Hello everyone(adsbygoogle = window.adsbygoogle || []).push({});

I have difficulties in understanding some stuff in Lagrangian and Hamiltonian mechanics. This concerns the equations :

[tex]\dot p = - \frac{\partial H}{\partial q}[/tex]

[tex]\frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q}[/tex]

First I have to say that I'm a math guy and I understand physics far better by considering things geometrically. Unfortunately, the above equations are 99% of times introduced / explained by decomposing quantities into coordinates : q becomes [itex](q_1, \cdots, q_n)[/itex], p becomes [itex](p_1, \cdots, p_n)[/itex] (even in so-called books of "physics for mathematicians"). I would prefere so much to have coordinate-free definitions because coordinate makes everything looks similar (to [itex]\mathbb{R}^n[/itex]) and make geometry-oriented thinking difficult.

Among the two equations of hamiltonian mechanics :

[tex]\dot q = \frac{\partial H}{\partial p}, \dot p = - \frac{\partial H}{\partial q}[/tex]

only the first is clear to me. Some might argue that the equations are symetric but according to me they are definitely not.

If H is a scalar field on [itex]\mathbb{R} \times T^*\mathcal{M}[/itex] where the manifold [itex]\mathcal{M}[/itex] is the configuration space, the informal derivative [itex]\frac{\partial H}{\partial p}[/itex] can be given a rigorous meaning. If time t and point q are given, [itex]p \mapsto H(t,q,p)[/itex] is a function from [itex]T^*_q\mathcal{M}[/itex] to [itex]\mathbb{R}[/itex]. Thus, it has a total derivative [itex]D( p \mapsto H(t,q,p) )[/itex] from [itex]T^*_q\mathcal{M}[/itex] to [itex]T_q\mathcal{M}[/itex]. So for a given trajectory [itex]\mathfrak{q}: \mathbb{R} \mapsto \mathcal{M}[/itex], the equation [itex]\dot{\mathfrak{q}}(t) = \frac{\partial H}{\partial p}(t,\mathfrak{q}(t),\mathfrak{p}(t))[/itex] has a precise meaning.

But for [itex]\frac{\partial H}{\partial q}[/itex], I don't understand what it could possibly mean to derivate H along q, with a constant p. When q is changing, you're moving from a fiber to another one, and the vectors p in different fibers are incomparable. Consequently, the phrase "constant p" sounds non-sensical (unless of course we have a tool to match the fibers but I have never seen any mention of a (pseudo)riemanian structure / connexion in this context)

For the same reason, I don't knwow how to interprete [itex]\frac{\partial L}{\partial q}[/itex]

I know that [itex]\mathcal{N} = T^*\mathcal{M}[/itex] is itself a manifold with interesting properties due to its canonical symplectic structure (such as the canonical isomorphism between [itex]T\mathcal{N} = TT^*\mathcal{M}[/itex] and [itex]T^*\mathcal{N} = T^*T^*\mathcal{M}[/itex]). So I have already considered a possible interpretation of [itex]\frac{\partial H}{\partial q}[/itex] as the (exterior) derivative of H, considered as a scalar field on [itex]\mathcal{N}[/itex], (the derivative [itex]dH : \mathcal{N} \mapsto T^*\mathcal{N}[/itex]). In a similar manner, for a "momentum trajectory" [itex]\mathfrak{p} : \mathbb{R} \mapsto \mathcal{N}[/itex], we can think about the derivated function [itex]\dot{\mathfrak{p}} : \mathbb{R} \mapsto T\mathcal{N}[/itex].

This could possibly match but it sounds so different from what the notation [itex]\frac{\partial H}{\partial q}[/itex] suggests ([itex]dH[/itex] is a really total derivative of H: it gives the variation of H for all "directions" of [itex]\mathcal{N} = T^*\mathcal{M}[/itex], including variations along q), that I can hardly believe the correct explanation is to be found this way.

What to think about all that ? Should I consider othe theorical entities like Poisson Brackets ?

Thank you

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Maths of Hamiltonian / Lagrangian mechanics

**Physics Forums | Science Articles, Homework Help, Discussion**