22. Analtycal Mechanics - Notes for Quantum Mechanics#

Contents
Topics and tools

  • Calculus of variations

  • Legendre transformation, in the definition of functions with different independent variables.

    Examples

    • From Lagrangian function to Hamiltonian function \(\mathcal{H}(\mathbf{q}, \mathbf{p}, t) := \mathbf{p} \cdot \dot{\mathbf{q}} - \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}},t)\)

    • From one generating function to another, e.g. \(F_2(\mathbf{q}, \mathbf{P}, t) := \mathbf{P} \cdot \mathbf{Q} + F_1(\mathbf{q}, \mathbf{Q}, t)\)

    • …

  • Properties of canonical transformations can be derived with a proper application of rules of derivation of composite functions, paying attention to the set of independent variables of these functions.

22.1. Lagrange and Hamiltonian Mechanics#

Lagrangian mechanics

Equations of motion follow the principle of stationary action functional

(22.1)#\[0 = \delta S[\mathbf{q}(t)] = \int_{t_0}^{t_1} \mathcal{L}( \mathbf{q}(t), \dot{\mathbf{q}}(t),t ) \, dt \ ,\]

with prescribed extreme values, so that \(\delta \mathbf{q}(t_0) = \delta \mathbf{q}(t_1) = \mathbf{0}\). Here \(\mathcal{L}\) is the Lagrangian function, with \(\mathbf{q}(t)\) the vector of generalized coordinates, and \(t\) the time. Equations of motion are the Lagrange equations

(22.2)#\[\dfrac{d}{dt}\left( \dfrac{\partial \mathcal{L}}{\partial \dot{\mathbf{q}}} \right) - \dfrac{\partial \mathcal{L}}{\partial \mathbf{q}} = \mathbf{0} \ .\]

The generalized momentum is defined as \(\mathbf{p} := \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{q}}}\). If the Lagrangian function doesn’t explicitly depend on the generalized coordinate \(q^k(t)\), the generalized momentum \(p_k(t) = \frac{\partial \mathcal{L}}{\partial q^k}\) is constant.

Hamiltonian mechanics.

Hamiltonian function is defined as \(\mathcal{H}(\mathbf{q}, \mathbf{p}, t) := \mathbf{p} \cdot \dot{\mathbf{q}} - \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, t)\), with \(\dot{\mathbf{q}}(\mathbf{q}, \mathbf{p}, t)\). It’s differential reads

(22.3)#\[\begin{split}\begin{aligned} d \mathcal{H} & = \dot{\mathbf{q}} \cdot d \mathbf{p} + \mathbf{p} \cdot d \dot{\mathbf{q}} - \partial_{\dot{\mathbf{q}}} \mathcal{L} \cdot d \dot{\mathbf{q}} - \partial_{\mathbf{q}} \mathcal{L} \cdot d \mathbf{q} - \partial_t \mathcal{L} dt = \\ & = \dot{\mathbf{q}} \cdot d \mathbf{p} - \partial_{\mathbf{q}} \mathcal{L} \cdot d \mathbf{q} - \partial_t \mathcal{L} dt \ , \end{aligned}\end{split}\]

by the definition of the generalized momentum, \(\mathbf{p} = \partial_{\dot{\mathbf{q}}} \mathcal{L}\). Hamilton equations immediately follows

\[\begin{split}\left\{ \begin{aligned} \dot{\mathbf{q}}(t) & = \partial_{\mathbf{p}} \mathcal{H}(\mathbf{q}(t), \mathbf{p}(t), t) \\ \dot{\mathbf{p}}(t) & =-\partial_{\mathbf{q}} \mathcal{H}(\mathbf{q}(t), \mathbf{p}(t), t) \\ \end{aligned} \right.\end{split}\]

with the relation

\[\left.\partial_t \mathcal{H}\right|_{\mathbf{q}, \mathbf{p}}(\mathbf{q}(t), \mathbf{p}(t),t) = - \left.\partial_t \mathcal{L}\right|_{\mathbf{q}, \dot{\mathbf{q}}}(\mathbf{q}(t), \dot{\mathbf{q}}(t),t) \ . \]

Using Hamilton’s equations, it’s immediately proved that \(d_t \mathcal{H} = \partial_t \mathcal{H}\). The Hamiltonian function is thus conserved - an integral of motion - if the Lagrangian function doesn’t explicitly depend on \(t\).

