Final revision

How to use this revision

Use this page after the chapters. It is a guide to reconstructing the arguments, not a replacement for the explanations and worked problems. The formula sheet includes symbol meanings and conditions; the derivation index links the complete proofs.

Coordinates and constraints

Describe the physical system before counting coordinates. Check which restrictions are independent and whether they are equalities, inequalities, or non-integrable velocity conditions. Write the transformations and test a simple configuration such as zero angle. Differentiate them once to obtain velocity, retaining support transport when present.

A virtual displacement freezes time. The omitted transport term is the difference from actual displacement for a moving constraint. Ideal reactions have zero total virtual work; they need not have zero actual work. Static equilibrium gives the virtual-work principle; subtracting momentum rate gives D’Alembert's principle.

For a derivation, be able to pass from particle force balance to the sum over independent generalized variations. For a calculation, be able to identify what a reaction does and why it disappears or remains.

Lagrangian mechanics

Choose coordinates, calculate $T$, choose a reference for $V$, form $L=T-V$, and compute the velocity partial derivative before taking its total time derivative. Any remaining force is obtained from virtual work and belongs on the right-hand side. Damping must oppose motion; its power should have the correct sign.

A cyclic coordinate has $L_{q_i}=0$. Its momentum $p_i=L_{\dot q_i}$ is conserved if $Q_i=0$. For time dependence, begin with $\mathcal E=\sum_i\dot q_ip_i-L$ and prove $\dot{\mathcal E}=\sum_iQ_i\dot q_i-L_t$. Identify $\mathcal E=T+V$ only under the stated kinetic and potential conditions.

Reproduce the kinetic expansion $T=T_2+T_1+T_0$, the symmetric coefficient matrix, and $\sum_i\dot q_iT_{\dot q_i}=2T_2+T_1$. Those properties explain both generalized inertia and the Hamiltonian energy distinction.

Hamiltonian mechanics

Calculate every $p_i$, invert the equations together, and substitute into $H=\sum_ip_i\dot q_i-L$. Check that no dynamical velocity remains. Then use $\dot q_i=H_{p_i}$ and $\dot p_i=-H_{q_i}+Q_i$ with all force terms expressed in the same variables.

A phase point represents a complete state. A turning point is not necessarily an equilibrium. An autonomous conservative oscillator follows an energy ellipse; damping makes it cross toward lower-energy curves. A pendulum separatrix separates oscillation from rotation.

Keep these two questions separate: does $H$ equal physical energy, and does it remain constant? A damped spring answers the first yes and second no. A rotating-wire reduction can answer the first no and second yes.

The full derivations of Lagrange's and Hamilton's equations are optional under the syllabus. Their construction and use, conservation proofs, and the conditions just listed remain part of the course.

Central-force mechanics

Start the property proofs from $\dot{\mathbf L}=\dot{\mathbf r}\times\mathbf p+\mathbf r\times\mathbf F$. Explain both zero terms. Use $\mathbf r\cdot\mathbf L=0$ for planarity, separating the radial $\ell=0$ case. Use the triangle area for $\dot S=\ell/(2\mu)$. Use the work–energy dot product for energy conservation under a time-independent radial force.

For two bodies, define center and relative coordinates and show the kinetic cross terms cancel. The relative inertial coefficient is $\mu$, and gravitational $k/\mu=G(m_1+m_2)$.

At fixed $\ell$, study $V_{\rm eff}=V+\ell^2/(2\mu r^2)$. Allowed radii satisfy $E\geq V_{\rm eff}$; turning radii solve equality; circular orbits also require zero slope. The sign of curvature decides linear radial stability, except that zero curvature is inconclusive.

Derive Binet using $u=1/r$, $\dot\theta=\ell u^2/\mu$, and $\dot r=-\ell u'/\mu$. Keep the signed-force convention consistent. State Bertrand's two potentials and the “every bounded orbit” condition; do not call a near-circular argument its complete proof.

Inverse-square motion and Kepler laws

Solve the constant-forced Binet equation, identify $p=\ell^2/(\mu k)$, and relate $e$ to energy. For an ellipse, $r_p=a(1-e)$, $r_a=a(1+e)$, $p=a(1-e^2)$, and $E=-k/(2a)$. Those relations let you move between initial data, geometry, and constants.

The first Kepler law follows from the conic geometry and negative energy. The second is the central-force area law. The third follows by dividing $\pi ab$ by $\ell/(2\mu)$ and eliminating $b$ and $\ell$. The semimajor axis in the two-body formula is the relative one.

For Runge–Lenz, define $\mathbf A=\mathbf p\times\mathbf L-\mu k\hat{\mathbf r}$, differentiate the radial unit vector, and show the force term cancels its derivative. Dotting with $\mathbf r$ produces the conic equation; squaring produces the energy–eccentricity relation. Explain why its direction is undefined for a circle.

Written work and error checks

A useful final rehearsal is to choose a mixed problem from the problem bank, solve it without consulting formulas, then explain the geometry, assumptions, and limiting cases in ordinary words. The separate solutions allow the mathematical result to be checked afterward.