Classical Mechanics

1. Newtonian Mechanics Review

1.1 Newton's Laws

1. First Law (Inertia): A body remains at rest or in uniform motion unless acted upon by a net force.
2. Second Law: $\mathbf{F} = m\mathbf{a} = \dot{\mathbf{p}}$ where $\mathbf{p} = m\mathbf{v}$.
3. Third Law: For every action, there is an equal and opposite reaction.

1.2 From Newton to Variational Principles

Newton's laws work well in Cartesian coordinates but become cumbersome in constrained systems or non-Cartesian coordinates. The Lagrangian and Hamiltonian formulations provide a more general and elegant framework based on energy principles.

The key insight: instead of tracking forces, track the energy of the system. The trajectory is the one that minimises (or more precisely, makes stationary) the action.

2. Generalised Coordinates and Constraints

2.1 Generalised Coordinates

A system with $n$ degrees of freedom can be described by $n$ generalised coordinates $q_1, q_2, \ldots, q_n$, which may be angles, arc lengths, or any other set of parameters that uniquely determines the configuration.

Example. A simple pendulum has one degree of freedom. We can use the angle $\theta$ from the vertical as the generalised coordinate, rather than the Cartesian coordinates $(x, y)$ of the bob.

2.2 Constraints

Holonomic constraints relate the coordinates by equations: $f(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N, t) = 0$

A holonomic constraint reduces the number of degrees of freedom.

Non-holonomic constraints involve inequalities or non-integrable differential relations.

Scleronomic constraints do not depend explicitly on time. Rheonomic constraints do.

Example. A bead sliding on a fixed wire: the constraint $f(x,y) = 0$ is holonomic and scleronomic. A bead on a wire that moves with time: holonomic and rheonomic.

2.3 Virtual Work and D'Alembert's Principle

A virtual displacement $\delta \mathbf{r}_i$ is an infinitesimal change in position consistent with the constraints at a fixed instant in time.

D'Alembert's Principle: For a system of $N$ particles:

$\sum_{i=1}^N (\mathbf{F}_i - m_i \ddot{\mathbf{r}}_i) \cdot \delta \mathbf{r}_i = 0$

where $\mathbf{F}_i$ includes both applied and constraint forces. For ideal constraints, the constraint forces do no virtual work, so only the applied forces contribute.

3. Lagrangian Mechanics

3.1 The Lagrangian

The Lagrangian of a system is defined as

$L(q_1, \ldots, q_n, \dot{q}_1, \ldots, \dot{q}_n, t) = T - V$

where $T$ is the kinetic energy and $V$ is the potential energy.

3.2 Euler-Lagrange Equation

Theorem 3.1 (Hamilton's Principle). The actual path of a system between times $t_1$ and $t_2$ is the one that makes the action

$S = \int_{t_1}^{t_2} L(q, \dot{q}, t)\, dt$

stationary.

Theorem 3.2 (Euler-Lagrange Equation). The path $q(t)$ that makes $S$ stationary satisfies

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right) - \frac{\partial L}{\partial q_j} = 0, \quad j = 1, \ldots, n$

Proof (for one degree of freedom). Consider a variation $q(t) + \epsilon \eta(t)$ where $\eta(t_1) = \eta(t_2) = 0$. The variation of the action:

$\delta S = \int_{t_1}^{t_2} \left(\frac{\partial L}{\partial q}\eta + \frac{\partial L}{\partial \dot{q}}\dot{\eta}\right) dt$

Integrating the second term by parts:

$\delta S = \int_{t_1}^{t_2} \left(\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}\right) \eta\, dt + \left[\frac{\partial L}{\partial \dot{q}}\eta\right]_{t_1}^{t_2}$

The boundary term vanishes since $\eta(t_1) = \eta(t_2) = 0$. For $\delta S = 0$ for all $\eta$, by the fundamental lemma of the calculus of variations:

$\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$

$\blacksquare$

3.3 Worked Example: Simple Pendulum

Problem. Derive the equation of motion for a simple pendulum of length $l$ and mass $m$.

Solution. Take $\theta$ as the generalised coordinate. The position of the bob is $(l\sin\theta, -l\cos\theta)$.

