Unit 1 · Review & revision · Classical Mechanics

How the ideas fit together

A constraint limits possible configurations or velocities. For regular independent holonomic constraints, count the remaining configuration degrees of freedom and choose that many independent generalized coordinates. Use transformations to obtain particle positions. The chain rule then gives physical velocity; a fixed-time variation gives virtual displacement. Force projected onto that virtual displacement defines generalized force. Ideal reactions disappear from total virtual work. Static balance gives the virtual-work principle; Newtonian dynamical balance gives D’Alembert's principle.

This is one chain of reasoning. If the final equation feels arbitrary, locate the earliest step you cannot explain and revisit its chapter. Do not replace that gap with memorization.

Unit 1 formula sheet

Formula	Meaning and symbols	Conditions
$n=3N-k$	$N$ particles, $k$ independent constraints, $n$ configuration freedoms	Regular independent holonomic equality constraints in 3D
$\mathbf r_a=\mathbf r_a(q,t)$	Position of particle $a$ in independent coordinates $q_i$	Valid local transformation
$\mathbf v_a=\sum_i\mathbf r_{a,i}\dot q_i+\mathbf r_{a,t}$	Physical velocity, including explicit transport	Differentiable transformation
$\partial\mathbf v_a/\partial\dot q_i=\mathbf r_{a,i}$	Velocity coefficient equals coordinate response	Position depends on $q,t$ , not $\dot q$
$d\mathbf r_{a,i}/dt=\partial\mathbf v_a/\partial q_i$	Useful mixed-derivative identity	Smooth transformations; partial derivative holds velocities fixed
$d\mathbf r_a=\sum_i\mathbf r_{a,i}dq_i+\mathbf r_{a,t}dt$	Actual differential along motion	Chain rule
$\delta\mathbf r_a=\sum_i\mathbf r_{a,i}\delta q_i$	Virtual displacement	Fixed time; holonomic coordinate description
$\sum_a\nabla_a f\cdot\delta\mathbf r_a=0$	Linearized constraint on a virtual variation	Smooth equality constraint at fixed time
$\delta W=\sum_a\mathbf F_a\cdot\delta\mathbf r_a$	Virtual work of specified forces	Admissible infinitesimal comparison
$Q_i=\sum_a\mathbf F_a\cdot\mathbf r_{a,i}$	Generalized force conjugate to $q_i$	Defined through virtual work
$\sum_a\mathbf R_a\cdot\delta\mathbf r_a=0$	Definition of ideal reaction system	Must hold for every admissible virtual variation
$\sum_iQ_i\delta q_i=0$ , hence $Q_i=0$	Static equilibrium in generalized coordinates	Independent variations; ideal bilateral regular constraints; reaction availability
$Q_i=-\partial V/\partial q_i$	Conservative generalized force	Forces represented by potential $V$
$\sum_a(\mathbf F_a-m_a\mathbf a_a)\cdot\delta\mathbf r_a=0$	D’Alembert's principle	Constant masses, inertial frame, ideal constraints
$Q_i=\sum_am_a\mathbf a_a\cdot\mathbf r_{a,i}$	Independent-coordinate dynamical equation	Additional independence of generalized variations

Here $\mathbf r_{a,i}$ abbreviates $\partial\mathbf r_a/\partial q_i$ , and $\mathbf r_{a,t}$ abbreviates the explicit-time derivative. These abbreviations do not introduce new physical quantities.

Derivation index

Derivation	Location
Degrees-of-freedom counting and assumptions	Chapter 2, Section 2
Generalized velocity formula	Chapter 4, Section 2
Two generalized derivative identities	Chapter 4, Section 3
Actual and virtual constraint differentials	Chapter 4, Section 4
Generalized-force formula	Chapter 5, Section 2
Virtual-work principle	Chapter 5, Section 4
Conservative generalized force	Chapter 5, Section 5
D’Alembert's principle	Chapter 6, Section 2
Generalized-coordinate dynamical form	Chapter 6, Section 3
Atwood and pendulum equations	Chapter 6, Section 4

