Classical mechanics Study Guide

📖 Core Concepts Classical mechanics: Study of forces and motion for macroscopic objects where quantum and relativistic effects are negligible. Statically vs. dynamically: Statics = equilibrium (no acceleration); Dynamics = motion with forces. Kinematics vs. dynamics: Kinematics describes motion (position, velocity, acceleration) without reference to forces; Dynamics adds the forces that cause the motion. Inertial frame: Reference frame where a body with zero net force moves at constant velocity; Newton’s second law $\mathbf{F}=m\mathbf{a}$ holds. Principle of least action: The true path of a system makes the action $S=\int L\,dt$ stationary (δS=0). Lagrangian ($L$): $L = T - V$ (kinetic minus potential energy). Hamiltonian ($H$): Legendre transform of $L$, often equals total energy $E = T+V$. Generalized coordinates ($qi$) & momenta ($pi$): Coordinates that may be angles, distances, etc.; momenta $pi = \partial L/\partial\dot qi$. Conservative force: Work depends only on start/end points; can be written $\mathbf{F} = -\nabla U$. Noether’s theorem: Every continuous symmetry of the Lagrangian yields a conserved quantity (e.g., time‑translation → energy). --- 📌 Must Remember Newton’s 2nd law (general): $\mathbf{F}= \dfrac{d\mathbf{p}}{dt}$, with $\mathbf{p}=m\mathbf{v}$ for constant $m$. Work‑energy theorem: $W{\text{total}} = \Delta K$, where $K=\tfrac12 mv^{2}$. Mechanical‑energy conservation (only conservative forces): $K+U = \text{const}$. Lagrange’s equations: $$\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot qi}\right)-\frac{\partial L}{\partial qi}=0$$ Hamilton’s equations: $$\dot qi = \frac{\partial H}{\partial pi},\qquad \dot pi = -\frac{\partial H}{\partial qi}$$ Galilean velocity addition: $v' = v - u$ (for frames moving with relative velocity $u$ along the $x$‑axis). Relativistic momentum: $\mathbf{p}= \gamma m \mathbf{v}$, $\displaystyle \gamma = \frac{1}{\sqrt{1-v^{2}/c^{2}}}$; reduces to Newtonian form when $v\ll c$. Center‑of‑mass motion: $\mathbf{R}{\text{CM}}$ follows Newton’s second law for the total mass as if all external forces act at the CM. --- 🔄 Key Processes Deriving equations of motion with Lagrange’s method Choose generalized coordinates $qi$. Write kinetic $T$ and potential $V$ energies → form $L=T-V$. Compute $\partial L/\partial qi$ and $\partial L/\partial \dot qi$. Apply Lagrange’s equation to obtain differential equations for each $qi$. Switching from Lagrangian to Hamiltonian Compute generalized momenta $pi = \partial L/\partial \dot qi$. Perform Legendre transform: $H = \sumi pi\dot qi - L$. Write Hamilton’s equations for $\dot qi$ and $\dot pi$. Applying the work‑energy theorem Identify all forces; split into conservative and non‑conservative. Compute work of conservative forces via $U$: $W{\text{cons}} = -\Delta U$. Compute work of non‑conservative forces directly (e.g., friction $Wf = \int \mathbf{F}f\cdot d\mathbf r$). Use $W{\text{total}} = \Delta K$ to solve for unknown speeds or distances. Transforming coordinates between inertial frames (Galilean) Position: $x' = x - u t$, $y' = y$, $z' = z$. Velocity: $v' = v - u$. Acceleration unchanged: $a' = a$. --- 🔍 Key Comparisons Statically vs. Dynamically Statically: $\sum \mathbf{F}=0$, $\sum \boldsymbol{\tau}=0$ → no acceleration. Dynamically: $\sum \mathbf{F}=m\mathbf{a}$ → acceleration present. Kinematics vs. Dynamics Kinematics: deals only with $\mathbf{r}(t)$, $\mathbf{v}(t)$, $\mathbf{a}(t)$. Dynamics: adds forces $\mathbf{F}$ and uses Newton’s laws. Newtonian vs. Relativistic Momentum Newtonian: $\mathbf{p}=m\mathbf{v}$. Relativistic: $\mathbf{p}= \gamma m\mathbf{v}$, $\gamma>1$ when $v$ approaches $c$. Conservative vs. Non‑conservative Forces Conservative: path‑independent work, $ \mathbf{F}= -\nabla U$. Non‑conservative: work depends on path (e.g., kinetic friction $\mathbf{F}f = -\lambda\mathbf{v}$). Lagrangian vs. Hamiltonian Formulation Lagrangian: uses coordinates & velocities; $L=T-V$. Hamiltonian: uses coordinates & momenta; $H=\sum pi\dot qi - L$, often $H=E$. --- ⚠️ Common Misunderstandings “Least” vs. “Stationary” action: The action need only be stationary (δS=0); the path may be a minimum, maximum, or