Handy Summary of Classical Mechanics

Key references

1. Goldstein, Herbert; "Classical Mechanics"; 1959; Addison-Wesley Publishing Company, Inc.; archive.org link to the 3rd edition (2002).
2. Lanczos, Cornelius; "The Variational Principles of Mechanics"; 4th ed.; 1970; Dover Publications, Inc..
3. Taylor, John R.; "Classical Mechanics"; 2005; University Science Books.
4. Susskind, Leonard & Hrabovsky, George; "The Theoretical Minimum: What You Need to Know to Start Doing Physics"; 2013; Basic Books.
5. Coopersmith, Jennifer; "The Lazy Universe"; 2017; Oxford University Press.
6. Goldstine, Herman H.; "A History of the Calculus of Variations from the 17th through the 19th Century"; 1980; Springer-Verlag;  archive.org link.

---

Lagrangian mechanics

Hamilton's principle

The actual path which a holonomic system follows between two points $1$ and $2$  in configuration space in a given time interval, $t_1$ to $t_2$, is such that the action integral

$$\begin{equation}\label{eqn:action:functional}S = \int_{t_1}^{t_2} \mathcal{L}dt\end{equation}$$

is stationary when taken along the actual path.

$\mathcal{L(\mathbf{q},\dot{\mathbf{q}},t)}$ is the Lagrangian, a function of the generalised coordinates $\mathbf{q}$, generalised velocities $\dot{\mathbf{q}}$, and time $t$, and satisfies the Euler-Lagrange equations

$$\begin{equation} \frac{d}{dt}\frac{\partial{\mathcal{L}}}{\partial{\dot{q_i}}} - \frac{\partial \mathcal{L}}{\partial q_i} = 0 \quad [i=1, \dots, n] \end{equation}$$

$n$ being the number of degrees of freedom.

The generalized force $F_i$ and generalized (or conjugate) momentum $p_i$ are respectively given by $$\begin{equation} \frac{\partial{\mathcal{L}}}{\partial q_i} = F_i, \quad \frac{\partial{\mathcal{L}}}{\partial \dot{q_i}} = p_i\end{equation}$$

The Euler-Lagrange equations can be rewritten as $F_i = \dot{p_i}$.

Historical Note

The Euler-Lagrange equations were first derived by Euler using a geometric method. They were subsequently refined by Lagrange using what is now known as The Calculus of Variations and d'Alembert's Principle of Virtual Work. The modern presentation based on the action integral is essentially the reformulation by Hamilton when he introduced his Principle of Least Action.

The coinage "Calculus of Variations" is due to Euler after he was impressed by Lagrange's analytical work and preferred it over his own method.

Refer Goldstine's text for more on the historical development.

Conservation and symmetry

If the Lagrangian does not contain a coordinate $q_i$ (though it may contain $\dot{q_i}$) then the coordinate is said to be ignorable and the corresponding conjugate momentum $p_i$ is conserved.

Conservation of Hamiltonian

If $\mathcal{L}$ does not depend explicitly on time ($\partial{\mathcal{L}}/\partial{t} = 0$), then the Hamiltonian

$$\begin{equation}\mathcal{H}(\mathbf{q},\mathbf{p},t) = (\sum_{i=1}^{n}\dot{q_i} p_i) - \mathcal{L}\end{equation}$$

is conserved.

For most systems, $\mathcal{H}$ is just the total energy and is frequently conserved. However, the identification of $\mathcal{H}$ as a constant of motion and as the total energy are two separate matters, and the conditions sufficient for one are not enough for the other.

Lagrangian for electromagnetism

The Lagrangian for a charge $q$ with mass $m$ in an electromagnetic field is $$\begin{equation}\mathcal{L}(\mathbf{r}, \dot{\mathbf{r}}, t) = \frac{1}{2}m\dot{\mathbf{r}}^2 - q(V-\dot{\mathbf{r}}\cdot\mathbf{A})\end{equation}$$

$V(\mathbf{r},t)$ and $\mathbf{A}(\mathbf{r},t)$ being the scalar and vector potentials respectively.

Physics and Intuition

One common criticism of modern Physics (i.e., Relativity and Quantum Mechanics) is that it is counter-intuitive. This is not unexpected: our evolutionary history in no way requires that we have an innate understanding of the mechanisms underpinning the Universe at all scales. That said, this criticism is not uniquely limited to modern Physics. In this post I explore concepts that we often take for granted today, and how those could have seemed counter-intuitive to people of earlier times.