Review of Classical Mechanics

Newtonian Mechanics

The basis of Newtonian Mechanics revolves around Newton’s second law of motion, i.e., $\vec{F}=m\vec{a}$.

Lagrangian Mechanics

A Lagrangian is defined as $L=T-U$, where

$T=\frac{1}{2}m({\mathring{x}}^2+{\mathring{y}}^2+{\mathring{z}}^2)\text{ and }U=U(x,y,z)$

Let us now define the “action” as $S={\int }_{t_1}^{t_2}L(y,\mathring{y})dt$.

It basically describes the variation of the Kinetic and Potential energies along the trajectory

• Euler-Lagrange Equation

$\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{dL}{d\mathring{q}}=0$, where $q$ denotes the coordinate.

• 4 Steps:

1. Pick Coordinates.
2. Find $T$, $U$ and hence $L$.
3. Use the Euler-Lagrange Equation for $L$.
4. Get “stuff”.

Legendre Transform

For a function $F(v)$, define $w=\frac{\partial F}{\partial v}$.

Then, $G(w)=w.v(w)-F(v(w))$ is the Legendre Transform of $F(v)$.

Basically, go from $[F(v),v]⟶[G(w),w]$.

Then,

$G^{\prime }=\frac{\partial G}{\partial w}=v+w\frac{\partial v}{\partial w}-\frac{\partial F}{\partial w}$

$=v+\frac{\partial F}{\partial v}.\frac{\partial v}{\partial w}+\frac{\partial F}{\partial v}.\frac{\partial v}{\partial w}$.

Hence, we obtain, $\frac{\partial G}{\partial w}=v$ which is the inverse of $w$.

Hamiltonian Mechanics

A Hamiltonian (total energy) is defined as $H=T+U$ which can also be written as (for say a pendulum),

$H=\frac{1}{2}m{l}^2{\mathring{\theta }}^2-mglcos\theta$.

Now, momentum $p=m{l}^2\mathring{\theta }⟹H=\frac{p^2}{2m{l}^2}-mglcos\theta$.

Also note that, $H=p\dot{\theta }-L$ or in general for any coordinate $q$, $H=p\dot{q}-L$ since $p\equiv \frac{\partial L}{\partial \dot{q}}$.

Hence, we obtain the Hamiltonian Equations as,

$\dot{q}=\frac{\partial H}{\partial p},\dot{p}=-\frac{\partial H}{\partial q}$.

Poisson Brackets and Time Evolution

This is a binary operator defined as,

$[A,B{]}_P=\frac{\partial A}{\partial p}\frac{\partial B}{\partial q}-\frac{\partial A}{\partial q}\frac{\partial B}{\partial p}$.

Now, for any operator $O=O(q,p,t)$,

$\frac{dO(q,p,t)}{dt}=\mathring{q}\frac{\partial O}{\partial q}+\mathring{p}\frac{\partial O}{\partial p}+\frac{\partial O}{\partial t}$.

But from the Hamiltonian Equations, we know $\dot{q}=\frac{\partial H}{\partial p},\dot{p}=-\frac{\partial H}{\partial q}$.

Hence, on substituting the above, we obtain

$\frac{dO}{dt}=\frac{\partial H}{\partial p}\frac{\partial O}{\partial q}-\frac{\partial H}{\partial q}\frac{\partial O}{\partial p}+\frac{\partial O}{\partial t}$.

By the definition of a Poisson Bracket,

$\frac{dO}{dt}=[H,O{]}_P+\frac{\partial O}{\partial t}$.

1. Any operator $O$ for which $[H,O{]}_P=-\frac{\partial O}{\partial t}$is conserved.
2. If $H$ is not explicitly dependent on $t$, $H$ is conserved.

Few Poisson brackets,

• $[x,x]=0=[p,p]$
• $[x,p]=1$
• $[x,H]=\dot{x},[p,H]=\dot{p}$.