Monday, 22 September 2008

From Lagrangian to Hamiltonian Mechanics

If the Lagrangian L does not depend explicitly on time, and varies with time only through the time-dependence of the coordinate q (and its time derivative), then we can define a constant of motion known as the Hamiltonian, H.

That is, if

\begin{align*}
L = L(q(t),\dot{q}(t)) \\
\mathrm{not}~~L = L(q(t), \dot{q}(t), t)
\end{align*}

then the Hamiltonian H can be defined as
H = \dot{q}\frac{\partial L}{\partial \dot{q}} - L


For the simplest example of a particle of mass m, moving through a potential V(x), with velocity v = dx/dt, we have
L = \frac{m\dot{x}^2}{2} - V(x) = T - V
. Using the above definition of the Hamiltonian,
H = \frac{m\dot{x}^2}{2} + V(x)
which can be identified as the total energy. Since H is a constant of motion, this result corresponds to conservation of energy!

More generally,
\frac{dH}{dt}=\frac{d}{dt}\left(\dot{x}\frac{\partial L}{\partial\dot{x}}\right)-\frac{\partial L}{\partial\dot{x}}\frac{d\dot{x}}{dt}-\frac{\partial L}{\partial x}\frac{dx}{dt}
= v\left( \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x}\right) = 0

since the last term inside the brackets vanishes (it's the Euler-Lagrange Equation!)

Note that if L depends explicitly on time, the above does not hold, and we lose conservation of energy; here, energy can be `used' in thermodynamically irreversible processes. Of course, if we expanded the Lagrangian to take into account a large enough system, we regain conservation of energy for the universe as a whole!

The Hamiltonian H is an integral of the motion, since
\frac{dH}{dt} = 0 \Rightarrow \int_{t_0}^{t_1}\frac{dH}{dt}\,dt = \left[H\right]_{t_0}^{t_1} = 0

It contains only first-order time derivatives of the coordinate q, whereas the Euler-Lagrange equation contains second-order time derivatives,
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}
.

As an example of determining the Hamiltonian from a Lagrangian, I'd like to look at the case of Special Relativity (SR). There are two reasons for this... first, the SR Lagrangian looks a bit different from the usual T - V form of classical mechanics. Secondly, Special Relativity will feature prominently in a number of future posts as I start to direct the methods of Lagrangian and Hamiltonian mechanics towards describing gauge theories.

The SR Lagrangian in a potential V(x) is given by
L = \frac{-mc^2}{\gamma} - V(x) = -mc^2\sqrt{1-\frac{v^2}{c^2}} - V(x)

where I have used
v = \dot{x} = \frac{dx}{dt}
and have referred to the rest mass as simply m, while many books use the notation m_0.

The Hamiltonian H can be found as
\begin{multiline*}
H = v\frac{\partial L}{\partial v} - L\\
= \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} + V(x)\\
= \gamma mc^2 + V(x)
\end{multiline*}

The last line is readily identified as the total relativistic energy of a particle of rest mass m in a potential V(x), so our interpretation of the Hamiltonian holds!

Canonical Momenta
I'd like to take a moment now to introduce some new nomenclature and notation, and to explain why it is useful here.

First, a reminder that the Euler-Lagrange equation for a coordinate q can be written as
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}


We can define a quantity p, known as the canonical momentum conjugate to the coordinate q, as follows
p=\frac{\partial L}{\partial \dot{q}}


and a quantity F, known as the canonical force conjugate to the momentum p, as
F = \frac{\partial L}{\partial q}

These follow the usual definitions such that
T = \frac{p^2}{2m}
and
F = -\nabla V
, since the Lagrangian L = T - V.

Written in this form, the Euler-Lagrange Equation directly represents Newton's 2nd Law:
F = \dot{\mathbf{p}}


For multiple coordinates
q_i
, the Hamiltonian H is given by

H = \sum_i p_i \dot{q}_i - L

where
p_i
is the canonical momentum conjugate to the coordinate
q_i
.

Introducing the concept of canonical momentum conjugate to a coordinate is essential for the future posts I'd like to make on this blog. The reason for this is that when I discuss quantum field theory, the concepts introduced above will play an essential role in the procedure. More on this later!

Meanwhile, although I provided a couple of classical mechanics examples for the use of the Lagrangian, in order to introduce the concept at a level that most people can understand from the world around them, I will refrain from posting similar examples making use of the Hamiltonian, since I wish to progress to more complicated topics.

In order to illuminate the path from here to a discussion of gauge theories, I'd like now to simply list some of the topics I must first cover. As such, the interested reader will notice any such posts! I anticipate that this list will be incomplete, and that I will need to cover certain topics in a lot more detail than others.
  1. Symmetries and Noether's Theorem (some of this)
  2. Special Relativity in four-vector formulation (lots of this)
  3. Relativistic electrodynamics (not much of this)
  4. Relativistic Quantum Mechanics (I've already discussed the Klein-Gordon equation, but I'll go over this and other aspects, again)
  5. Spin & Relativity (Pauli matrices and the gamma matrices)
  6. The Dirac Equation (in some detail!)
  7. U(1) symmetries and the Gauge Principle
  8. Quantum Electrodynamics
  9. SU(2) and Electroweak Unification
  10. Aside on Superconductivity (which may be omitted)
  11. Spontaneous Symmetry Breaking and the Higgs Mechanism


I expect those posts to take some considerable time. I've spent at least two weeks discussing Lagrangian mechanics, and we have a long way to go before I can introduce the Higgs mechanism, which is the ultimate goal of this entire series of posts. The idea is that anyone with a basic grounding in physics and mathematics should be able to learn enough about Lagrangian & Hamiltonian Mechanics, Quantum Mechanics and Special Relativity to be able to appreciate the formulation of the gauge theories of the Standard Model of Particle Physics, and to go a little beyond that and get a look at the Higgs mechanism. Let's hope I make it to the end!

1 comment:

  1. i fully agree with you, and hope that you are able to prove the point you raised in the last paragraph about how it should be possible for anyone with a basic but firm background in mathematics and physics to get an idea about gauge theories.
    i am in 2nd year of undergrad college, and i've been usinf classical mech, classical field theory, special relativity and quantum mechanics to do quantum field theory, leading to (non-abelian) gauge theories, and i find it is VERY doable.

    ReplyDelete