22.2. Canonical transformations#

Definition

A canonical transformation is defined as a change of coordinates from \((\mathbf{q}, \mathbf{p})\) to \((\mathbf{Q}, \mathbf{P})\) so that the equations of motion have the same expression

(22.4)#\[\begin{split} \left\{ \begin{aligned} \dot{\mathbf{q}} & = \partial_{\mathbf{p}} \mathcal{H} \\ \dot{\mathbf{p}} & =-\partial_{\mathbf{q}} \mathcal{H} \\ \end{aligned} \right. \qquad , \qquad \left\{ \begin{aligned} \dot{\mathbf{Q}} & = \partial_{\mathbf{P}} \mathscr{H} \\ \dot{\mathbf{P}} & =-\partial_{\mathbf{Q}} \mathscr{H} \\ \end{aligned} \right. \end{split}\]
Simplectic structure

Let’s define

\[\begin{split} \boldsymbol\eta = \begin{pmatrix} \mathbf{q} \\ \mathbf{p} \end{pmatrix} \qquad , \qquad \boldsymbol\zeta = \begin{pmatrix} \mathbf{Q} \\ \mathbf{P} \end{pmatrix} \end{split}\]

whose relation is assumed to be invertible for each time \(t\),

(22.5)#\[\boldsymbol\zeta( \boldsymbol\eta, t) \qquad , \qquad \boldsymbol\eta ( \boldsymbol\zeta, t) \ . \]

Hamilton equations. Hamilton equations (22.4) can be recast as

(22.6)#\[ \dot{\boldsymbol{ \eta}} = \mathbf{J} \nabla_{\boldsymbol{ \eta}} \mathcal{H} \qquad , \qquad \dot{\boldsymbol{\zeta}} = \mathbf{J} \nabla_{\boldsymbol{\zeta}} \mathscr{H} \ , \]

with the matrix \(\mathbf{J} = \begin{pmatrix} \mathbf{0} & \mathbf{I} \\ -\mathbf{I} & \mathbf{0} \end{pmatrix}\).

Recasting Hamilton equations in new coordinates, using derivatives of composite functions. As the the vector of the new coordinates \(\boldsymbol\zeta(t)\) can be written as a function of the old coordinates \(\boldsymbol\eta(t)\) through the function (22.5)

\[\begin{split}\begin{aligned} \dfrac{d}{dt} \boldsymbol{\zeta} \left(\boldsymbol\eta(t), t\right) & = \dot{\boldsymbol\eta} \cdot \nabla_{\boldsymbol\eta} \boldsymbol\zeta|_t + \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} = \\ & = \mathbf{M}^T(\boldsymbol\eta(t), t) \dot{\boldsymbol\eta} + \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} = \\ & = \mathbf{M}^T \, \mathbf{J} \, \nabla_{\boldsymbol\eta} \mathcal{H} + \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} \ , \end{aligned}\end{split}\]

and changing from \((\boldsymbol\zeta, t)\) to \((\boldsymbol\eta, t)\) variables

