Handy Summary of Classical Mechanics

Key references

1. Goldstein, Herbert; "Classical Mechanics"; 1959; Addison-Wesley Publishing Company, Inc..
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.

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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

$$\begin{array}{}\text{(1)}& S={\int }_{t_1}^{t_2}\mathcal{L}dt\end{array}$$

is stationary when taken along the actual path.

$\mathcal{L}\mathcal{(}\mathbf{q}\mathcal{,}\dot{\mathbf{q}}\mathcal{,}\mathcal{t}\mathcal{)}$ 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

$$\begin{array}{}\text{(2)}& \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q_i}}-\frac{\partial \mathcal{L}}{\partial q_i}=0[i=1,\dots ,n]\end{array}$$

$n$ being the number of degrees of freedom.

The generalized force $F_i$ and generalized (or conjugate) momentum $p_i$ are respectively given by $$\begin{array}{}\text{(3)}& \frac{\partial \mathcal{L}}{\partial q_i}=F_i,\frac{\partial \mathcal{L}}{\partial \dot{q_i}}=p_i\end{array}$$

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

$$\begin{array}{}\text{(4)}& \mathcal{H}(\mathbf{q},\mathbf{p},t)=(\sum _{i=1}^n\dot{q_i}{p}_i)-\mathcal{L}\end{array}$$

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 $$\begin{array}{}\text{(5)}& \mathcal{L}(\mathbf{r},\dot{\mathbf{r}},t)=\frac{1}{2}m{\dot{\mathbf{r}}}^2-q(V-\dot{\mathbf{r}}\cdot \mathbf{A})\end{array}$$

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

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Hamiltonian mechanics

Given the $2n$-dimensional phase space of $n$ generalised coordinates $q_i$ and $n$ conjugate momenta $p_i$ then

$$\begin{array}{}\text{(6)}& \dot{q_i}=\frac{\partial \mathcal{H}}{\partial p_i}\text{and}\dot{p_i}=-\frac{\partial \mathcal{H}}{\partial q_i}[i=1,\dots ,n]\end{array}$$

are the canonical equations of Hamilton.

If $\mathcal{H}$ is independent of $q_i$ then the conjugate momentum $p_i$ is conserved and the coordinate $q_i$ is ignorable.

A phase-space orbit is the path traced in phase space by a system as the system evolves in time.

Hamiltonian for electromagnetism

The Hamiltonian for a charge $q$ in an electromagnetic field is

$$\begin{array}{}\text{(7)}& \mathcal{H}=\frac{(\mathbf{p}-q\mathbf{A}{)}^2}{(2m)}+qV\end{array}$$

Canonical transformations

Coordinate transformations given by

$$\begin{array}{}\text{(8)}& Q_i& =Q_i(\mathbf{q},\mathbf{p},t)\text{(9)}& P_i& =P_i(\mathbf{q},\mathbf{p},t)\end{array}$$

that satisfy

$$\begin{array}{}\text{(10)}& \dot{Q_i}=\frac{\partial K}{\partial P_i},\dot{P_i}=-\frac{\partial K}{\partial Q_i}\end{array}$$

where $K(\mathbf{Q},\mathbf{P},t)$ is some function are said to be canonical (or contact) transformations. The transformation equations are completely specified by the generating function $F$ of the transformation which can be written as a function of independent variables in one of only four forms:

$$F_1(\mathbf{q},\mathbf{Q},t),F_2(\mathbf{q},\mathbf{P},t),F_3(\mathbf{p},\mathbf{Q},t),F_4(\mathbf{p},\mathbf{P},t)$$

For $F_1$,

$$p_i=\frac{\partial F_1}{\partial q_i},P_i=-\frac{\partial F_1}{\partial Q_i},K=H+\frac{\partial F_1}{\partial t}$$

For $F_2$,

$$p_i=\frac{\partial F_2}{\partial q_i},Q_i=\frac{\partial F_2}{\partial P_i},K=H+\frac{\partial F_2}{\partial t}$$

