Physics Basics – Advanced Study Notes
Program: Nagaland Patwari / Lekhpal
Subject: General Awareness & Current Affairs
Topic: Physics Basics (MIT‑Level, First Principles)
These notes are meant for self‑study and exam preparation.
They assume a solid grounding in high‑school mathematics (limits, vectors, differential equations) and aim to expose students to the rigorous, first‑principles viewpoint that underpins modern physics.
Table of Contents
- Classical Mechanics
1.1. Newtonian Framework
1.2. Lagrange’s Principle
1.3. Hamiltonian Dynamics
1.4. Constraints & Generalized Coordinates - Electromagnetism
2.1. Maxwell’s Equations (Differential & Integral Form)
2.2. Electromagnetic Potentials & Gauge Freedom
2.3. Energy–Momentum Tensor
2.4. Edge Cases: Causality & Superluminal Propagation - Thermodynamics & Statistical Mechanics
3.1. Laws of Thermodynamics (Microscopic View)
3.2. Entropy & Phase Space Volume
3.3. Canonical & Microcanonical Ensembles
3.4. Edge Cases: Negative Temperatures & Non‑Extensive Systems - Quantum Mechanics – Foundations
4.1. Wavefunction & Born Rule
4.2. Schrödinger Equation & Operators
4.3. Uncertainty Principle & Commutation Relations
4.4. Edge Cases: Non‑Hermitian Operators & PT Symmetry - Special Relativity
5.1. Lorentz Transformations & Invariants
5.2. Relativistic Dynamics
5.3. 4‑Vectors & Minkowski Space
5.4. Edge Cases: Tachyons & Causality Violations - Problem Set
6.1. Challenging Problem (with Hint) - Further Reading & References
1. Classical Mechanics
1.1 Newtonian Framework
| Quantity | Symbol | Units | Notes |
|---|---|---|---|
| Position | (\mathbf{r}) | m | Vector in Euclidean space |
| Velocity | (\mathbf{v} = \dot{\mathbf{r}}) | m s(^{-1}) | Time derivative |
| Acceleration | (\mathbf{a} = \ddot{\mathbf{r}}) | m s(^{-2}) | |
| Force | (\mathbf{F}) | N | Sum of all external forces |
| Mass | (m) | kg | Invariant in Newtonian mechanics |
Newton’s Second Law (in differential form):
[
\boxed{\mathbf{F} = m \mathbf{a} = m \ddot{\mathbf{r}}}
]
Rigorous derivation: Starting from the principle of least action (see §1.2) ensures that this form holds in all inertial frames.
1.2 Lagrange’s Principle
Principle of Least Action:
[
\delta \int_{t_1}^{t_2} L(\mathbf{r},\dot{\mathbf{r}},t),dt = 0
]
where (L = T - V) is the Lagrangian, (T) kinetic energy, (V) potential energy.
Euler–Lagrange Equation: [ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i}=0 ] for generalized coordinate (q_i).
Proof Sketch: Variation of the action (\delta S = 0) with fixed endpoints leads to the above differential equation.
Edge Case – Dissipative Forces:
For forces depending on velocity (e.g., drag), the Lagrangian is not strictly (T - V). One introduces a Rayleigh dissipation function or uses non‑conservative generalized forces (Q_i): [ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)-\frac{\partial L}{\partial q_i}=Q_i ]
1.3 Hamiltonian Dynamics
Define canonical momentum: [ p_i = \frac{\partial L}{\partial \dot{q}_i} ] Hamiltonian: [ H(q,p,t) = \sum_i p_i \dot{q}_i - L(q,\dot{q},t) ] Hamilton’s equations: [ \dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i} ]
Hamiltonian as Energy: For time‑independent (L), (H) equals total mechanical energy (E = T + V).
Edge Case – Canonical Transformations:
Any transformation preserving the form of Hamilton’s equations (i.e., symplectic transformation) is canonical. This underlies the powerful method of action–angle variables.
1.4 Constraints & Generalized Coordinates
- Holonomic Constraints: (f(q_1,\dots,q_n,t)=0).
- Non‑holonomic Constraints: Linear in velocities (a_i(q,t)\dot{q}_i + b(q,t)=0).
For holonomic constraints, reduce degrees of freedom; for non‑holonomic, introduce Lagrange multip