Consider a particle of mass \(m\) moving in two dimensions in the \(x-y\) plane, constrained to a rail-track whose shape is described by an arbitrary function \(y=f(x)\). There is no gravity acting on the particle.

(a) Write the Lagrangian in terms of the \(x\) degree of freedom only.

(b) Consider some general transformation of the form

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== dx = g(x), dt = 0; where g(x) is an arbitrary function of x. Assuming that this transformation is a symmetry of the system such that S = 0, show that it implies the following differential equation relating f(x) and g(x) 1 d g' g 2(1 +12) dx (12) ; where prime stands for derivative with respect to x (not t). (c) Write a general expression for the associated conserved charge in terms of f(x), g(x), and x. (d) We will now specify a certain g(x), and try to find the laws of physics obeying the prescribed symmetry; i.e., for given g(x), we want to find the shape of the rail-track f(x). Let g(x) go = where go is a constant. Find the corresponding f(x) such that this g(x) yields a symmetry. Sketch the shape of the rail-track. (Hint: h(x) = f.)