How to demonstrate if a dynamical system can be Hamiltonian

To demonstrate whether a dynamical system can be Hamiltonian, you check a small set of strict mathematical conditions. Below is a clear, technical checklist, from fastest rejection to constructive proof.

Start from the system form 

\[ \dot{x} = f(x), \qquad x \in \mathbb{R}^{2n}. \]

The system is Hamiltonian if and only if it can be written as

\[ \dot{x} = J \nabla H(x), \]

where the canonical symplectic matrix is 

\[ \dot{x} = J \nabla H(x), \] where \[ J = \begin{pmatrix} 0 & I_n \\ - I_n & 0 \end{pmatrix} \] is the canonical symplectic matrix, and \( H(x) \) is a scalar Hamiltonian function. 

Necessary condition: zero divergence 

 Hamiltonian flows preserve phase--space volume (Liouville's theorem): \[ \nabla \cdot f(x) = 0. \] If \( \nabla \cdot f(x) \neq 0 \), the system is not Hamiltonian. If \( \nabla \cdot f(x) = 0 \), Hamiltonian structure is possible. Example: damped pendulum \[ \dot q = \frac{p}{m\ell^2}, \qquad \dot p = -m g \ell \sin q - c p \] The divergence is \[ \nabla \cdot f = \frac{\partial \dot q}{\partial q} + \frac{\partial \dot p}{\partial p} = 0 - c = - c \neq 0. \] Therefore, the system cannot be Hamiltonian. 

Canonical Hamiltonian form 

 For a two--dimensional system \[ \dot q = f(q,p), \qquad \dot p = g(q,p), \] a Hamiltonian \( H(q,p) \) exists if \[ \dot q = \frac{\partial H}{\partial p}, \qquad \dot p = -\frac{\partial H}{\partial q}. \] 

Integrability condition 

 Taking mixed derivatives yields the consistency condition \[ \frac{\partial f}{\partial q} = -\frac{\partial g}{\partial p}. \] If this condition fails, no Hamiltonian exists. If it holds, a Hamiltonian can be constructed. 

Hamiltonian construction 

 If the integrability condition holds, define \[ H(q,p) = \int f(q,p)\, dp + C(q), \] and enforce \[ \frac{\partial H}{\partial q} = -g(q,p) \] to determine \( C(q) \). 

The Hamiltonian is unique up to an additive constant. 

Example: undamped pendulum 

 \[ \dot q = \frac{p}{m\ell^2}, \qquad \dot p = -m g \ell \sin q \] Check: \[ \frac{\partial \dot q}{\partial q} = 0, \qquad -\frac{\partial \dot p}{\partial p} = 0. \] Hamiltonian: \[ H(q,p) = \frac{p^2}{2 m \ell^2} + m g \ell (1 - \cos q). \] 

Non-Hamiltonian example: damped pendulum 

 \[ \dot p = -m g \ell \sin q - c p \] \[ -\frac{\partial \dot p}{\partial p} = c \neq 0. \] Hence, no scalar Hamiltonian \( H(q,p) \) exists. 

Jacobian matrix test 

 Let \( Df(x) \) be the Jacobian of \( f(x) \). Hamiltonian systems satisfy \[ Df(x)\,J + J\,Df(x)^{\mathsf T} = 0. \] Violation of this condition implies the system is not Hamiltonian. 

Energy conservation 

For Hamiltonian systems, \[ \frac{dH}{dt} = \nabla H^{\mathsf T} J \nabla H = 0. \] Energy conservation is necessary but not sufficient for Hamiltonian structure.