I searched for a definition of Hamiltonian system on Huang and Tuckerman text but have not found anything precise.

So intuitively I suppose:

Hamiltonian system= a system which admits a complete description via Hamilton equations.

NON Hamiltonian system= a system which cannot be described via Hamilton equation.

Often I have heard things like "An hamiltonian system is a system which conserves energy".

But I know that an Hamiltonian system conserves energy if and only if its Hamiltonian does not depend explicitly on time: $$\frac{dH}{dt}=\frac{\partial H}{\partial t}$$ Up to now I have never found an example of an Hamiltonian system which does not conserve energy. My first question is: could someone give me an example of an Hamiltonian system which does not conserve energy?

Reading this: An example of non-Hamiltonian systems it says:

Hamiltonian mechanics describes reversible dynamics.

If this is true, could someone give me a reference in which this is explained?


A mechanical system is Hamiltonian if it is described by a scalar function $H:\Gamma\to\mathbb R$, where $\Gamma$ is an even-dimensional manifold, which physically corresponds to the cotangent bundle to the configuration space (this interpretation might actually lack any physical interpretation, cf. http://www.ltcc.ac.uk/courses/BioMathematics/LTCC_LV2010.pdf on the Hamiltonian approach to the Lotka-Volterra equations). The dynamics arises through the local flow defined by the Hamiltonian vector field $X_H$, which comes from taking the 1-form $\text d H$ and taking it to the tangent bundle through the natural symplectic structure (of symplectic form $\omega$) on $\Gamma$ by $$\text d H := \iota_{X_H}\omega.$$ A remarkable property of Hamiltonian systems is then given by Liouville's theorem, which states that the volume of a closed surface in $\Gamma$ is preserved by the flow generated by $X_H$. Roughly speaking, this means that every point inside such a surface, when taken as an initial condition for the equation of motions, evolves in such a way that the volume of the closed surface remains constant at any time, although it might change shape through the time evolution. Mechanical systems that do not exhibit this phenomenon cannot possibly be Hamiltonians, and therefore the equations cannot be derived, as described above, from a single scalar function.


You are mixing two different definitions of "Hamiltonian system". The standard definition is "a system that obeys the Hamilton equations and whose Hamiltonian is time-independent". It is only under this definition that the statements "Hamiltonian systems conserve energy" and "Hamiltonian systems describe reversible dynamics" are true. You can't find an example of a Hamiltonian system (in this sense) which does not conserve energy, because Hamiltonian systems conserve energy by definition.

Under your weaker, nonstandard definition - "A Hamiltonian system is a system which obeys the Hamilton equations" - it is quite simple to find examples that do not conserve energy: just write down a conservative system plus a friction term. The most familiar example would be the dampled harmonic oscillator.


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