(22.7)#\[\begin{aligned} \dot{\boldsymbol{\zeta}} & = \mathbf{M}^T \mathbf{J} \mathbf{M} \, \nabla_{\boldsymbol\zeta} \left.\mathcal{H}\right|_t + \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} \ , \end{aligned}\]

with \(\left\{ \mathbf{M} \right\}_{ij} = \frac{\partial \zeta_j}{\partial \eta_i}(\boldsymbol\eta, t)\), and the rule of transformation of the gradient of a scalar function,

\[f = f( \boldsymbol\eta, t) = f( \boldsymbol\eta(\boldsymbol\zeta, t), t) = \mathscr{f}( \boldsymbol\zeta, t) = \mathscr{f}( \boldsymbol\zeta( \boldsymbol\eta, t), t) \ ,\]
\[\begin{aligned} \nabla_{\boldsymbol\eta} f(\boldsymbol\eta(\boldsymbol\zeta(t), t), t) = \left.\nabla_{\boldsymbol\eta} \boldsymbol\zeta\right|_t \cdot \left.\nabla_{\boldsymbol\zeta} \mathscr{f} \right|_t = \mathbf{M} \nabla_{\boldsymbol\zeta} f \ . \end{aligned}\]

The equation (22.7) must be compared with the Hamilton equations \(\dot{\boldsymbol{\zeta}} = \mathbf{J} \nabla_{\boldsymbol\zeta} \mathscr{H}\) from (22.6).

Comparison between different forms of Hamilton equations - canonical transformations are pure kinematics, independent from the physics of the system of interest

From the comparison

\[\begin{split}\left\{ \begin{aligned} \dot{\boldsymbol{\zeta}} & = \mathbf{M}^T \mathbf{J} \mathbf{M} \, \nabla_{\boldsymbol\zeta} \left.\mathcal{H}\right|_t + \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} \\ \dot{\boldsymbol{\zeta}} & = \mathbf{J} \nabla_{\boldsymbol\zeta} \mathscr{H} \end{aligned} \right.\end{split}\]

it follows

\[ \mathbf{M}^T \mathbf{J} \mathbf{M} \, \nabla_{\boldsymbol\zeta} \left.\mathcal{H}\right|_t + \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} = \mathbf{J} \nabla_{\boldsymbol\zeta} \mathscr{H} \]

System 0. Let the system zero be a physical system whose Hamiltonian \(\mathcal{H}_0(\mathbf{q}, \mathbf{p}, t) := 0\). Remark This is the functional definition of the Hamiltonian, not 1-dimensional constraint between variables. It follows that

\[ \partial_t \boldsymbol\zeta|_{\boldsymbol\eta} = \mathbf{J} \nabla_{\boldsymbol\zeta} \mathscr{H}_0 \ . \]

Generic system. Exploiting the latter relation, for a generic system

\[ \mathbf{M}^T \mathbf{J} \mathbf{M} \, \nabla_{\boldsymbol\zeta} \left.\mathcal{H}\right|_t = \mathbf{J} \nabla_{\boldsymbol\zeta} \left( \mathscr{H} - \mathscr{H}_0 \right) \ . \]

As this relation must hold for any system, two relations follow:

  • the simplectic condition

    \[\mathbf{M}^T \mathbf{J} \mathbf{M} = \mathbf{J} \ .\]

  • a relation between the Hamiltonian functions

    \[\mathcal{H} - \mathscr{H} + \mathscr{H}_0 = f(t)\]

A comparison with the partial derivatives (22.9) of of type-2 generating function \(F_2(\mathbf{q}, \mathbf{P}, t)\), it follows that1

\[\left.\partial_t F_2\right|_{\mathbf{q}, \mathbf{P}} = \mathscr{H} - \mathcal{H} =\mathscr{H}_0 - f(t) \ .\]

22.3. Generating functions#

Generating functions

The pair of Hamilton equations (22.4) can be derived from the same variational principle of Lagrangian mechanics for two different choices of the generalized variables,

\[\begin{split}\begin{aligned} 0 & = \delta \int_{t_0}^{t_1} \mathcal{L}(\mathbf{q}(t), \dot{\mathbf{q}}(t), t) \, dt \\ 0 & = \delta \int_{t_0}^{t_1} \mathscr{L}(\mathbf{Q}(t), \dot{\mathbf{Q}}(t), t) \, dt \end{aligned}\end{split}\]

