortest path, \(y(x)\).
