Liouville's theorem (Hamiltonian)

Liouville's theorem, in the context of Hamiltonian mechanics, is a fundamental theorem stating that the phase-space distribution function is constant along trajectories in phase space. In other words, the density of a collection of system points (representing possible states of a physical system) in phase space remains constant as the system evolves according to Hamilton's equations of motion. This means that the volume occupied by the system points is preserved.

More formally, consider a Hamiltonian system described by generalized coordinates q_i and conjugate momenta p_i, where i ranges from 1 to N, representing the number of degrees of freedom. The state of the system is then represented by a point in a 2N-dimensional phase space with coordinates (q₁, q₂, ..., q_N, p₁, p₂, ..., p_N).

Let ρ(q, p, t) be the phase-space distribution function, where q and p represent the vectors of generalized coordinates and conjugate momenta, respectively, and t represents time. The distribution function ρ(q, p, t) represents the density of system points at a given point in phase space and at a given time.

Liouville's theorem states that the total time derivative of the phase-space distribution function is zero:

dρ/dt = ∂ρ/∂t + Σ_i (∂ρ/∂q_i) (dq_i/dt) + Σ_i (∂ρ/∂p_i) (dp_i/dt) = 0

Using Hamilton's equations of motion, dq_i/dt = ∂H/∂p_i and dp_i/dt = -∂H/∂q_i, where H is the Hamiltonian function, the total time derivative can be rewritten as:

dρ/dt = ∂ρ/∂t + Σ_i (∂ρ/∂q_i) (∂H/∂p_i) - Σ_i (∂ρ/∂p_i) (∂H/∂q_i) = 0

This can also be written in terms of the Poisson bracket:

∂ρ/∂t + {ρ, H} = 0

where {ρ, H} is the Poisson bracket of ρ and H.

The conservation of phase-space volume has several important consequences. For example, it is crucial in statistical mechanics, as it ensures that the microcanonical ensemble, representing a system with a fixed energy, is stationary (time-independent). It also underlies the ergodic hypothesis, which postulates that, over long times, the trajectory of a system in phase space will explore all regions of phase space consistent with its energy.

It's crucial to remember that Liouville's theorem applies to Hamiltonian systems, meaning systems where the equations of motion can be derived from a Hamiltonian function. The theorem relies on the conservation of energy and the symplectic structure of phase space. For non-Hamiltonian systems with dissipative forces, the theorem generally does not hold, and phase-space volume is not conserved.