In order to get the same equations of motion, the two Lagrangian functions may differ by a time derivative \(\frac{d F_1}{dt}(\mathbf{q}(t), \mathbf{Q}(t), t)\), whose integral reads \(F_1(\dots,)\) evaluated in the extremes of integration, and whose variation is thus zero. Thus

\[\mathcal{L}(\mathbf{q}(t), \dot{\mathbf{q}}(t), t) = \mathscr{L}(\mathbf{Q}(t), \dot{\mathbf{Q}}(t), t) + \frac{d F_1}{dt}(\mathbf{q}(t), \mathbf{Q}(t), t)\]

or as a function of the Hamiltonian functions,

\[\mathbf{p} \cdot \dot{\mathbf{q}} -\mathcal{H}(\mathbf{q}, \mathbf{p}, t) = \mathbf{P} \cdot \dot{\mathbf{Q}} - \mathscr{H}(\mathbf{Q}, \mathbf{P}, t) + \frac{d F_1}{dt}(\mathbf{q}(t), \mathbf{Q}(t), t) \ ,\]

and thus

\[\begin{split}\begin{aligned} \dfrac{d F_1}{dt}(\mathbf{q}(t), \mathbf{Q}(t), t) & = \mathbf{p} \cdot \dot{\mathbf{q}} - \mathbf{P} \cdot \dot{\mathbf{Q}} - \mathcal{H}(\mathbf{q}, \mathbf{p}, t) + \mathscr{H}(\mathbf{Q}(t), \mathbf{P}(t),t) = \\ & = \partial_\mathbf{q} F_1 \cdot \dot{\mathbf{q}} + \partial_\mathbf{Q} F_1 \cdot \dot{\mathbf{Q}} + \partial_t F_1 \end{aligned}\end{split}\]

so that - using the arbitariness of the result on the choice of coordinates and the physics of the system - its differential reads

\[d F_1 = \mathbf{p} \cdot d \mathbf{q} - \mathbf{P} \cdot d \mathbf{Q} + ( \mathscr{H} - \mathcal{H} ) dt \ .\]

and its partial derivatives

(22.8)#\[ \left.\partial_{\mathbf{q}} F_1\right|_{\mathbf{q}, t } = \mathbf{p} \qquad , \qquad \left.\partial_{\mathbf{Q}} F_1\right|_{\mathbf{Q}, t } = -\mathbf{P} \qquad , \qquad \left.\partial_{t } F_1\right|_{\mathbf{q}, \mathbf{Q}} = \mathscr{H} - \mathcal{H} \ . \]

This is Type-1 generating function. Type-2 generating function is defined as

\[F_2(\mathbf{q}, \mathbf{P}, t) = \mathbf{P} \cdot \mathbf{Q} + F_1(\mathbf{q}, \mathbf{P}, t) \ ,\]

and its differential gives

\[d F_2 = \mathbf{p} \cdot d \mathbf{q} + \mathbf{Q} \cdot d \mathbf{P} + ( \mathscr{H} - \mathcal{H} ) dt \ , \]

and its partial derivatives

(22.9)#\[ \left.\partial_{\mathbf{q}} F_2\right|_{\mathbf{P}, t} = \mathbf{p} \qquad , \qquad \left.\partial_{\mathbf{P}} F_2\right|_{\mathbf{q}, t} = \mathbf{Q} \qquad , \qquad \left.\partial_{t } F_2\right|_{\mathbf{q}, \mathbf{P}} = \left.\partial_t F_1\right|_{\mathbf{q}, \mathbf{Q}} = \mathscr{H} - \mathcal{H} \ . \]

Type-3.

Type-4.