For $F_3$,

$$q_i=-\frac{\partial F_3}{\partial p_i},P_i=-\frac{\partial F_3}{\partial Q_i},K=H+\frac{\partial F_3}{\partial t}$$

For $F_4$,

$$q_i=-\frac{\partial F_4}{\partial p_i},Q_i=\frac{\partial F_4}{\partial P_i},K=H+\frac{\partial F_4}{\partial t}$$

A transformation of time occurs automatically in the $4$-dimensional relativistic Hamiltonian formulation; the invariant parameter of the system is the proper time $\tau$ and the ordinary time $t$ appears as one of the coordinates of the particle.

The motion of a system point in time is simply a particular contact transformation of the canonical coordinates in phase space.

The particular form for $F_2$ below generates the identity transformation:

$$F_2=\sum _i{q}_i{P}_i,p_i=P_i,Q_i=q_i,K=H$$

Integral invariants of Poincaré

The integral

$$J_n=\int \dots \int d{q}_1\dots d{q}_{n}d{p}_1\dots d{p}_n$$

over any arbitrary region of phase space is invariant under canonical transformations. The invariance of $J_n$ is equivalent to the statement that volume in phase space is invariant under canonical transformations.

Lagrange brackets

(Note: The bracket notation used in this post is from Goldstein. In particular, the notation for Poisson brackets directly translates to the commutator of Quantum Mechanics for which the Poisson brackets are a direct analogue.)

The Lagrange bracket of $u$ and $v$ defined by

$$\begin{array}{}\text{(11)}& \{u,v{\}}_{\mathbf{q},\mathbf{p}}=\sum _i(\frac{\partial q_i}{\partial u}\frac{\partial p_i}{\partial v}-\frac{\partial p_i}{\partial u}\frac{\partial q_i}{\partial v})\end{array}$$

is invariant under canonical transformations.

$$\{u,v\}=-\{v,u\}$$

The fundamental Lagrange brackets are

$$\begin{array}{}\text{(12)}& \{q_i,q_j\}& =0\text{(13)}& \{p_i,p_j\}& =0\text{(14)}& \{q_i,p_j\}& ={\delta }_{ij}\end{array}$$

Poisson brackets

The Poisson bracket of $u$ and $v$ is defined by

$$\begin{array}{}\text{(15)}& [u,v{]}_{\mathbf{q},\mathbf{p}}=\sum _k(\frac{\partial u}{\partial q_k}\frac{\partial v}{\partial p_k}-\frac{\partial u}{\partial p_k}\frac{\partial v}{\partial q_k})\end{array}$$

with the identity

$$\begin{array}{}\text{(16)}& [u,v]=-[v,u]\end{array}$$

The fundamental Poisson brackets

$$\begin{array}{}\text{(17)}& [q_i,q_j]& =0\text{(18)}& [p_i,p_j]& =0\text{(19)}& [q_i,p_j]& ={\delta }_{ij}\end{array}$$

are canonical invariants.

All Poisson brackets are independent of the set of canonical coordinates they are expressed in.

The canonical equations of motion written using Poisson brackets are

$$\begin{array}{}\text{(20)}& [q_i,\mathcal{H}]=\frac{\partial \mathcal{H}}{\partial p_i}& =\dot{q_i}\text{(21)}& [p_i,\mathcal{H}]=-\frac{\partial \mathcal{H}}{\partial q_i}& =\dot{p_i}\end{array}$$

In general,

$$\begin{array}{}\text{(22)}& \frac{d}{dt}u(\mathbf{q},\mathbf{p},t)=[u,\mathcal{H}]+\frac{\partial u}{\partial t}\end{array}$$

If $u$ is a constant of motion then

$$\begin{array}{}\text{(23)}& [u,\mathcal{H}]=0\end{array}$$

Jacobi's identity

$$\begin{array}{}\text{(24)}& [u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0\end{array}$$

If $u$ and $v$ are two constants of motion and $w$ is taken to be $\mathcal{H}$ then the relation reduces to

$$\begin{array}{}\text{(25)}& [\mathcal{H},[u,v]]=0\end{array}$$

Thus, the Poisson bracket of two constants of motion is itself a constant of the motion (Poisson's theorem).