$T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) = \frac{1}{2}ml^2\dot{\theta}^2$

$V = -mgl\cos\theta$

$L = T - V = \frac{1}{2}ml^2\dot{\theta}^2 + mgl\cos\theta$

Euler-Lagrange equation:

$\frac{\partial L}{\partial \theta} = -mgl\sin\theta, \quad \frac{\partial L}{\partial \dot{\theta}} = ml^2\dot{\theta}$

$\frac{d}{dt}(ml^2\dot{\theta}) + mgl\sin\theta = 0 \implies \ddot{\theta} + \frac{g}{l}\sin\theta = 0$

For small angles ($\sin\theta \approx \theta$): $\ddot{\theta} + \frac{g}{l}\theta = 0$, giving simple harmonic motion with $\omega = \sqrt{g/l}$. $\blacksquare$

3.4 Lagrange Multipliers for Constraints

When holonomic constraints cannot be eliminated by coordinate choice, introduce Lagrange multipliers $\lambda_a$:

$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j} = \sum_a \lambda_a \frac{\partial f_a}{\partial q_j}$

The multipliers $\lambda_a$ are proportional to the constraint forces.

4. Hamiltonian Mechanics

4.1 Generalised Momentum

The generalised momentum conjugate to $q_j$ is

$p_j = \frac{\partial L}{\partial \dot{q}_j}$

4.2 The Hamiltonian

The Hamiltonian is defined by the Legendre transform:

$H(q_1, \ldots, q_n, p_1, \ldots, p_n, t) = \sum_{j=1}^n p_j \dot{q}_j - L$

When the transformation is regular (i.e., the Hessian $\partial^2 L / \partial \dot{q}_j \partial \dot{q}_k$ is non-singular), this is well-defined.

If $L$ does not depend explicitly on time and $V$ is velocity-independent, then $H = T + V$ (total energy).

4.3 Hamilton's Equations

Theorem 4.1 (Hamilton's Equations). The equations of motion in Hamiltonian form are

$\dot{q}_j = \frac{\partial H}{\partial p_j}, \quad \dot{p}_j = -\frac{\partial H}{\partial q_j}$

These are $2n$ first-order ODEs (compared to $n$ second-order ODEs in the Lagrangian formulation).

Proof. From $H = \sum p_j \dot{q}_j - L$:

$dH = \sum \dot{q}_j\, dp_j + \sum p_j\, d\dot{q}_j - \sum \frac{\partial L}{\partial q_j}\, dq_j - \sum \frac{\partial L}{\partial \dot{q}_j}\, d\dot{q}_j - \frac{\partial L}{\partial t}\, dt$

Since $p_j = \partial L / \partial \dot{q}_j$, the $d\dot{q}_j$ terms cancel:

$dH = \sum \dot{q}_j\, dp_j - \sum \dot{p}_j\, dq_j - \frac{\partial L}{\partial t}\, dt$

Comparing with $dH = \sum \frac{\partial H}{\partial p_j} dp_j + \sum \frac{\partial H}{\partial q_j} dq_j + \frac{\partial H}{\partial t} dt$:

$\dot{q}_j = \frac{\partial H}{\partial p_j}, \quad \dot{p}_j = -\frac{\partial H}{\partial q_j}, \quad \frac{\partial H}{\partial t} = -\frac{\partial L}{\partial t}$

$\blacksquare$

4.4 Phase Space

Hamiltonian mechanics naturally lives in phase space: the $2n$-dimensional space with coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$. Each point in phase space represents a complete state of the system (positions and momenta).

Liouville's Theorem. The flow in phase space is incompressible: the phase space volume is conserved along trajectories.

5. Noether's Theorem and Conservation Laws

5.1 Statement of Noether's Theorem

Theorem 5.1 (Noether's Theorem). For every continuous symmetry of the action, there is a corresponding conserved quantity.

More precisely: if the Lagrangian is invariant under the transformation $q_j \to q_j + \epsilon f_j(q, t)$ (for infinitesimal $\epsilon$), then

$Q = \sum_j p_j f_j$

is a constant of motion.

5.2 Specific Conservation Laws

Time translation invariance ($L$ does not depend explicitly on $t$):

$\frac{dH}{dt} = -\frac{\partial L}{\partial t} = 0 \implies H = \mathrm{const}$

This gives conservation of energy.

Spatial translation invariance ($L$ is invariant under $x \to x + \epsilon$):

$\frac{\partial L}{\partial x} = 0 \implies p_x = \frac{\partial L}{\partial \dot{x}} = \mathrm{const}$

This gives conservation of linear momentum.