22.4. Hamilton-Jacobi formulation of mechanics#

Let a set of canonical variables \((\mathbf{Q}, \mathbf{P})\) exist so that \(\mathscr{H}(\mathbf{Q}, \mathbf{P}, t) = 0\).2 Remark This is the definition of the function, not a 1-dimensional constraint. For this set of canonical variables, Hamilton’s equations implies that \(\mathbf{Q}\), and \(\mathbf{P}\) are integrals of motion

\[\begin{split}\left\{ \begin{aligned} \dot{\mathbf{Q}} & = \nabla_{\mathbf{P}} \mathscr{H} && \qquad \rightarrow \qquad \mathbf{Q}(t) = \overline{\mathbf{Q}} \\ \dot{\mathbf{P}} & =-\nabla_{\mathbf{Q}} \mathscr{H} && \qquad \rightarrow \qquad \mathbf{P}(t) = \overline{\mathbf{P}} \\ \end{aligned} \right.\end{split}\]

Using type-2 generating function, and its partial derivatives (22.9), \(\mathbf{p} = \partial_{\mathbf{q}} F_2(\mathbf{q}, \mathbf{P}, t)\),

\[\begin{aligned} 0 & = \partial_t F_2\left( \mathbf{q}, \mathbf{P}, t \right) - \underbrace{\mathscr{H}(\mathbf{Q}, \mathbf{P}, t)}_{=0} + \mathcal{H}(\mathbf{q}, \mathbf{p}, t) \ , \end{aligned}\]

or, calling \(F_2(\mathbf{q}(t), \overline{\mathbf{P}}, t) := S(\mathbf{q}(t), t; \overline{\mathbf{P}})\) - as it can be seen as the action functional as a function of the ending time \(t\) and generalized coordinates \(\mathbf{q}(t)\), see below - the common form of the Hamilton-Jacobi equation follows

(22.10)#\[\begin{aligned} 0 = \partial_t S\left( \mathbf{q}(t), t; \overline{\mathbf{P}} \right) + \mathcal{H}\left(\mathbf{q}(t), \partial_{\mathbf{q}} S(\mathbf{q}(t), t; \overline{\mathbf{P}}) , t \right) \ . \end{aligned}\]

Stationary problems. For problems whose Hamiltonian is not explicitly function of time, \(\mathcal{H}(\mathbf{q}, \mathbf{p})\), the Hamiltonian is constant and equal to the energy of the system, \(\mathcal{H}(\mathbf{q}, \mathbf{p}) = E\). Integration in \(t\) gives

\[S\left(\mathbf{q}(t), t; \overline{\mathbf{P}} \right) = - E t + W\left(\mathbf{q}(t); \overline{\mathbf{P}} \right) \ .\]

The constant energy \(E\) is a function of the integrals of motion \(\overline{\mathbf{P}}\) only.

Introducing this expression in the Hamilton-Jacobi equation (22.10) gives an equation for \(W\left( \mathbf{q}, \overline{\mathbf{P}} \right)\),

\[E = \mathcal{H}\left( \mathbf{q}, \partial_{\mathbf{q}} W\left(\mathbf{q}, \overline{\mathbf{P}} \right) \right) \ .\]

22.4.1. Action functional and type-2 generating function#

22.5. Angle-action variables#

Let the system of interest be a conservative system using the original set of variables \((\mathbf{q}, \mathbf{p})\), so that the Hamiltonian doesn’t explicitly depend on time \(t\), and it’s constant \(\mathcal{H}(\mathbf{q}, \mathbf{p}) = E\). Let a set of canonical variables \((\mathbf{Q}, \mathbf{P}) =: (\mathbf{w}, \mathbf{J})\) exist so that \(\mathbf{J}(t) = \overline{\mathbf{J}}\) are integrals of motion, and \(\mathscr{H}(\mathbf{w}, \overline{\mathbf{J}}, t) = \mathcal{H}(\mathbf{q}, \mathbf{p}) = E\), trajectory-wise.

\[\begin{split}\left\{ \begin{aligned} \dot{\mathbf{w}} & = \nabla_{\mathbf{J}} \mathscr{H} \\ \dot{\mathbf{J}} & =-\nabla_{\mathbf{w}} \mathscr{H} = \mathbf{0} && \qquad \rightarrow \qquad \mathscr{H}(\mathbf{\overline{J}}, t) \\ \end{aligned} \right.\end{split}\]

