Unit 2 · Review · Classical Mechanics

Lagrangian mechanics in one argument

Unit 1 incorporated geometry into independent coordinates and projected away ideal reactions. The Lagrangian method organizes those projected equations through $T$ and $V$ . Generalized momentum then connects cyclic coordinates to conservation laws, while the energy function handles time dependence and external work.

The principal equations are

L=T-V,\qquad p_i=\frac{\partial L}{\partial\dot q_i},\qquad \frac d{dt}\frac{\partial L}{\partial\dot q_i}-\frac{\partial L}{\partial q_i}=Q_i,

(P.1)

Q_i=\sum_a\mathbf F_a^{\rm rem}\cdot\frac{\partial\mathbf r_a}{\partial q_i},\qquad \mathcal E=\sum_i\dot q_ip_i-L,\qquad \dot{\mathcal E}=\sum_iQ_i\dot q_i-L_t.

(P.2)

For a cyclic coordinate with $Q_i=0$ , $p_i$ is constant. For autonomous unforced motion, $\mathcal E$ is constant. Only after checking the kinetic-energy structure should it be identified with $T+V$ .

Definitions and proofs

Subject	What a complete explanation contains	Chapter
Lagrangian	Energies, coordinate dependence, meaning of the minus sign	7
Remaining generalized force	Virtual-work projection and no double counting	8
Cyclic coordinate	Absence of $q_i$ , possible presence of $\dot q_i$ , force qualification	9
Symmetry and momentum	Translation and rotation arguments with identified conserved quantities	9
Energy theorem	Product rule, cancellation, explicit-time and force terms	9
Kinetic-energy function	$T_2+T_1+T_0$ , symmetric inertia matrix, Euler identity	10
Moving and coupled systems	Laboratory velocities, mixed terms, chosen approximations	11

The full derivation of Lagrange's equation is optional under the syllabus. The conservation and kinetic-energy proofs in this table are not made optional by that exception.

Written problems and numerical checks

Reconstruct the pendulum and Atwood equations without looking at their worked examples. Then derive the moving-pivot equation and explain why support acceleration survives while constant support velocity does not. For a damping problem, calculate power as well as acceleration. For coupled motion, test whether the proposed coordinates change more than one particle's position.

The exercises in Chapters 7–11 include definitions, direct calculations, proof questions, extended applications, and model limitations. Their solutions are separate. The course problem bank includes problems that connect this unit with Hamiltonian and orbital methods.

Checks before leaving the unit

A coordinate has been chosen successfully when its transformation satisfies the constraints and distinguishes nearby configurations. An equation has been obtained correctly when its partial and total derivatives have been taken with the right variables fixed. A conservation statement is complete when its force and time-dependence assumptions are visible. These three checks prevent most errors in applications.