Rotational invariance ($L$ is invariant under rotation about an axis):

$\frac{\partial L}{\partial \theta} = 0 \implies p_\theta = \frac{\partial L}{\partial \dot{\theta}} = \mathrm{const}$

This gives conservation of angular momentum.

5.3 Worked Example

Problem. A particle moves in a central potential $V(r)$. Show that angular momentum is conserved.

Solution. In spherical coordinates $(r, \theta, \phi)$ with $V = V(r)$:

$L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2\sin^2\theta\,\dot{\phi}^2) - V(r)$

Since $L$ does not depend on $\phi$ (rotational symmetry about the $z$-axis):

$p_\phi = \frac{\partial L}{\partial \dot{\phi}} = mr^2\sin^2\theta\,\dot{\phi} = \mathrm{const}$

This is the $z$-component of angular momentum. By Noether's theorem, the full angular momentum vector is conserved for any central potential. $\blacksquare$

6. Central Force Problems

6.1 Reduction to One Dimension

For a particle of mass $m$ in a central potential $V(r)$, using polar coordinates $(r, \phi)$ in the plane of motion:

$L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\phi}^2) - V(r)$

The angular momentum $l = mr^2\dot{\phi}$ is conserved. Substituting $\dot{\phi} = l/(mr^2)$:

$L = \frac{1}{2}m\dot{r}^2 + \frac{l^2}{2mr^2} - V(r)$

The effective potential is $V_{\mathrm{eff}}(r) = V(r) + \frac{l^2}{2mr^2}$, where the second term is the centrifugal barrier.

6.2 The Kepler Problem

For $V(r) = -k/r$ (gravitational or Coulomb potential):

Bertrand's Theorem: The only central potentials that give closed orbits for all bound states are $V(r) \propto 1/r$ (Kepler/Coulomb) and $V(r) \propto r^2$ (harmonic oscillator).

Key results for Kepler orbits:

• Orbits are conic sections (ellipses, parabolas, or hyperbolas).
• The semi-major axis $a$ satisfies $E = -k/(2a)$ for bound orbits.
• The period is $T = 2\pi\sqrt{ma^3/k}$ (Kepler's third law).

7. Small Oscillations and Normal Modes

7.1 Equilibrium and Small Oscillations

At a stable equilibrium, $V$ has a local minimum. Expanding around equilibrium ($q_j = q_j^0 + \eta_j$):

$L \approx \frac{1}{2}\sum_{j,k} T_{jk}\dot{\eta}_j\dot{\eta}_k - \frac{1}{2}\sum_{j,k} V_{jk}\eta_j\eta_k$

where $T_{jk} = \frac{\partial^2 T}{\partial \dot{q}_j \partial \dot{q}_k}\bigg|_0$ is the (constant) mass matrix and $V_{jk} = \frac{\partial^2 V}{\partial q_j \partial q_k}\bigg|_0$ is the (constant) stiffness matrix.

7.2 Normal Modes

Assuming solutions of the form $\eta_j = a_j e^{i\omega t}$, the eigenvalue problem is:

$(V - \omega^2 T)\mathbf{a} = \mathbf{0}$

The normal mode frequencies $\omega_\alpha$ are solutions to $\det(V - \omega^2 T) = 0$.

Theorem 7.1. For a stable system, all normal mode frequencies are real and positive. The normal modes are orthogonal with respect to both $T$ and $V$.

7.3 Worked Example: Double Pendulum (Small Oscillations)

For two equal masses $m$ on massless rods of length $l$:

The kinetic and potential energy matrices (to second order) give the eigenvalue problem with solutions $\omega_1 = \sqrt{g/l}(2 - \sqrt{2})^{1/2}$ and $\omega_2 = \sqrt{g/l}(2 + \sqrt{2})^{1/2}$.

The corresponding normal modes are:

• Mode 1: both pendulums swing in the same direction (in phase).
• Mode 2: the pendulums swing in opposite directions (out of phase).

Common Pitfall

The Lagrangian and Hamiltonian formulations are equivalent only when the Legendre transform from $L$ to $H$ is regular. If $\det(\partial^2 L / \partial \dot{q}_i \partial \dot{q}_j) = 0$, the system has constraints and the Hamiltonian formulation requires special treatment (Dirac brackets or constraint analysis).