The differences in mechanics may be defined as ,
Theoretical Mechanics
This is a
term used to differentiate between experimental mechanics (bouncing little
balls off each other) and theoretical mechanics (trying to derive equations how
little balls bounce off each other). As such, it encompasses classical
mechanics, analytical mechanics and rational mechanics.
Classical Mechanics
Classical Mechanics is used in two different contexts: First, it is the
antonym to Quantum Mechanics. As such, we usually assume a
strictly deterministic world ruled by certain differential equations, as opposed to a
quantum mechanical view where probability densities evolve according to the
Schrödinger/Heisenberg equation (iℏ∂tΨ=HΨiℏ∂tΨ=HΨ or A˙=iℏ[H,A]+∂tAA˙=iℏ[H,A]+∂tA).
On the other hand, the term ‘classical mechanics‘ is
sometimes used to describe Newtonian Mechanics as opposed to the later
developments by Euler, Lagrange, Hamilton and Jacobi. Newtonian mechanics rely
on the equation F=p˙=maF=p˙=ma (assuming m˙=0m˙=0) to describe the movement of a point particle of mass m. Since a=x¨a=x¨, they usually require two integrations to solve for the
trajectory of the particle.
Analytical Mechanics
Analytical Mechanics form the ‘other’ branch of classical mechanics and build upon
Newtonian mechanics. It can be divided into three major steps: Lagrangian Mechanics, Hamiltonian Mechanics and
mechanics based on the Hamilton-Jacobi equation.
Lagrangian Mechanics
Lagrangian Mechanics starts off with the definition of the
Lagrangian LL which acts as a measure for the difference between kinetic
and potential energy. It is a function of the coordinates and velocities of all
particles:
L(q1,q2,…,qN,q˙1,q˙2,…,q˙N,t)=T−UL(q1,q2,…,qN,q˙1,q˙2,…,q˙N,t)=T−U
Lagrange then postulated that the actual trajectories of
the particles between time t1t1 and time t2t2 are these that minimise the action SS as defined by
S=∫t2t1LdtS=∫t1t2Ldt
which leads to a variation problem: Find {qi,q˙i}{qi,q˙i} such that
δS=∫t2t1δLdt=0δS=∫t1t2δLdt=0
where δXδX describes the variation of XX by its arguments (coordinates and velocities, in our
case). As it happens, there is an equation that describes when a given quantity
fulfills this requirement, namely the Euler-Lagrange equations. These are:
∂qiL−ddt∂q˙iL=0∂qiL−ddt∂q˙iL=0
where ∂x=∂∂x∂x=∂∂x and I dropped the argumets of LL. Note that these equations are of
first order in {qi,q˙i}{qi,q˙i}, as opposed to Newton’s F=p¨F=p¨. Furthermore, note that I used qiqi to denote the coordinate(s) of
the ii-th particle rather than xixi: This is because Lagrangian mechanics makes it very easy to
implement generalised coordinates. This is best shown by
example:
Assume (in two dimensions) that you have a bolt of length ll fixed at the origin (0,0)(0,0) and a mass mm at the other end of the string.
Furthermore assume that the bolt has always the same length. To then describe
the movement of the mass using the standard coordinates, we need to introduce yy and xx and integrate each of them and
do all sorts of ugly things and, most importantly, always have to take care
that y2+x2=l2y2+x2=l2. However, we notice
that there is only one degree of freedom: the angle. By then introducing
a generalised coordinate qq, we can implement the requirement y2+x2=l2y2+x2=l2 by simply not
admitting any other coordinates. We set
x=lcos(q)y=lsin(q)x=lcos(q)y=lsin(q)
and can be sure that the bolt always has the same length.
Assuming a constant gravitational potential (i.e. potential energy m⋅g⋅xm⋅g⋅x), we can write
L(q,q˙,t)=12mlq˙2−m⋅g⋅lcos(q)L(q,q˙,t)=12mlq˙2−m⋅g⋅lcos(q)
and hence
m⋅g⋅lsin(q)−ddt1mlq˙=0.m⋅g⋅lsin(q)−ddt1mlq˙=0.
You might notice that there’s still a q¨q¨ hidden there. That’s where Hamiltonian Mechanics comes
in.
Hamiltonian Mechanics
Hamilton noticed that Lagrangian mechanics is still
basically Newtonian mechanics with a nicer dress, but by applying a Legendre
transformation to LL, we can actually get rid off q˙q˙ (and therefore q¨q¨).
To this end, we introduce the ‘canonically conjugated
momentum‘ pj=∂q˙jLpj=∂q˙jL and the Hamiltonian HH which is a function of qq, pp (and, rarely, tt), defined by:
H(q1,q2,…,qN,p1,p2,…,pN,t)=∑iq˙ipi−L=T+U.H(q1,q2,…,qN,p1,p2,…,pN,t)=∑iq˙ipi−L=T+U.
You might want to verify that HH does not depend on q˙iq˙i, but only on the canonically conjugated coordinates and their
momentums {qi,pi}{qi,pi}. We can then rewrite the Euler-Lagrange equations as follows:
q˙i=∂piH;p˙i=−∂qiH.q˙i=∂piH;p˙i=−∂qiH.
You can memorise these by setting H(q,p,t)=T(p)+U(q)H(q,p,t)=T(p)+U(q), the second term then becomes ‘something like’ ∇U=−F=−p˙i∇U=−F=−p˙i.
These
equations are still asymmetric (hence the rule above), but by introducing
Poisson brackets:
{A,B}=∑i[∂qiA∂piB−∂piA∂qiB]{A,B}=∑i[∂qiA∂piB−∂piA∂qiB]
we can
actually fix that. Observe:
q˙i={qi,H};p˙i={pi,H}q˙i={qi,H};p˙i={pi,H}
since ∂qipj=0∀i,j∂qipj=0∀i,j.
Furthermore, we can now leap to quantum mechanics with
relative ease: Simply add a ‘hat’ to HH and replace {⋅,⋅}{⋅,⋅} by −iℏ[⋅,⋅]−iℏ[⋅,⋅], where [A,B]=AB−BA[A,B]=AB−BA denotes the commutator :)
If you
feel particularly nasty, look up the Hamilton-Jacobi equation, it is really not
nice (or the most beautiful thing ever seen, depending on your POV).
Analytical
mechanics is a branch of classical mechanics that is not vectorial mechanics .
Analytical mechanics uses two scalar properties of motion, the kinetic and
potential energies, instead of vector forces, to analyse the motion. Analytical
mechanics includes Lagrangian mechanics, Hamiltonian mechanics, Routhian
mechanics...
Theoretical mechanics is a branch of
mechanics which employs mathematical models and abstractions of physics to
rationalize, explain and predict mechanical phenomena. This is in contrast to
experimental mechanics, which uses experimental tools to probe these phenomena.
Rational mechanics is a branch of
theoretical mechanics characterized by a purely axiomatic approach, where some
few axioms are selected and then the rest of the theory logically derived as
theorems and corollaries. This branch is usually more mathematically oriented
than others.
Classical mechanics is that branch
of mechanics that ignores quantum effects. Classical mechanics can be either
relativistic or non-relativistic, although in older literature classical
mechanics often means pre-relativistic classical mechanics.
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