# Physical interpretation of the symplectic property for particle systems

This this Wikipedia page states that symplecticness (being symplectic) is a property of particle systems governed by Hamilton's equation. I am thinking of classical-mechanics point particles described by position and momentum, nothing particularly 'fancy'.

I can see how Hamilton's equations can govern the evolution of a system of particles. However, I lack an understanding of the physical meaningfulness of symplecticness as a descriptive property, and of what could go paradoxical if this property is not respected.

The resources I found along the way lean to the side of mathematical formalism [1-4]. After curiosity I looked for the etymology of the word; although I found no authoritative source, the word symplectic could have to do with 'hitting together', so matters involving friction or being complementary.

Would someone please provide some explanation and examples to show why symplecticness is important for describing the evolution of particle systems?

From this community, but still too jargonistic for me:

• – Qmechanic May 4 '20 at 11:16

## 1 Answer

In my experience, symplectic structures are a neat mathematical way of describing what essentially is a rotational phenomenon. The generator of rotations in $$\mathbb R^2$$ is the symplectic $$2\times 2$$ matrix. Similarly, the rotations in $$\mathbb C$$ are generated by $$i$$. Invariance under rotations implies that some quantities are conserved, like angular momentum. So, more generally, when the symplectic structure is invariant, you can expect there to be invariant quantities. In Hamiltonian mechanics, the invariance of the symplectic structure is related to the Hamiltonian function being a first integral. You also get other interesting results as a consequence of this symplectic structure, like Liouville's theorem, Poincaré recurrence etc...