Infinitesimal contact transformations and generators

$$\begin{array}{}\text{(26)}& Q_i& =q_i+\delta q_i=q_i+ϵ\frac{\partial G}{\partial p_i}\text{(27)}& P_i& =p_i+\delta p_i=p_i-ϵ\frac{\partial G}{\partial q_i}\end{array}$$

where $ϵ$ is some infinitesimal parameter of the transformation and $G$ is given by

$$F_2=\sum _i{q}_i{P}_i+ϵG(\mathbf{q},\mathbf{P})$$

It is customary to designate $G$ as the generating function even though, strictly speaking, the term refers to $F$.

When $G=\mathcal{H}(\mathbf{q},\mathbf{p})$ and $ϵ$ is a small time interval $dt$,

$$\delta q_i=dt\dot{q_i}=d{q}_i,\delta p_i=-dt(-\dot{p_i})=d{p}_i$$

Thus, the Hamiltonian is the generator of the system motion with time.

For any function $u(\mathbf{q},\mathbf{p}),$ we have $\delta u=ϵ[u,G]$ which implies $\delta \mathcal{H}=ϵ[\mathcal{H},G]$. If $G$ is a constant of motion, then $[\mathcal{H},G]=0$ and, hence, $\delta \mathcal{H}=0$. Thus, the constants of motion are the generating functions of those infinitesimal canonical transformations which leave the Hamiltonian invariant. One can therefore determine all the constants of motion by an examination of the symmetry properties of the Hamiltonian!

Liouville's theorem

The density of systems in the neighbourhood of some given system in phase space remains constant in time.

Helmholtz circulation theorem

Vortices cannot be created or destroyed in phase space.

Noether's theorem

Every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law.

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Hamilton-Jacobi theory

$$\mathcal{H}(\mathbf{q},\mathbf{p},t)+\frac{\partial S}{\partial t}=0$$

is the Hamilton-Jacobi equation with $S$ being the action functional (equation $\text{1}$). $S$, also called Hamilton's principal function, happens to be the (type 2) generating function that characterises a canonical transformation to new coordinates $(\mathbf{Q},\mathbf{P},t)$ where either all the $Q_i$ and $P_i$ are constant or the $Q_i$ are cyclic (making the conjugate momenta $P_i$ constant). The corresponding Hamiltonian $\mathcal{H}(\mathbf{Q},\mathbf{P},t)$ equals $0$.

$$\mathcal{H}(q_1,\dots ,q_n,\frac{\partial S}{\partial q_1},\dots ,\frac{\partial S}{\partial q_n},t)+\frac{\partial S}{\partial t}=0$$ where the individual momenta $p_i=\partial S/\partial q_i$ are constants.

The solutions yield $\mathbf{q}$ and $\mathbf{p}$ in terms of their starting values ${\mathbf{q}}_0$ and ${\mathbf{p}}_0$ at time $t_0$.

$$\begin{array}{}\text{(28)}& q_i& =q_i({\mathbf{q}}_0,{\mathbf{p}}_0,t)\text{(29)}& p_i& =p_i({\mathbf{q}}_0,{\mathbf{p}}_0,t)\end{array}$$

Geometrical interpretation

Hyper-surfaces of constant $S$ represent 'wave-fronts' that evolve in time making the Hamilton-Jacobi theory a direct analogue of the 'matter waves' in Quantum Mechanics. The motion of a single particle can be represented by a wave. This played an important role in the development of Schrödinger's wave mechanics.

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Continuous systems (Fields)

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