When calculating the Lyapunov exponents it is usual to average over initial conditions.
Yes and no – depends on what you consider an initial condition.
You regularly rescale your separation to be able to approximate the $t→∞$ in the definition of Lyapunov exponents. You could however say that every rescaling corresponds to a different initial condition. In the following I assume that initial condition does not refer to this.
Now, you usually calculate Lyapunov exponents along one trajectory defined by some initial condition. Whether you average over one long trajectory starting from one initial condition or several trajectories starting from different initial conditions makes no difference, if the latter lie on the same irreducible invariant manifold: If you cut up the long trajectory into several small ones, you have essentially the same situation.
If your dynamics has an attractor (which does not happen in Hamiltonian systems), its easy to ensure that different initial conditions lie on the same irreducible invariant manifold (the attractor) and this may facilitate distributed computing and similar. For Hamiltonian systems, it is not that easy.
In a Hamiltonian system is it correct to average of energy as well, or do we pick an ensembles of trajectories with the the same energy, and understand the Lyapunov spectra to be a function of energy.
You usually calculate Lyapunov exponents for a given irreducible invariant manifold¹ and thus averaging over different initial conditions in an arbitrary manner will average over different irreducible invariant manifolds. This in turn may obfuscate what you actually want to see. For example, for the double pendulum, you may have chaotic and regular invariant manifolds in the same system (with different energies). Therefore, I would not average over different initial conditions, but rather see whether results for different initial conditions (even with the same energy) agree with each other. To get a better statistic, you can always increase the integration time for a given initial condition.
Furthermore when calculating the Lyapunov exponents do we consider the divergence of trajectories that differ in the value of energy, or do we only calculate the Lyapunov exponents within an equal-energy manifold? eg. If we have a point (x,y), and (x+dx,y) has the same energy whereas (x,y+dy) has a different energy, do I consider divergence of trajectories with initial perturbation in both of these directions? or just the equal-energy direction?
I have never seen that such a distinction being made. However, if you try to estimate the Lyapunov exponent of a regular dynamics with finite perturbations, I can see the problem of the perturbed dynamics having a different frequency which yields a spurious linearly growing separation. However, this problem should vanish with more advanced methods like Benettin et al’s that consider infinitesimal differences.
Is it the same story for other integrals of motion?
None of the above thoughts is specific to energy. Note in particular that a considerable part of non-linear dynamics deals with non-Hamiltonian systems in which no integrals of motion exist or are known.
¹ You may also look at the Lyapunov exponents of a transient dynamics, but you do not have those in Hamiltonian systems anyway.
