It is known that conservative force, defined as those which are the gradient of a scalar function, are curl free. This is a consequence of basic vector calculus
\[\nabla \times {\bf F}_{\rm cons} = - \nabla \times \nabla V = 0\]The motion of a particle under conservative motion (motion arising under the action of a conservative force) can equally be described in the Hamiltonian mechanics framework (reviewed within here). This Hamiltonian is most often in the form of kinetic terms + potential terms, namely
\[\label{eq:hamiltonian} \tag{1} \mathcal{H} = \frac{ {\bf p}^2}{2}+ V({\bf r})\]when considering just one particle in d-dim. Hamiltons equations give
\[\dot{\bf r} = \frac{\partial \mathcal{H}}{\partial {\bf p}} = {\bf p}\] \[\dot{\bf p} = -\frac{\partial \mathcal{H}}{\partial {\bf r}} = -\nabla V\]and when combined imply that the equaton of motion for \({\bf r}\) alone is
\[\label{eq:ode} \tag{2} \ddot {\bf r} = -\nabla V({\bf r})\]Note this choice isn’t unique - we could have easily defined, for example, the Hamiltonian
\[\mathcal{H}' = \frac{ {\bf p'}^2}{2 \eta}+ \eta V({\bf r'})\]for constant \(\eta\), which would also lead us to \eqref{eq:ode}. However, one should note that the definition of the conjugate momenta changes with this rewriting - the Hamiltonian still represents the systems true energy (up to a factor of \(\eta\)).
Some equations of motion, admittedly often arising from some effective description, may contain forces which are non-conservative. Clearly taking a Hamiltonian of the form of \eqref{eq:hamiltonian} can not lead to a force with a curl. Despite this the phase space of \({\bf r}\) and \({\bf v = \dot r}\) is still volume-preserving since
\[\nabla_{\bf r} \dot {\bf r} + \nabla_{\bf v} \dot{\bf v} = \nabla_{\bf r} {\bf v} + \nabla_{\bf v} {\bf F}({\bf r}) = 0\]which is a statement of Liouville’s theorem when the system is Hamiltonian with conjugate momenta given by \({\bf p = \dot r}\), but holds despite the form of the conjugate momenta (or the existence of a Hamiltonian). Thus the dynamics are non-dissipative too. [2]
Can we create other forms of the Hamiltonian that will produce a desired (non curl-free) force? The answer is maybe. We can immediately see that there are some choices of Hamiltonian that give us forces with some amount of curl. Take for example the Hamiltonian in [1] with anisotropic kinetic energy
\[\mathcal{H}_1 = \frac{1}{2} \alpha p_x^2 + \beta p_x p_y + \frac{1}{2} \gamma p_y^2 + V(x,y)\]The equation of motion for \({\bf r} = (x,y, z)\) is
\[\label{eq:curl-ode} \tag{3} \ddot {\bf r} = - {\bf M} \nabla V \qquad {\rm where\ }\quad {\bf M} = \begin{pmatrix} \alpha & \beta & 0 \\ \beta & \gamma & 0 \\ 0 & 0 & 0 \end{pmatrix}\]The \(z\) directed curl is given by
\[(\alpha - \gamma) \partial_{xy} V + \beta ( \partial_{yy} -\partial_{xx})V\]which is non zero for general \(V\), so at least some curl forces can be embedded in a Hamiltonian without doing crazy things like doubling the degrees of freedom. The Hamiltonian is obviously conserved throughout the dynamics. We confirm this by playing around with \eqref{eq:curl-ode}. Projecting down to \((x,y)\), and assuming \({\bf M}\) invertible
\[\begin{equation} \begin{aligned} {\bf M}^{-1} \ddot {\bf r} &= - \nabla V \\ \dot {\bf r} \cdot {\bf M}^{-1} \ddot {\bf r} &= - \dot {\bf r} \cdot \nabla V \\ \implies 0 &= \frac{d}{dt} \Big( \frac{1}{2} \dot{\bf r}^T {\bf M}^{-1} \dot {\bf r} + V({\bf r}) \Big) \end{aligned} \end{equation}\]which, after the identification of \(\dot {\bf r} = {\bf M} {\bf p}\), is equivalent in form to the Hamiltonian.
Another pertinent example is of a pure shear force
\[\ddot {\bf r} = f(y) {\bf e}_x\]This force has some curl for general \(f\). Take
\[\mathcal{H}_{\rm shear} = \frac{1}{2} p_{x}^2 + p_{y} v_{y}(0) - x f(y)\]Hamilton’s equations yield
\[\dot p_x = f(y) \qquad \dot x = p_x\] \[\dot p_y = x f'(y) \qquad \dot y = v_y(0)\]which leads to
\[\ddot x = f(y) \qquad \ddot y = 0\]as required. Again the Hamiltonian is conserved, although its form is sufficiently foreign that one is hesitant to use the term “energy” to describe its constant value. Can verify constancy directly
\[\begin{equation} \begin{aligned} \dot{\mathcal{H}} &= \dot{p}_x p_x + \dot{p}_y v_y(0) - \dot{x} f(y) - \dot{y} x f'(y) \\ &= \dot{x} f(y) + x f'(y) v_y(0) -\dot{x} f(y) - v_y(0) x f'(y) \\ &= 0 \end{aligned} \end{equation}\]The Hamiltonian has the peculiarity of depending on an initial condition.
A final example - 2-dim propelled motion
\[\ddot \theta = 0\] \[\ddot {\bf r} = v {\bf n}(\theta) - \nabla V({\bf r})\]The dynamics is roughly that of a particle being pushed around at a constant self propulsion speed \(v\), with an evolving angular direction \(\theta \in [0, 2\pi)\).
Consider
\[\mathcal{H}_{\rm prop} = \frac{1}{2} {\bf p \cdot p} + v_{\theta}(0) p_{\theta} - v {\bf r \cdot n} + V({\bf r})\]yielding
\[\dot {\bf r} = {\bf p} \qquad \dot{\bf p} = v {\bf n} - \nabla V\] \[\dot{\theta} = v_{\theta}(0) \qquad \dot{p}_{\theta} = - v {\bf r} \cdot {\bf e}_{\phi}(\theta)\]And yet again we have defined an effective Hamiltonian for the dynamics.
[References]
1. Berry, M. V. & Shukla, P. Hamiltonian curl forces. Proc. R. Soc. A. 471, 20150002 (2015).
2. Berry, M. V. & Shukla, P. Classical dynamics with curl forces, and motion driven by time-dependent flux. J. Phys. A: Math. Theor. 45, 305201 (2012).