saddle point. Friction as a “force”: It is velocity‑dependent; using $Ff = -\lambda v$ leads to an exponential decay of speed, not a constant deceleration. Inertial frames and fictitious forces: In a non‑inertial frame you must add fictitious forces (Coriolis, centrifugal) to recover Newton’s second law; they are not real interactions. Energy conservation with non‑conservative forces: Mechanical energy is not conserved when friction or air resistance does work; total energy (including thermal) is conserved. Legendre transform misuse: Forgetting to express $\dot qi$ in terms of $pi$ before forming $H$ yields an incorrect Hamiltonian. --- 🧠 Mental Models / Intuition “Rubber sheet” for potentials: Visualize $U(\mathbf{r})$ as a landscape; a particle “rolls downhill” under conservative forces (steepest‑descent direction = $-\nabla U$). Generalized coordinates as “independent sliders”: Each $qi$ isolates a single degree of freedom, making the math of constraints trivial. Action as “economy of motion”: The true path uses the “least” (stationary) amount of “effort” measured by $L$ over time. Phase‑space flow: In Hamiltonian mechanics, think of the system as a fluid moving through $(q,p)$ space, preserving volume (Liouville’s theorem). --- 🚩 Exceptions & Edge Cases High‑speed (relativistic) regime: Replace $m\mathbf{a}$ with $\dfrac{d}{dt}(\gamma m\mathbf{v})$; Newton’s second law no longer holds in its simple form. Very massive bodies: General relativity required; Newtonian gravity gives inaccurate predictions (e.g., Mercury’s perihelion precession). Chaotic systems: Even with deterministic equations, tiny uncertainties in initial conditions make long‑term predictions unreliable. Non‑central forces: Lorentz force $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ violates Newton’s third law in its strong form; only total momentum is conserved. --- 📍 When to Use Which Newtonian/Force‑based approach → simple point‑mass problems, clear force diagrams, constant mass, non‑relativistic speeds. Energy‑based (work‑energy theorem) → problems where forces are hard to resolve but start/end speeds are asked; especially with conservative forces. Lagrangian → systems with constraints, non‑Cartesian coordinates, or where forces are derived from potentials (e.g., pendulum, double pendulum). Hamiltonian → when phase‑space analysis, canonical transformations, or connection to quantum mechanics is needed; also convenient for time‑independent total energy. Galilean transformation → switching between inertial frames moving at low speeds relative to each other. Relativistic formulas → any situation where $v \gtrsim 0.1c$ or precision requires $O(v^2/c^2)$ corrections. --- 👀 Patterns to Recognize Conserved quantity ⇔ symmetry: time‑translation → energy, spatial translation → linear momentum, rotation → angular momentum. Quadratic kinetic energy: $T = \tfrac12 \sum mi \dot{\mathbf r}i^{\,2}$ → leads to linear momenta $pi = mi\dot{\mathbf r}i$. Potential energy depends only on relative coordinates → forces are internal and obey Newton’s third law (action–reaction). Separable Lagrangian: $L(q,\dot q)=T(\dot q)-V(q)$ → equations of motion often simplify to familiar forms (e.g., simple harmonic oscillator). Hamiltonian equals total energy when $L$ has no explicit time dependence → $dH/dt=0$ → energy conservation. --- 🗂️ Exam Traps “Minimum” action wording: Test writers may use “least action” but the correct principle is “stationary action”. Neglecting fictitious forces: In rotating frames, forgetting Coriolis/centrifugal terms leads to wrong accelerations. Assuming all forces are conservative: Problems with friction or air resistance require accounting for non‑conservative work; mechanical energy is not constant. Mixing frames without transforming forces: Using Galilean velocity addition but keeping forces unchanged is fine (forces invariant), but acceleration must remain unchanged; a common mistake is to add $u$ to acceleration. Incorrect sign in potential gradient: $\mathbf{F} = -\nabla U$; flipping the sign reverses the direction of the force. Using $p=mv$ in relativistic regime: Leads to underestimation of momentum and kinetic energy at high speeds. ---