Key references
- Goldstein, Herbert; "Classical Mechanics"; 1959; Addison-Wesley Publishing Company, Inc..
- Lanczos, Cornelius; "The Variational Principles of Mechanics"; 4th ed.; 1970; Dover Publications, Inc..
- Taylor, John R.; "Classical Mechanics"; 2005; University Science Books.
- Susskind, Leonard & Hrabovsky, George; "The Theoretical Minimum: What You Need to Know to Start Doing Physics"; 2013; Basic Books.
- Coopersmith, Jennifer; "The Lazy Universe"; 2017; Oxford University Press.
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 $F_i = \dot{p_i}$.
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.
