Unit 3 · Review · Classical Mechanics

States, momenta, and evolution

Hamiltonian mechanics replaces generalized velocities by their conjugate momenta. The replacement is locally valid when the velocity Hessian is nonsingular. The Hamiltonian then generates paired first-order equations in a $2n$ -dimensional phase space.

p_i=L_{\dot q_i},\qquad H(q,p,t)=\sum_ip_i\dot q_i-L,

(P.1)

\dot q_i=H_{p_i},\qquad\dot p_i=-H_{q_i}+Q_i,\qquad \dot H=H_t+\sum_iQ_i\dot q_i.

(P.2)

In the definition of $H$ , all dynamical velocities must be eliminated. In its derivatives, the other canonical coordinates and momenta are held fixed. In the force term, any velocities must also be expressed in canonical variables.

Construction and interpretation

Question	Required reasoning	Chapter
Can the transformation be made?	Invert the momentum equations; check regularity	12
What is a mechanical state?	Configuration plus momenta; dimensions and initial data	13
What does a phase portrait show?	Energy contours, direction, equilibria, turning points, separatrices	13
Does $H$ equal mechanical energy?	Check stationary coordinates and velocity-independent potential	14
Is $H$ conserved?	Evaluate $H_t$ and remaining-force power	14
How is a cyclic coordinate used?	Conserve its momentum if unforced, then reduce at fixed momentum	15

The complete derivation of Hamilton's equations is optional. Constructing a Hamiltonian, stating and applying its equations, and explaining its physical meaning are core syllabus material.

Comparisons to practise

For the pendulum, derive the same angular equation from a tangent force balance, a Lagrangian, and a Hamiltonian. For a polar particle, show why the derivative of $H$ with respect to $r$ must hold angular momentum fixed. For a damped spring, draw the distinction between the conservative energy ellipses and the inward dissipative trajectories. For a rotating wire, calculate both the conserved reduced $H$ and laboratory energy.

The chapter exercises and solutions develop each calculation separately. The comparison table collects the assumptions and variables of the three formulations.

Interpreting a completed calculation

A Hamiltonian is not a universal synonym for energy, and an autonomous expression is not enough for conservation when remaining forces act. Phase space records the state of the entire system, not another physical orbit. Once these distinctions are clear, the method becomes a direct way to exploit cyclic momenta and reduce central-force motion.