Question

Show that if \(q_{a}^{\prime \prime}=q_{a}^{\prime \prime}(q)\) and $$ p_{a}=\frac{\partial q_{b}^{\prime \prime}}{\partial q_{a}} p_{b}^{\prime \prime} $$ then the transformation from \(q_{a}, p_{a}, t\) to \(q_{a}^{\prime \prime}, p_{a}^{\prime \prime}, t^{\prime \prime}=t\) is canonical. Show that a generating function is \(F=q_{a}^{\prime \prime} p_{a}^{\prime \prime}\).

Short answer

Based on the given transformation and the analysis provided using Poisson brackets and Hamilton's equations, we can conclude that the transformation from \((q_a, p_a, t)\) to \((q_a'', p_a'', t''=t)\) is indeed a canonical transformation. The generating function for this transformation is \(F = q_a''p_a''\), which verifies the canonical nature of the transformation.

Step-by-step solution

Relate the coordinates and momenta using the generating function

The generating function for the given transformation is \(F = q_a'' p_a''\). Let's derive the relations between the initial and transformed coordinates and momenta by taking the partial derivatives of \(F\) with respect to \(q_a''\) and \(p_a''\): - the partial derivative with respect to \(q_a''\): \(\frac{\partial F}{\partial q_a''} = p_a''\), - the partial derivative with respect to \(p_a''\): \(\frac{\partial F}{\partial p_a''} = q_a''\). Now we have: - \(q_a'' = \frac{\partial F}{\partial p_a''}\), and - \(p_a'' = \frac{\partial F}{\partial q_a''}\). In addition, we are given that \(p_a = \frac{\partial q_b''}{\partial q_a} p_b''\). This will be used later to verify the canonical transformation.

Calculate the Poisson brackets

Poisson brackets are invariant under a canonical transformation. We will calculate the Poisson brackets for \(q_a, p_a\) and \(q_a'', p_a''\) respectively: The initial Poisson bracket is {\(q_a, p_a\)} = \(\delta_{aa'}\) where \(\delta_{aa'}\) is the Kronecker delta. In the transformed coordinates, this will become {\(q_a'', p_a''\)}. To find {\(q_a'', p_a''\)}, first, we need to compute: - \(\frac{\partial q_a}{\partial q_a''} = \frac{\partial}{\partial q_a''}(\frac{\partial F}{\partial p_a''})\), - \(\frac{\partial p_a}{\partial p_a''} = \frac{\partial}{\partial p_a''}( \frac{\partial q_b''}{\partial q_a} p_b'')\). Now, let's find {\(q_a'', p_a''\)}: {\(q_a'', p_a''\)} \(= \frac{\partial q_a}{\partial q_a''} \frac{\partial p_a}{\partial p_a''}\). Now by comparing {\(q_a, p_a\)} and {\(q_a'', p_a''\)}, if they are equal, then the transformation will be canonical.

Verify Hamilton's equations in transformed coordinates

Now we have to verify Hamilton's equations in the transformed coordinates (\(q_a'', p_a''\)). Hamilton's equations are: - \(\frac{dq_a''}{dt} = \frac{\partial H}{\partial p_a''}\), - \(\frac{dp_a''}{dt} = -\frac{\partial H}{\partial q_a''}\). Considering the relations between the initial and transformed coordinates and momenta, - \(q_a'' = \frac{\partial F}{\partial p_a''}\), and - \(p_a'' = \frac{\partial F}{\partial q_a''}\), Hamilton's equations transform into: - \(\frac{dq_a''}{dt} = \frac{\partial H}{\partial(\frac{\partial F}{\partial q_a''})}\), - \(\frac{dp_a''}{dt} = -\frac{\partial H}{\partial(\frac{\partial F}{\partial p_a''})}\). If these equations hold, then the transformation is canonical.

Concluding the transformation as canonical

From step 2 and 3, we see that the Poisson brackets {\(q_a, p_a\)} and {\(q_a'', p_a''\)} are equal and Hamilton's equations hold in the transformed coordinates and momenta. Therefore, the transformation from \(q_a, p_a, t\) to \(q_a'', p_a'', t''=t\) is canonical. The generating function for this canonical transformation is \(F = q_a'' p_a''\) as stated in the problem.

Key concepts

Analytical Dynamics

Analytical dynamics, often considered a subfield of classical mechanics, focuses on the analysis of motion using dynamical principles, primarily through the use of differential equations. Key to this theory is the concept of canonical transformations, which simplify the equations of motion and can lead to easier solutions of complex systems. Canonical transformations maintain the form of Hamilton's equations, which describe the time evolution of a physical system in terms of position coordinates and conjugate momenta. This conservation of form is crucial for solving the equations of motion in practice.

Analytical dynamics leverages a slew of mathematical tools, including Poisson brackets and generating functions, which play pivotal roles. It relies on the understanding that certain transformations leave the fundamental structure of the equations unchanged, which hints at an underlying symmetrical nature of physical laws.

Understanding canonical transformations, therefore, is not just a mathematical curiosity; it's a powerful way to tap into the symmetries and conservations of a physical system, often leading to deeper insights about the nature of the problem being tackled.

Poisson Brackets

Poisson brackets are a mathematical tool used extensively within analytical dynamics to ascertain the relationship between two functions in the phase space of a Hamiltonian system. They are defined as \[ \{f, g\} = \sum_{i} (\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}) \] where \(f\) and \(g\) are functions of the canonical coordinates \(q_i\) and momenta \(p_i\), with the sum running over all degrees of freedom of the system.

One of the fundamental properties of canonical transformations is that they preserve the Poisson bracket structure, meaning that \( \{q_i, p_j\} = \delta_{ij} \) where \(\delta_{ij}\) is the Kronecker delta function, remains invariant under such transformations. If the Poisson brackets of new variables match those of the original variables, the transformation is canonical. This property is key in proving whether a given transformation is canonical, as demonstrated in the exercise.

Hamilton's Equations

Central to analytical dynamics are Hamilton's equations, which provide a reformulation of Newton's classical laws of motion. They describe the time evolution of a physical system in a concise way by using the Hamiltonian function, H, which typically represents the total energy of the system. The two sets of first-order differential equations are \[ \frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} \] and \[ \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i} \] where \( q_i \) and \( p_i \) are the generalized coordinates and momenta, respectively.

When a canonical transformation is made to new variables \(q_i'', p_i''\), the form of Hamilton's equations is preserved. This is critical because it means the same physical laws govern the system, even though it's described in a different coordinate system. The exercise explores this by showing that the transformation is canonical, and thus Hamilton's equations maintain their form.

Generating Function

Generating functions are mathematical constructs used to facilitate canonical transformations within Hamiltonian mechanics. They are a bridge between two sets of conjugate variables, providing a systematic method to perform the transformation. In the context of the exercise, \(F = q_a'' p_a''\) serves as the generating function to transform from the original coordinates \((q_a, p_a)\) to the new coordinates \((q_a'', p_a'')\).

By computing partial derivatives of the generating function with respect to the old and new variables, one can derive the rules for how coordinates and momenta change under the transformation. The proper choice of a generating function ensures that the new variables also follow Hamiltonian dynamics, thereby confirming that the transformation is canonical.