14.1 Hamiltonian Mechanics and Canonical Transformations

Hamiltonian mechanics offers a powerful approach to classical dynamics. It uses the Hamiltonian function to describe a system's total energy and introduces equations of motion that simplify problem-solving in many cases.

This formulation bridges classical and quantum mechanics. It introduces concepts like canonical transformations and the Hamilton-Jacobi equation, which are crucial for understanding more advanced topics in physics and mathematics.

Hamiltonian Mechanics

Hamiltonian function and equations

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• Hamiltonian function $H(q, p, t)$ represents the total energy of a system (kinetic + potential) and is defined as the Legendre transform of the Lagrangian $L(q, \dot{q}, t)$
  • Legendre transform: $H(q, p, t) = \sum_{i} p_i \dot{q}_i - L(q, \dot{q}, t)$
  • Generalized coordinates $(q_1, q_2, ..., q_n)$ describe the system's configuration
  • Generalized momenta $(p_1, p_2, ..., p_n)$ are defined as $p_i = \frac{\partial L}{\partial \dot{q}_i}$
• Hamilton's equations of motion are equivalent to Lagrange's equations but expressed in terms of generalized coordinates $q$ and momenta $p$
  • $\dot{q}_i = \frac{\partial H}{\partial p_i}$ relates the time derivative of the coordinate to the partial derivative of the Hamiltonian with respect to the corresponding momentum
  • $\dot{p}_i = -\frac{\partial H}{\partial q_i}$ relates the time derivative of the momentum to the negative partial derivative of the Hamiltonian with respect to the corresponding coordinate
• Advantages of the Hamiltonian formulation include the symmetry between coordinates and momenta, easier identification of conserved quantities (energy), and its suitability for quantum mechanics

Canonical transformations in dynamics

• Canonical transformations are changes of variables from $(q, p)$ to $(Q, P)$ that preserve the form of Hamilton's equations, allowing for the simplification of dynamical systems
  • Generated by a generating function $F$, which can be a function of $q$ and $Q$, $q$ and $P$, $p$ and $Q$, or $p$ and $P$, along with time $t$
  • Types of canonical transformations include point transformations ($Q_i = Q_i(q, t)$, $P_i = P_i(q, p, t)$) and extended point transformations ($Q_i = Q_i(q, p, t)$, $P_i = P_i(q, p, t)$)
• Generating functions and their relationships are described by the equation $p_i dq_i - H dt = P_i dQ_i - K dt + dF$, where $K(Q, P, t)$ is the transformed Hamiltonian
• Canonical transformations can simplify dynamical systems by eliminating time dependence in the Hamiltonian, reducing the number of degrees of freedom, or identifying cyclic coordinates (ignorable coordinates) that do not appear in the transformed Hamiltonian

Hamilton-Jacobi Theory

Hamilton-Jacobi equation applications

• The Hamilton-Jacobi equation, $\frac{\partial S}{\partial t} + H(q, \frac{\partial S}{\partial q}, t) = 0$, is a partial differential equation for the Hamilton's principal function (action) $S(q, t)$
  • To solve the equation, separate variables: $S(q, t) = W(q) - Et$, where $E$ is a constant (energy)
  • Solve the resulting time-independent Hamilton-Jacobi equation for $W(q)$, which satisfies $\frac{\partial W}{\partial q_i} = p_i$
• Applications of the Hamilton-Jacobi equation include finding equations of motion, determining trajectories in configuration space, identifying conserved quantities, and solving problems with separable Hamiltonians (additively separable into kinetic and potential energy terms)

Hamiltonian vs quantum mechanics

• The correspondence principle states that in the limit of large quantum numbers, quantum mechanics reduces to classical mechanics, and expectation values of quantum observables behave like classical variables
• Canonical quantization is the process of replacing classical variables $(q, p)$ with quantum operators $(\hat{q}, \hat{p})$, which satisfy the commutation relations $[\hat{q}_i, \hat{p}_j] = i\hbar \delta_{ij}$, where $\hbar$ is the reduced Planck's constant and $\delta_{ij}$ is the Kronecker delta
• The Schrödinger equation, $i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi$, is the quantum analog of the Hamilton-Jacobi equation, describing the time evolution of the wave function $\Psi$
• Ehrenfest's theorem relates the time evolution of expectation values of quantum observables to classical equations of motion:
  1. $\frac{d}{dt} \langle \hat{q}_i \rangle = \langle \frac{\partial \hat{H}}{\partial \hat{p}_i} \rangle$ for the expectation value of position
  2. $\frac{d}{dt} \langle \hat{p}_i \rangle = -\langle \frac{\partial \hat{H}}{\partial \hat{q}_i} \rangle$ for the expectation value of momentum