Classical Mechanics

1. Newton's Laws of Motion

The axiomatic starting point. Three laws that define the concept of force and inertia:

In various coordinate systems:

Newton's formulation is limited by the need to resolve constraint forces and by the difficulty of working in non-Cartesian coordinates. The Lagrangian approach solves both problems elegantly.

2. Principle of Least Action (Hamilton's Principle)

The most profound principle in physics: The physical trajectory \(q(t)\) between fixed endpoints \(q(t_1)\) and \(q(t_2)\)is the one that makes the action functional stationary:

Functional variation explained: Consider a variation\(q(t) \to q(t) + \epsilon\,\eta(t)\) where \(\eta(t_1) = \eta(t_2) = 0\)(endpoints are fixed). The action becomes a function of \(\epsilon\):

The stationarity condition \(dS/d\epsilon\big|_{\epsilon=0} = 0\) for all choices of \(\eta(t)\) yields the Euler-Lagrange equations (see next entry).

The Lagrangian is \(L = T - V\) for mechanical systems. This principle extends to fields (\(S = \int \mathcal{L}\, d^4x\)), general relativity (Einstein-Hilbert action), and quantum mechanics (Feynman path integral: weight each path by \(e^{iS/\hbar}\)).

3. Euler-Lagrange Equations

Full derivation from \(\delta S = 0\): Expand\(L(q+\epsilon\eta, \dot{q}+\epsilon\dot{\eta}, t)\) to first order in \(\epsilon\):

Integrate the second term by parts. The boundary term vanishes since \(\eta(t_1) = \eta(t_2) = 0\):

Combining and using the fundamental lemma of calculus of variations (the integrand must vanish for arbitrary \(\eta\)):

One equation for each generalized coordinate \(q_i\). For \(L = \frac{1}{2}m\dot{x}^2 - V(x)\), this reduces to Newton's second law: \(m\ddot{x} = -dV/dx\). The beauty is that it works in any coordinate system and automatically handles constraints.

4. Conjugate Momentum and Cyclic Coordinates

Definition: The canonical or conjugatemomentum associated with coordinate \(q_i\) is:

With this definition, the Euler-Lagrange equation becomes simply \(\dot{p}_i = \partial L / \partial q_i\).

Cyclic coordinates and conservation: If \(q_i\)does not appear explicitly in L (only \(\dot{q}_i\) does), we call \(q_i\) cyclic or ignorable. Then:

Examples: For a free particle in Cartesian coordinates, all coordinates are cyclic and linear momentum is conserved. For a central force, the azimuthal angle \(\phi\) is cyclic, yielding conservation of angular momentum \(p_\phi = mr^2\dot{\phi} = L_z\). Note: \(p_i\) is not always \(m\dot{q}_i\) -- e.g., for a charged particle in an EM field, \(\mathbf{p} = m\mathbf{v} + q\mathbf{A}\).

5. The Hamiltonian (Legendre Transform)

Construction via Legendre transformation: We want to replace the velocity variables \(\dot{q}_i\) with the conjugate momenta \(p_i = \partial L/\partial\dot{q}_i\). The Legendre transform defines:

where \(\dot{q}_i\) on the right is understood as a function of \((q, p)\) obtained by inverting \(p_i = \partial L/\partial\dot{q}_i\).

H = T + V for natural systems: If the kinetic energy is a homogeneous quadratic in \(\dot{q}_i\) (i.e., \(T = \frac{1}{2}\sum_{ij}M_{ij}\dot{q}_i\dot{q}_j\)) and \(L = T - V\) with V independent of velocities, then by Euler's theorem for homogeneous functions: \(\sum_i \dot{q}_i\,\partial T/\partial\dot{q}_i = 2T\), so:

When the coordinate transformation does not depend explicitly on time and the potential is velocity-independent, the Hamiltonian equals the total energy. This is the most common case, but not universal -- e.g., for a charged particle in an EM field, \(H = (\mathbf{p}-q\mathbf{A})^2/(2m) + q\phi\).

