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I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where the $q_i$'s are constrained to (in this case it is simply $\mathbb{R}^n$), then the $(\dot q_1, \dot q_2,... \dot q_n) $'s lie in the tangent space of $M$ at the $(q_1,q_2,...q_n)$'s. So Lagrangian can be considered as a function on the tangent bundle $T_M$. The transformation from a Lagrangian to a Hamiltonian converts $\dot q_i$ dependence into $p_i = \frac{\partial L}{\partial \dot q_i} $ dependence.

Here comes my confusion. The transformation from a Lagrangian to a Hamiltonian converts $\dot q_i$ dependence into $p_i = \frac{\partial L}{\partial \dot q_i} $ dependence. If we consider $\dot q_i$s as components of the velocity vector $\vec{v}$, then the $p_i$'s are components of a vector in the DUAL space, and the Hamiltonian is a function on the cotangent bundle $T^*_M$ . However, the conventional definition of momentum in classical mechanics is $\vec{p} = m\vec{v} $ which implies that $p$ is a vector. So it seems that canonical momentum conjugates are dual to the conventional momentum. What do I make of that? This distinction is not necessary in $\mathbb{R}^n$ since vectors and one-forms have the same components. But what happens in a general manifold? Should I attach any significance to the fact that canonical momentum conjugates are one-forms?

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  • $\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/176555/2451 and links therein. $\endgroup$ – Qmechanic Oct 7 '15 at 6:47
  • $\begingroup$ @Qmechanic I think the link you provided addresses the question of "why the canonical momentum conjugates are dual to velocity vectors" which I already understand. I guess my question is: when do you need to pay attention to the fact that canonical momentum conjugate is in the dual space? How is that important? $\endgroup$ – Zhengyan Shi Oct 7 '15 at 17:39
  • $\begingroup$ So since momentum and velocity live in dual spaces, one needs a metric to raise and lower indices between them. That metric is often chosen to be the unit metric. $\endgroup$ – Qmechanic Oct 7 '15 at 18:19
  • $\begingroup$ Wait I thought momentum and velocity live in the same space, which is dual to the space that canonical momentum conjugates live in. That is my major confusion... canonical momentum and momentum don't live in the same space anymore... $\endgroup$ – Zhengyan Shi Oct 7 '15 at 18:21
  • $\begingroup$ Well, now we enter the issue of conventions & semantics. This is not constructive to discuss. Different authors use different conventions. The conventions used must be read from context. $\endgroup$ – Qmechanic Oct 7 '15 at 18:32

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