From Classical Trajectories to Quantum Commutation Relations

Abstract

In describing a dynamical system, the greatest part of the work for a theoretician is to translate experimental data into differential equations. It is desirable for such differential equations to admit a Lagrangian and/or an Hamiltonian description because of the Noether theorem and because they are the starting point for the quantization. As a matter of fact many ambiguities arise in each step of such a reconstruction which must be solved by the ingenuity of the theoretician. In the present work we describe geometric structures emerging in Lagrangian, Hamiltonian and Quantum description of a dynamical system underlining how many of them are not really fixed only by the trajectories observed by the experimentalist.

To Alberto on his 60th birthday.

Appendix: Inverse Problem for Implicit Differential Equations

In Sect. 9.2 we saw how, in general, experimental data lead the theoretician to build a submanifold of $\mathbf {TT}\mathcal {Q}$, that is, an implicit differential equation on (a submanifold of) the tangent bundle $\mathbf T \mathcal {Q}$. Within this context the inverse problem is much more complicated to address than it is in the explicit case. Essentially, this is due to the fact that the Euler-Lagrange equations are formulated by means of an implicit differential equation on $\mathbf T ^*\mathcal {Q}$ rather than by means of an implicit differential equation on $\mathbf T \mathcal {Q}$, even though the Lagrangian function is defined on the tangent bundle $\mathbf T \mathcal {Q}$. Indeed, given the Lagrangian function $\mathscr {L}$, we have $\mathrm {d}\mathscr {L}:\mathbf T \mathcal {Q}\rightarrow \,\mathbf T ^{*}{} \mathbf T \mathcal {Q}$, and thus $\mathrm {d}\mathscr {L}(\mathbf T \mathcal {Q})$ is a submanifold of $\mathbf T ^{*}{} \mathbf T \mathcal {Q}$. By means of the inverse of the canonical Tulczyjew isomorphism $\tau :\mathbf {TT}^*\mathcal {Q}\rightarrow \mathbf T ^*\mathbf T \mathcal {Q}$ [25], we obtain a submanifold of $\mathbf {TT}^*\mathcal {Q}$, i.e., an implicit differential equation on $\mathbf T ^*\mathcal {Q}$. Explicitly, we have

$$\begin{aligned} \mathrm {d}\mathscr {L}\;\; |\;\; \mathbf T \mathcal {Q} \rightarrow \mathbf T ^*\mathbf T \mathcal {Q} \;\; : \;\; (q^i,\, v^i) \mapsto \left( q^i,\,v^i,\, \frac{\partial \mathscr {L}}{\partial q^i},\, \frac{\partial \mathscr {L}}{\partial v^i} \right) \,, \end{aligned}$$

(9.71)

while the Tulczyjew isomorphism between $\mathbf {TT}^*\mathcal {Q}$ and $\mathbf T ^*\mathbf T \mathcal {Q}$ is defined by

$$\begin{aligned} \tau \;\;|\;\; \mathbf {TT}^*\mathcal {Q} \rightarrow \mathbf T ^*\mathbf T \mathcal {Q} \;\;:\;\; (q^i,\,p_i,\,v_q^i,\, {v_p}_i) \mapsto (q^i,\,v_q^i,\,{v_p}_i,\,p_i) \end{aligned}$$

(9.72)

(see [25], Sect. 3). This is, in fact, a symplectomorphism mapping the canonical symplectic form over $\mathbf T ^*\mathbf T \mathcal {Q}$, i.e., $\mathrm {d}q^i \wedge \mathrm {d}{p_q}_i + \mathrm {d}v^i \wedge \mathrm {d}{p_v}_i$, into the canonical symplectic form over $\mathbf {TT}^*\mathcal {Q}$, i. e., $\mathrm {d}v_q^i \wedge \mathrm {d}{p_v}_i + \mathrm {d}q^i \wedge \mathrm {d}{p_q}_i$. By composing $\mathrm {d}\mathscr {L}$ and $\tau ^{-1}$ we obtain a submanifold of $\mathbf T ^*\mathbf T \mathcal {Q}$, say $\Sigma$, given by

$$\begin{aligned} \Sigma :=\left( \tau ^{-1} \circ \mathrm {d}\mathscr {L}\right) (\mathbf T \mathcal {Q}) =\left\{ \, (q^i,p_i,\,{v_q}^i ,\, {v_p}_i)\in \mathbf {TT}^*\mathcal {Q} \;|\; p_i = \frac{\partial \mathscr {L}}{\partial v^i}\,, \; {v_p}_i = \frac{\partial \mathscr {L}}{\partial q^i} \, \right\} \,. \end{aligned}$$

