3

I'm trying to solve and understand a provided solution to a problem in Rovelli - Quantum Loop Gravity on page 56, eq. number (2.86).

In the problem among other things I'm asked to compute the equations of motion for $a$ and $b$ using following covariant Lagrangian: $$ {\cal{L}} \left( a, b, \dot{a}, \dot{b} \right) = \sqrt{\left(2E - a^2 - b^2\right) \left( \dot{a}^2 + \dot{b}^2 \right)}. \tag{2.86} $$ The hint says that:

  1. The action is minimized for a geodesic of $$ds^2 = \left(2E - a^2 - b^2 \right) \left( da^2 + db^2 \right).$$
  2. Gauge fixing can be applied to set $\dot{a}^2 + \dot{b}^2 = 1$.
  3. Variables can be separated.

Furthermore, I'm supposed to show that the system is formally similar to two harmonic oscillators with total energy $E$.

Hamiltonian associated to this Lagrangian is trivially zero (as it should be for covariant theory). Applying gauge-fixing leads to trivial equations of motion $a = 0$, $b = 0$ so I assume this should not be done as the first step. Otherwise the equations of motion derived from:

$$ p_a = \frac{\partial{\cal{L}}}{\partial \dot{a}} = \dot{a} \sqrt{\frac{2E - a^2 - b^2}{\dot{a}^2 + \dot{b}^2}}, \\ \frac{\partial{\cal{L}}}{\partial{a}} = -a \sqrt{\frac{\dot{a}^2 + \dot{b}^2}{2E - a^2 - b^2}} $$ are inseparable and cumbersome with no solution in sight.

There is a Hamiltonian constraint provided in the text that can be easily verified without clear steps to derivation: $$ C = \frac{1}{2} \left( p_a^2 + a^2 + p_b^2 + b^2 \right) - E = 0.\tag{2.87} $$

Technically, it should play the role of equations of motion, but substituting $p_a$ from above and similarly $p_b$ gives trivial zero yet again.

What's even more confusing is that the answers that Rovelli provides: $$\begin{align} a(\tau) =& A \sin(\tau + \phi_a),\tag{2.90} \cr b(\tau) =& B \sin(\tau + \phi_b) \tag{2.91} \end{align}$$ does not seem to be a solution either (unless I'm mistaken).

Does anyone know how this could be solved?

Qmechanic
  • 201,751
Darkseid
  • 1,420

1 Answers1

2

Hints: The point is to construct the corresponding Hamiltonian formulation of the Lagrangian (2.86). The Dirac-Bergmann analysis reveals a primary constraint $$C\approx 0.\tag{2.87}$$ [When we try to isolate the two velocities $\dot{q}^i$ in the equation for momenta $p_i=f_i(q,\dot{q},t)$, we discover that the two momenta $p_i$ are not independent.] OP has already noted that the original Hamiltonian $H_0$ vanishes, due to world-line (WL) reparametrization, cf. e.g. this Phys.SE post$^1$.

So the full Hamiltonian becomes just Lagrange multiplier times constraint: $$H~=~\lambda C.\tag{1}$$ Note that the Hamiltonian (1) is completely separable in the original phase space variables and takes the form of two independent harmonic oscillators. This explains the solution (2.90) & (2.91).

--

$^1$ The Lagrangian (2.86) is of the form of the Lagrangian for a massive relativistic point particle in a curved target space, cf. e.g. this Phys.SE post. [The only difference is that the 2D target space in eq. (2.86) has Euclidean signature, which produces some different sign-factors.] The solutions to the Euler-Lagrange (EL) equations are geodesics.

Qmechanic
  • 201,751