Consider a system with one degree of freedom and Hamiltonian \(h(q, p, t) .\) Show that the dynamical trajectories in \(P T=\mathbb{R}^{3}\) are tangent to the vector field $$ \boldsymbol{x}=\frac{\partial h}{\partial p} i-\frac{\partial h}{\partial q} j+\boldsymbol{k} $$ where \(\boldsymbol{i}, \boldsymbol{j}\), and \(\boldsymbol{k}\) are unit vectors along the \(q, p\), and \(t\) axes. Let \(\Sigma\) be a surface in \(\mathbb{R}^{3}\) given by $$ p=\frac{\partial S}{\partial q} $$ where \(S=S(q, t)\). Show that if \(S\) is a solution of the HamiltonJacobi equation, then \(\boldsymbol{x}\) is tangent to \(\Sigma\). Show conversely that if \(\boldsymbol{x}\) is tangent to \(\Sigma\), then $$ \frac{\partial}{\partial q}\left[\frac{\partial S}{\partial t}+h\left(q, \frac{\partial S}{\partial q}, t\right)\right]=0 $$

Understanding Hamiltonian mechanics is crucial for diving deep into advanced topics in classical mechanics and quantum mechanics. It provides a powerful framework for modeling the dynamic behavior of physical systems.

To grasp the core of Hamiltonian mechanics, imagine the entire state of a system being described by its position and momentum, typically denoted as 'q' for generalized coordinates and 'p' for the conjugate momenta. The Hamiltonian function, denoted by 'H' or 'h', encapsulates the total energy of the system, which is often the sum of kinetic and potential energies.

Hamilton's Equations of Motion

In essence, Hamilton's equations elegantly describe how a system evolves over time. For a system with one degree of freedom, the equations are given by:
\[\frac{dq}{dt} = \frac{\partial h}{\partial p},\qquad \frac{dp}{dt} = -\frac{\partial h}{\partial q}\]
These equations are symmetrical and provide the time evolution of both position and momentum, offering a complete picture of the system's dynamics.

Appreciating the symmetry and beauty of these equations can allow students to understand the evolution of systems not just in momentum or position separately, but as an intertwined dance of both variables, fully describing a physical system’s moment-by-moment existence.

The Hamilton-Jacobi equation is a reformation of classical mechanics that is as fundamental as Newton's laws and Hamilton's equations, but it provides a different perspective. It is closely related to the principle of least action and has implications for both classical and quantum mechanics.

The equation is given by:
\[\frac{\partial S}{\partial t} + h\left(q, \frac{\partial S}{\partial q}, t\right) = 0\]
where 'S' is the action, which is a function of coordinates and time. Solving the Hamilton-Jacobi equation for 'S' can sometimes be simpler than solving Hamilton's equations directly, particularly in systems with symmetries.

Connecting to Trajectories

In the context of the exercise, if 'S' is a solution to the Hamilton-Jacobi equation for a given Hamiltonian, then the vector field describing the dynamical trajectories of the system in phase-space is always tangent to a surface determined by the derivative of the action 'S'. This connection is profound because it links the concept of action, fundamental to the principle of least action, to the actual paths the system can take throughout its evolution.

Dynamical trajectories represent the pathways through phase space that a system traverses over time as it evolves according to the prescribed laws of motion. In Hamiltonian mechanics, these trajectories can be visualized as curves in a coordinate system made up of generalized coordinates and momenta.

By understanding the tangent vector field described by:
\[\boldsymbol{x}=\frac{\partial h}{\partial p} \boldsymbol{i}-\frac{\partial h}{\partial q} \boldsymbol{j}+\boldsymbol{k}\]
students learn that this field represents the velocity at which the system moves through its phase space. Notably, the 'i' and 'j' components are governed by Hamilton's equations of motion.

The Tangency Condition

In the exercise, if the vector field 'x' is tangent to a surface 'Σ', this tangency condition implies certain constraints on the system's dynamics. Specifically, if the system's action 'S' fulfills the Hamilton-Jacobi equation, the trajectories—those curves in phase space—conform precisely to the evolution described by 'x' being tangent to \'Σ'. Conversely, through mathematical demonstration, the condition of tangency can illustrate that 'S' indeed solves the Hamilton-Jacobi equation.

For students, imagining dynamical trajectories as actual paths a particle might follow is a valuable way to link the abstract mathematical expressions to the physical movements we observe in nature, such as the orbit of planets or the swing of a pendulum.