Introduction

Abstract

We briefly review the Euclidean geometry, Galilean transformations, the Lagrangian formalism, and Newton’s gravity. While the reader is supposed to be already familiar with all these concepts, it is convenient to summarize them here because they are used or generalized in the next chapters for the theories of special and general relativity. We end the chapter pointing out the inconsistency between Galilean transformations and Maxwell’s equations and how this issue led to the theory of special relativity between the end of the 19th century and the beginning of the 20th century.

Problems

1.1

Verify Eq. (1.14).

1.2

The transformation between spherical coordinates $(r,\theta ,\phi )$ and cylindrical coordinates $(\rho ,z,\phi ')$ is

$$\begin{aligned} \rho = r \sin \theta \, , \quad z = r \cos \theta \, , \quad \phi ' = \phi \, , \end{aligned}$$

(1.127)

with inverse

$$\begin{aligned} r = \sqrt{\rho ^2 + z^2} \, , \quad \theta = \arctan \left( \frac{\rho }{z}\right) \, , \quad \phi = \phi ' \, . \end{aligned}$$

(1.128)

Write the metric tensor $g_{ij}$ and then the line element dl in cylindrical coordinates.

1.3

Consider the transformation $x^i \rightarrow x'^i$ described by the Galilean transformation in Eq. (1.36). Show that the expression of the Euclidean metric $\delta _{ij}$ does not change.

1.4

Consider the transformation $x^i \rightarrow x'^i$ described by the rotation in the xy plane $R_{xy}$ in Eq. (1.42). Show that the expression of the Euclidean metric $\delta _{ij}$ does not change.

1.5

The Lagrangian of a free point-like particle in spherical coordinates is in Eq. (1.77) and the transformations between spherical coordinates $(r,\theta ,\phi )$ and cylindrical coordinates $(\rho ,z,\phi ')$ are given by Eqs. (1.127) and (1.128). Write the Lagrangian in cylindrical coordinates and then the corresponding Euler–Lagrange equations.

1.6

From the Euler–Lagrange equations obtained in the previous exercise, write the Christoffel symbols in cylindrical coordinates.

1.7

The Lagrangian of a free point-like particle of mass m moving on a spherical surface of radius R is

$$\begin{aligned} L = \frac{1}{2} m R^2 \left( \dot{\theta }^2 + \sin ^2\theta \dot{\phi }^2 \right) \, . \end{aligned}$$

(1.129)

Note that here the Lagrangian coordinates are $(\theta ,\phi )$, while R is a constant. Write the Euler–Lagrange equations for the Lagrangian in (1.129).

1.8

Let us consider the following Lagrangian

$$\begin{aligned} L = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right) \, , \end{aligned}$$

(1.130)

where x and y are the Lagrangian coordinates. It is the Lagrangian of a particle moving in a 2-dimensional space and subject to the potential $V = k ( x^2 + y^2 )/2$. Find the constant(s) of motion and then write the corresponding Euler–Lagrange equations.

1.9

Show that the Maxwell equations are not invariant under Galilean transformations.