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Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function defined on? (I know that the $\dot p$ is there for names sake only).

My stab at an answer is it is a product space between $\mathcal L:TQ\times T^*Q\times \mathbb R\to \mathbb R$ since we are dealing with both velocity and momentum (where $Q$ is a configuration manifold). However this makes zero intuitive sense to me. If indeed it is phase space, my understanding was that taking the fibre derivative with respect to a function on the tangent bundle changed the velocity coordinates to momentum coordinates - the Legendre transform?

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Let $X$ be the phase space. Then $L_\text{ph}(q,p,\dot{q},\dot{p},t)$ is a function on $TX\times \mathbb{R}$1, since the coordinates of $TX\times\mathbb{R}$ are precisely the coordinates of $X$, i.e. $(q,p)$ and their derivatives $(\dot{q},\dot{p})$ (and time $t$).

If Hamilton's equations are fulfilled, there are relations among $q,\dot{q},p,\dot{p}$ (the defining relations of the Legendre transformation) that reduce $L_\text{ph}(q,\dot{q},p,\dot{p},t)$ to the usual Lagrangian $L(q,\dot{q},t)$.


1In full analogy to the usual Lagrangian $L(q,\dot{q},t)$ being a function on $TQ\times\mathbb{R}$, where $Q$ is the $q$-space ("configuration space").

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  1. If $Q$ is configuration space, then the Lagrangian is a function $L: TQ\times \mathbb{R}\to \mathbb{R}$.

  2. Let the cotangent bundle $M:=T^{\ast}Q$ be the corresponding phase space.

  3. The Hamiltonian/phase space Lagrangian is a function $L_H: TM\times \mathbb{R}\to \mathbb{R}$.

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