01 May 2026

Classical Mechanics Syllabus Map

A compact map of the SEM-I classical mechanics syllabus from Lagrangian mechanics to small oscillations.

msc semester-i classical-mechanics concept-map

The classical mechanics part of the SEM-I syllabus should be read as a connected development, not as separate formulas. It begins with constrained motion, moves to variational principles, then to phase space, and finally to symmetry and small oscillations.

Flow of ideas

1. D’Alembert principle removes ideal constraint forces through virtual work.
2. Lagrange equations describe dynamics using generalized coordinates.
3. Hamilton principle derives Lagrange equations from stationary action.
4. Conservation theorems connect cyclic coordinates and symmetries with conserved quantities.
5. Hamilton equations move the theory into phase space $(q,p)$.
6. Hamilton-Jacobi equation converts dynamics into a first-order PDE for $S$.
7. Canonical transformations preserve Hamilton’s form and are generated by functions.
8. Infinitesimal generators produce translations, rotations, and time evolution.
9. Poisson brackets encode time evolution and canonical structure.
10. Small oscillations linearize motion near equilibrium and lead to normal modes.

The logical thread is that each new formulation keeps the same physics but gives a more powerful language for solving problems.

Core formulas

D’Alembert principle:

\[\sum_i(\mathbf F_i-m_i\ddot{\mathbf r}_i)\cdot\delta\mathbf r_i=0.\]

Lagrange equation:

\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_i}\right) -\frac{\partial L}{\partial q_i}=0.\]

Hamilton equations:

\[\dot q_i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q_i}.\]

Hamilton-Jacobi equation:

\[H\left(q_i,\frac{\partial S}{\partial q_i},t\right) +\frac{\partial S}{\partial t}=0.\]

Poisson evolution:

\[\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}.\]

Normal-mode equation:

\[(K-\omega^2M)\mathbf a=0.\]

Revision strategy

Read the notes in this order:

1. D’Alembert principle and Lagrange equations
2. Hamilton principle and calculus of variations
3. Least action and Hamilton equations
4. Hamilton-Jacobi equation
5. Canonical transformations and generating functions
6. Poisson brackets
7. Poisson theorems and angular momentum
8. Conservation theorems and infinitesimal generators
9. Small oscillations and normal modes

Standard derivations

For a long-answer question, the following derivations should be prepared in full:

1. Lagrange’s equation from D’Alembert’s principle.
2. Lagrange’s equation from Hamilton’s principle.
3. Hamilton’s equations from the Legendre transform.
4. Hamilton-Jacobi equation from Hamilton’s principal function.
5. Poisson-bracket form of Hamilton’s equations.
6. Normal-mode equation for small oscillations.

Answer sequence

A good mechanics answer usually has this order:

1. state the principle or definition;
2. write the mathematical form;
3. identify the coordinates and constraints;
4. derive the required equation;
5. interpret the conserved quantity, frequency, or physical result.

Main points

• The syllabus moves from constraints to variational methods, then to phase space.
• Symmetry appears first through cyclic coordinates and later through generators.
• Poisson brackets provide the shortest notation for Hamiltonian dynamics.
• Small oscillations turn nonlinear mechanics into an eigenvalue problem.

Practice questions

1. Arrange the main topics of SEM-I classical mechanics in logical order.
2. List three conservation laws and the symmetries associated with them.
3. Write the Hamilton-Jacobi equation and explain the meaning of $S$.
4. State the Poisson-bracket form of Hamilton’s equations.
5. Why do small oscillations lead to an eigenvalue problem?