(9.73)

Writing $i_{\Sigma }$ for the canonical immersion of $\Sigma$ into $\mathbf {TT}^*\mathcal {Q}$, it follows that $\Sigma$ is a Lagrangian (or simply isotropic if L is not regular) submanifold of $\mathbf {TT}^*\mathcal {Q}$ because

$$\begin{aligned} i_{\Sigma }^{*}\left( \mathrm {d}v_q^i \wedge \mathrm {d}{p_v}_i + \mathrm {d}q^i \wedge \mathrm {d}{p_q}_i \,\right) \, = \, \mathrm {d}v_q^i \wedge \,\mathrm {d} \left( \frac{\partial \mathscr {L}}{\partial v^i} \right) + \mathrm {d}q^i \wedge \mathrm {d}\left( \frac{\partial \mathscr {L}}{\partial q^i} \right) \,=\, 0 \end{aligned}$$

(9.74)

since $\mathscr {L}$ depends only on $(q^i,\,{v_q}^i)$. Note that, with the prescription that ${v_p}_i \,=\, \frac{d}{dt}p_i$, it immediately follows that $\Sigma$ as defined in (9.73) is the submanifold of $\mathbf {TT}^*\mathcal {Q}$ on which the Euler-Lagrange equations

$$\begin{aligned} \frac{d}{dt}\frac{\partial \mathscr {L}}{\partial v^i} \,=\, \frac{\partial \mathscr {L}}{\partial q^i} \end{aligned}$$

(9.75)

are identically satisfied.

Remark 5

A similar construction is possible for the Hamiltonian case where it clearly emerges that the submanifold of $\mathbf {TT}^*\mathcal {Q}$ one obtains is the graph of a vector field, that is, the equations are always explicit ones. Consider the cotangent bundle $\mathbf T ^*\mathcal {Q}$ of the configuration space $\mathcal {Q}$ and a Hamiltonian function $\mathscr {H} \;\;|\;\; \mathbf T ^*\mathcal {Q} \rightarrow \mathbb {R}$. Via its differential

$$\begin{aligned} \mathrm {d}\mathscr {H}\;\;|\;\;\mathbf T ^*\mathcal {Q} \rightarrow \mathbf T ^*\mathbf T ^*\mathcal {Q} \,, \end{aligned}$$

(9.76)

we can define a submanifold $\Sigma$ of $\mathbf T ^*\mathbf T ^*\mathcal {Q}$ by setting $\Sigma :=\mathrm {d}\mathscr {H}(\mathbf T ^*\mathcal {Q})$. Differently from the Lagrangian case, we have that $\mathbf T ^*\mathbf T ^*\mathcal {Q}$ is isomorphic to $\mathbf {TT}^*\mathcal {Q}$ because the Poisson structure $\Lambda$ associated with the canonical symplectic structure over $\mathbf T ^*\mathcal {Q}$ define an isomorphism between differential forms and vector fields on $\mathbf T ^*\mathcal {Q}$. With an evident abuse of notation, we denote this isomorphism with $\Lambda$. By means of $\Lambda$, we obtain the implicit differential equation $\Lambda (\Sigma )$ on $\mathbf {TT}^*\mathcal {Q}$. Denoting by $X_{\mathscr {H}}$ the Hamiltonian vector field associated with $\mathscr {H}$ by means of the Poisson tensor $\Lambda$, that is, $X_{\mathscr {H}}=\Lambda (\mathrm {d}\mathscr {H})$, the following diagram may be defined:

and it follows that $\Lambda (\Sigma )$ is precisely $X_{\mathscr {H}}(\mathbf T ^*\mathcal {Q})$. Consequently, being $\Lambda (\Sigma )$ the image of $\mathbf T ^*\mathcal {Q}$ through a vector field, it emerges that the dynamics is always an explicit one in the Hamiltonian case, as it is also clear from the standard form of the Hamilton equations.