Partial derivative of \(F_1(\mathbf{q}, \mathbf{Q}, t)\) w.r.t. time \(t\) is therefore zero

\[\partial_t F_1|_{\mathbf{q}, \mathbf{w}} = \mathscr{H} - \mathcal{H} = 0 \qquad \rightarrow \qquad F_1(\mathbf{q}, \mathbf{w}) \ .\]

and alos the hamiltonian \(\mathscr{H}\) can’t be a function of time, \(\mathcal{H}(\mathbf{q}, \mathbf{p}) = \mathscr{H}(\overline{\mathbf{J}}) = E(\overline{\mathbf{J}})\), trajectory-wise. Thus the gradient of the Hamiltonian \(\mathscr{H}\) w.r.t. \(\mathbf{J}\) is constant as well \(\nabla_{\mathbf{J}} \mathscr{H} =: \overline{\boldsymbol\omega}\), and the evolution of the generalized coordinate is linear (including periodic orbits, if the coordinates are cycical),

\[\mathbf{w}(t) = \mathbf{w}_0 + \overline{ \boldsymbol{\omega} } \, t \ .\]

Differentials of the generating functions. With this choice of conjugated variables, the differential of the type-1 generating function becomes

\[d F_1(\mathbf{q}, \mathbf{w}, t) = \mathbf{p} \cdot d \mathbf{q} - \overline{\mathbf{J}} \cdot d \mathbf{w} \ ,\]

and the integration along a closed trajectory \(\gamma\) in the space \((\mathbf{q}, \mathbf{w})\) gives, if the function \(F_1\) is regular in the region where integration occurs, so that \(\oint d F_1 = 0\).

\[\begin{split}\begin{aligned} \oint_{\gamma_k} d F_1 & = \oint_{\gamma_k} \mathbf{p} \cdot d \mathbf{q} - \oint_{\gamma_k} \overline{\mathbf{J}} \cdot \overline{\boldsymbol{\omega}} dt = \\ & = \oint_{\gamma_k} \mathbf{p} \cdot d \mathbf{q} - \overline{\mathbf{J}} \cdot \overline{\boldsymbol{\omega}} \, T_{k} = \\ & = \oint_{\gamma_k} \mathbf{p} \cdot d \mathbf{q} - \dfrac{1}{2\pi} \sum_{i} \overline{J}_i \, \frac{T_{k}}{T_i} \ , \end{aligned}\end{split}\]

with \(T_{\gamma}\) the period of the trajectory \(\gamma\). The differential of the type-2 generating function reads

\[\begin{split}\begin{aligned} d F_2 & = \mathbf{J} \cdot d \mathbf{w} + \mathbf{w} \cdot d \mathbf{J} + d F_1 = \\ & = \mathbf{p} \cdot d \mathbf{q} + \mathbf{w} \cdot d \mathbf{J} \end{aligned}\end{split}\]

and its integration over a closed trajectory in the \((\mathbf{q}, \mathbf{J})\) space, with \(\mathbf{J} = \overline{\mathbf{J}}\), \(\mathbf{w} = \mathbf{w}_0 + \overline{\boldsymbol\omega} \, t\), gives

\[\oint d F_2 = \oint d F_1 + \oint \overline{\mathbf{J}} \cdot \overline{\boldsymbol\omega} \, dt = \oint \mathbf{p} \cdot d \mathbf{q} \ . \]

Remark. todo Discussion about the regularity of \(F_2\) and the domain of integration. Integration of an exact differential over a closed path different from zero implies that something “strange” is happening.

todo

  • Separable systems

  • definition of action variables in separable systems

1

The link between simplectic structure and generating functions can be derived with a proper use of derivatives of composit functions.

2

Does this set of variables exist? For short-time? For long-time?