Definition index

Term	Precise working definition
Actual displacement	Infinitesimal position change along an actual trajectory over elapsed time
Admissible variation	Infinitesimal comparison consistent with the specified instantaneous constraints
Bilateral constraint	Equality restriction with locally two-sided allowed variations within its surface
Configuration	Complete specification of particle positions at an instant
Configuration space	Set of all admissible configurations; one point represents one complete arrangement
Constraint	Restriction on admissible positions, velocities, or both
Constraint force	Reaction associated with enforcing the modeled restriction
D’Alembert's principle	Zero total virtual work of applied forces minus momentum rates under ideal constraints
Degree of freedom	One independent parameter needed to specify configuration locally
Generalized coordinate	Independent parameter used to label admissible configurations
Generalized force	Coefficient of a generalized variation in virtual work
Generalized velocity	Time derivative of a generalized coordinate
Holonomic constraint	Restriction expressible or integrable into a position-and-time equality
Ideal constraint	Constraint whose reaction system has zero total virtual work for all admissible variations
Non-holonomic constraint	Restriction not reducible to an equivalent position-and-time equality; the main examples here are non-integrable velocity restrictions
Rheonomous constraint	Constraint with explicit time dependence in the chosen description
Scleronomous constraint	Constraint without explicit time dependence in the chosen description
Transformation equations	Relations giving physical particle positions from generalized coordinates and time
Unilateral constraint	One-sided restriction, commonly represented by an inequality
Virtual displacement	Infinitesimal admissible configuration change at fixed time
Virtual work	Sum of force dot virtual-displacement products
Virtual-work principle	Applied-force virtual work vanishes for every admissible variation in equilibrium under the stated ideal-constraint assumptions

A reliable problem-solving method

Specify the physical model. Fixed support or moving support? Smooth or rough? Rod or string? Prescribed motion or a dynamical unknown?
Count and choose. Identify independent constraints, count freedoms, and select coordinates.
Define conventions. Draw axes, positive directions, angles, and reference lengths.
Write transformations. Verify that the positions satisfy the constraints.
Construct virtual displacements. Hold time fixed and keep admissibility explicit.
Identify forces. Separate applied forces from reactions, then justify which reaction works vanish or cancel.
Choose equilibrium or dynamics. Use virtual work for static equilibrium and D’Alembert for motion.
Collect independent variations. Set their coefficients to zero only when independence is justified.
Interpret and check. Verify dimensions, signs, special cases, and tension/contact conditions.

Unit 1 problems

These problems connect constraint geometry, virtual work, and equations of motion. Attempt them before consulting the formula sheet. Separate hints and solutions follow the chapter solutions.

R1 · Translating circular hoop. A bead is threaded on a horizontal circular hoop of fixed radius $a$ , whose centre translates as $X(t)=Ut$ , $Y=0$ . Write the constraints in three dimensions, classify them, count freedoms, and give transformations.

R2 · Configuration degrees of freedom. Compare the freedoms of a free planar particle, a pendulum with prescribed changing length, a pendulum with dynamically variable length, and a planar double pendulum. Explain every count.

R3 · Ideal reactions and actual power. For the translating hoop of R1, derive physical velocity and virtual displacement. With a radial reaction of signed magnitude $N_r$ , show that its virtual work vanishes while its actual power need not.

R4 · Spring equilibrium on an incline. Prove the virtual-work principle from static force balance and define every assumption. Apply it to a mass on a smooth incline supported by an uphill spring of stiffness $k$ , using extension $q$ measured down the slope. Find equilibrium extension.

R5 · Dynamics of connected masses. Derive D’Alembert's principle and use it for an Atwood machine with masses $4\,\mathrm{kg}$ and $1\,\mathrm{kg}$ . Find acceleration and string tension with $g=9.8\,\mathrm{m\,s^{-2}}$ .

R6 · Coordinate and velocity identities. Derive the generalized velocity formula and the two associated derivative identities. Explain why $\partial/\partial\dot q_i$ holds coordinates fixed even though a real trajectory relates coordinate and velocity.

R7 · Motion on a helical wire. A bead slides on a smooth fixed helix $\mathbf r(\phi)=(R\cos\phi,R\sin\phi,b\phi)$ under downward gravity. Derive its equation for $\phi(t)$ using D’Alembert's principle. Interpret the case $b=0$ .

R8 · Constraint models and admissibility. Critique these claims: “Every velocity restriction is non-holonomic”; “ideal constraint forces never do work”; “zero virtual work for one variation proves equilibrium.” Give a counterexample or the missing condition for each.

Reviewing the physical argument

Begin by stating the definitions and the distinctions between them. Next reproduce the velocity and virtual-work derivations without looking. Then solve the pendulum and Atwood problems from a blank diagram. Finish by explaining the moving-wire counterexample and the independence requirement aloud.

If you only remember formulas, use the Physical Meaning paragraphs to rebuild the argument. If you understand the ideas but make errors, use the worked examples to practise signs and derivatives. If you can solve familiar systems, try R7 and Problem 6.16 to test transfer to a new constraint geometry.

From constrained motion to Lagrangian mechanics

You should now be able to explain and calculate each of these without guessing: independent coordinates; explicit time dependence; actual versus virtual displacement; generalized force and its units; ideal-reaction elimination; static versus dynamical balance; and the limitations of independent-variation arguments.

Unit 2 organizes the same mechanics through kinetic and potential energy. The full derivation of Lagrange’s equations is optional under the syllabus; their meaning and applications are developed in the main chapters.