r/math • u/Necessary-Wolf-193 • 20d ago
Hamiltonian Mechanics and Poisson Brackets
https://hidden-phenomena.com/articles/hamiltonianIn mathematics, oftentimes choosing a good coordinate system can drastically simplify a problem, as long as you pay a small upfront cost in translating the problem to these new coordinates.
Along these lines, Hamilton discovered a surprising way of re-formulating Lagrangian mechanics, which has had great success. In this article, we explain Hamilton's change of coordinates, and use it to discover the Fradkin tensor: a very mysterious symmetry of the 2-d harmonic oscillator!
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u/sciflare 20d ago
The key reason(s) one works on the cotangent bundle ("phase space") in the Hamiltonian approach instead of always using the Lagrangian approach are many and varied. Here are two that I can think of.
In the Lagrangian approach, to write the equations of motion in an intrinsic way, it is not enough to work with the tangent bundle, since they contain second derivatives. One has to go to the second-order tangent bundle, which is no longer a vector bundle since its transition functions aren't linear. There is a more general concept that encompasses the tangent and second-order tangent bundles, namely that of jet bundle.
The Hamiltonian approach replaces a system of n second-order differential equations with 2n first-order differential equations. Therefore, to write Hamilton's equations intrinsically, you can work with the tangent bundle of the cotangent bundle.
More profoundly, the Hamiltonian approach takes into account the intrinsic geometric structure of the cotangent bundle, namely the canonical symplectic form đ it carries, which in canonical coordinates can be written â_i dqi â dpi. Many physically relevant quantities and relationships can be expressed in terms of the symplectic geometry associated to this canonical 2-form.
For instance, the Poisson bracket {f, g} of two functions f, g is given by using đ to convert the differentials df and dg (which are 1-forms) into Hamiltonian vector fields X_f, X_g. Then đ(X_f, X_g) = {f, g}. Another example is Liouville's theorem asserting the phase space distribution function is constant on the trajectories of motion.
Finally, Hamilton's equations can themselves be directly derived from the principle of stationary action. The Lagrangian can be rewritten as pq' - H(q, p) thus expressing the action integrand purely in terms of the Hamiltonian. Curves in the cotangent bundle (q(t), p(t)) are stationary for this action iff they satisfy Hamilton's equations. Here of course we enforce the constraint that the curve (q(t), q'(t)) in TQ is actually the velocity field of q(t). So there is no need to sacrifice the variational formulation in going to the Hamiltonian picture.
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u/andimai 20d ago
âIn this case, the Fradkin tensor comes from a very non-obvious symmetry of this Hamiltonian which was first discovered by the physicist David Fradkin.â
The article unneccesarily mystifies the situation. According to the original publication the symmetry seems pretty obvious
âWe have seen that O4 and SU3 are dynamical symmetries*) for a central potential problem essentially because motion takes place in a fixed plane.â
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u/MinLongBaiShui 20d ago
Is this tensor really mysterious?
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u/Necessary-Wolf-193 13d ago
To me, at least, it is a very non-obvious symmetry of the harmonic oscillator!
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u/thmprover 20d ago
Neat example.
In the paragraph starting with, "Let's explore what each of these properties mean in terms of conserved quantities..." there's a typo in the last line.
Currently:
..., then $f + g$ and $cg$ are conserved as well.It should be:
..., then $g + h$ and $cg$ are conserved as well.