6. Hamilton's Equations of Motion

Derivation from the modified Hamilton's principle: Treat\(q_i(t)\) and \(p_i(t)\) as independent variables. The action in phase space is:

Taking the total differential of H and requiring \(\delta S = 0\) with independent variations \(\delta q_i\) and \(\delta p_i\):

Since \(\delta q_i\) and \(\delta p_i\) are arbitrary, each coefficient must vanish:

These are 2n first-order ODEs (vs. n second-order Euler-Lagrange equations). They reveal the symplectic structure of mechanics and provide the foundation for canonical quantization: in QM, \(q\) and \(p\) become operators with \([\hat{q},\hat{p}] = i\hbar\).

7. Poisson Brackets

Definition: For any two phase-space functions \(f(q, p, t)\) and \(g(q, p, t)\):

Time evolution: The total time derivative of any observable \(f\) along a trajectory is:

Fundamental brackets: The canonical coordinates satisfy:

Poisson brackets satisfy the same algebraic properties as quantum commutators: bilinearity, antisymmetry, the Jacobi identity, and the Leibniz rule. Canonical quantization is the replacement \(\{f,g\} \to \frac{1}{i\hbar}[\hat{f},\hat{g}]\). A quantity \(f\) is conserved if and only if \(\{f,H\} = 0\) and \(\partial f/\partial t = 0\).

8. Noether's Theorem

Statement: Every continuous symmetry of the action corresponds to a conserved quantity (and vice versa).

Derivation: Consider an infinitesimal transformation\(q_i \to q_i + \epsilon\,\delta q_i\), \(t \to t + \epsilon\,\delta t\)that leaves the action invariant (\(\delta S = 0\)). The variation of the Lagrangian is:

Using the Euler-Lagrange equations and rearranging, the conserved Noether charge is:

Fundamental examples:

In field theory, Noether's theorem extends to continuous symmetries of the Lagrangian density, yielding conserved currents \(j^\mu\) satisfying \(\partial_\mu j^\mu = 0\). Gauge symmetries in the Standard Model give rise to conserved charges like electric charge, color charge, and weak isospin.

9. Central Force Problem and Effective Potential

Setup: A particle of mass m moves under a central force \(F(r)\). The Lagrangian in polar coordinates is:

Since \(\theta\) is cyclic, \(p_\theta = mr^2\dot{\theta} = \ell\) (angular momentum) is conserved. Eliminating \(\dot{\theta} = \ell/(mr^2)\), the radial equation becomes a 1D problem with an effective potential:

The radial energy equation is:

The centrifugal term \(\ell^2/(2mr^2)\) creates a barrier at small r, preventing collapse for \(\ell \neq 0\). For the gravitational potential\(V = -GMm/r\), the orbits are conic sections (Kepler's first law). Circular orbits occur at \(V'_{\text{eff}} = 0\); their stability requires \(V''_{\text{eff}} > 0\).

10. Small Oscillations and Normal Modes

Taylor expansion near equilibrium: Let \(q_0\)be a stable equilibrium point where \(V'(q_0) = 0\) and \(V''(q_0) > 0\). Setting \(x = q - q_0\):

where \(k = V''(q_0)\). This gives the simple harmonic oscillator with frequency\(\omega = \sqrt{k/m}\).

Coupled oscillators and normal modes: For a system with n degrees of freedom near equilibrium, expand T and V to quadratic order:

The equations of motion form the generalized eigenvalue problem:

The eigenvalues \(\omega_k^2\) give the normal mode frequencies; the eigenvectors \(\mathbf{a}_k\) give the normal mode shapes. In normal coordinates \(\eta_k\), the system decouples into n independent harmonic oscillators: \(\ddot{\eta}_k + \omega_k^2\eta_k = 0\). This technique extends to vibrations of molecules, crystals (phonons), and fields (Fourier modes).