We can say that Euler-Lagrange equations force us to work with a Lagrangian submanifold of $\mathbf {TT}^*\mathcal {Q}$, while, on the one hand, we saw in Sect. 9.2 how experimental data would naturally lead us to build a submanifold of $\mathbf {TT}\mathcal {Q}$. Consequently, the following question is unavoidable: how can these two seemingly uncompatible instances be related? The essential difficulty is due to the absence of a natural, “pre-existing”, symplectic structure on $\mathbf T {} \mathbf T \mathcal {Q}$. To be able to formulate the inverse problem for the submanifold of $\mathbf {TT}\mathcal {Q}$ we construct out of the trajectories on $\mathcal {Q}$, we would need a map:

$$\begin{aligned} \phi \;\; : \;\; \mathbf T \mathcal {Q} \rightarrow \mathbf T ^*\mathcal {Q} \end{aligned}$$

(9.78)

so that we would be able to map the submanifold of $\mathbf {TT}\mathcal {Q}$, that we constructed out of trajectories, onto a submanifold of $\mathbf T {} \mathbf T ^*\mathcal {Q}$ by means of the tangent map $T\phi$:

It may happen that the Lagrangian function itself could provide us with the map $\phi$ by means of the fiber derivative

$$\begin{aligned} \mathscr {F}\mathscr {L}\;|\;\; \mathbf T \mathcal {Q} \rightarrow \mathbf T ^*\mathcal {Q} \;:\;\; (q^i, \, v^i) \mapsto \pi _\mathbf{T ^*}\, \circ \, \tau \, \circ \, \mathrm {d}\mathscr {L}= \left( q^i, \frac{\partial \mathscr {L}}{\partial v^i} \right) \,. \end{aligned}$$

(9.80)

This map coincides with the map defined by the following diagram:

By means of the fiber derivative, the canonical symplectic structure $\Omega = \mathrm {d}p_i \wedge \mathrm {d}q^i$ and its potential $\theta = p_i \mathrm {d}q^i$ on $\mathbf T ^*\mathcal {Q}$ can be pulled-back on $\mathbf T \mathcal {Q}$ to obtain

$$\begin{aligned} \begin{aligned}&\theta _{\mathscr {L}} = (\mathscr {F}\mathscr {L})^* \theta = \partial ^v_i \mathscr {L}\mathrm {d}q^i \, \\ \Omega _{\mathscr {L}} = (\mathscr {F}\mathscr {L})^*&\Omega = \frac{\partial ^2 \mathscr {L}}{\partial q^i \partial v^j} \mathrm {d}q^i \wedge \mathrm {d}q^j + \frac{\partial ^2 \mathscr {L}}{\partial v^i \partial v^j} \mathrm {d}v^i \wedge \mathrm {d}q^j\,. \end{aligned} \end{aligned}$$

(9.82)

By means of its tangent map, $T\mathscr {F}\mathscr {L}$, the symplectic structure on $\mathbf {TT}^*\mathcal {Q}$ and its potential, $\dot{\Omega }$ and $\dot{\theta }$, can be pulled-back on $\mathbf {TT}\mathcal {Q}$:

$$\begin{aligned} \begin{aligned} \dot{\theta }_{\mathscr {L}} = (T\mathscr {F}\mathscr {L})^* \dot{\theta } \\ \dot{\Omega }_{\mathscr {L}} = (T\mathscr {F}\mathscr {L})^*\dot{\Omega }\,. \end{aligned} \end{aligned}$$

(9.83)

Note that, in general, $\Omega _{\mathscr {L}}$ and $\dot{\Omega }_{\mathscr {L}}$ are no longer symplectic form because they may present a kernel which depends on $\mathscr {L}$.

In conclusion, the Lagrangian plays a double role within the formulation of the inverse problem for implicit differential equations. First, it defines a Lagrangian submanifold $\Sigma$ of $\mathbf {TT}^*\mathcal {Q}$, which represents the Lagrangian formulation of the dynamics. Second, it allows for the definition of a fiber derivative $\mathscr {F}\mathscr {L}$ which, if suitable regularity conditions on $\mathscr {L}$ are satisfied, would make it possible to impose that the pre-image of $\Sigma$ through $T\mathscr {F}\mathscr {L}$ coincide with the submanifold of $\mathbf {TT}\mathcal {Q}$ on which the experimental data naturally live. See also Sect. 2.1 in [17] for another discussion about the inverse problem for implicit differential equations.