Dynamics §

The second law is the fundamental equation. All other equations of motion (Lagrangian, Hamiltonian) are equivalent reformulations. The first law defines inertial frames; the third law constrains the force model (internal forces cancel in a closed system).

For a system of $N$ particles:

Summing over all particles, internal forces cancel (third law), giving:

where $M$ is total mass and $\vec{a}_{\text{CM}}$ is the center-of-mass acceleration.

The momentum form is more general than $F = ma$ (it handles variable-mass systems like rockets). Conservation of total momentum follows when $\vec{F}_{\text{net}}^{\text{ext}} = 0$.

The angular momentum $\vec{L} = \vec{r} \times \vec{p}$ is a pseudovector perpendicular to the plane of motion. Its magnitude is $L = mvr\sin\alpha$ where $\alpha$ is the angle between $\vec{r}$ and $\vec{p}$.

The torque equation $\vec{\tau} = d\vec{L}/dt$ is the rotational analogue of $\vec{F} = d\vec{p}/dt$. For a system of particles, the total angular momentum about the center of mass is conserved when the net external torque vanishes.

In the Lagrangian formulation, conservation of $\vec{L}$ follows from rotational symmetry via Noether’s theorem (see below). The rigid-body extension $\vec{L} = \mathbf{I}\,\vec{\omega}$ (inertia tensor times angular velocity) is treated in Rigid-Body Mechanics.

Example: A figure skater pulling in their arms reduces the moment of inertia $I$, so $\omega$ increases to keep $L = I\omega$ constant.

A force is conservative if $\nabla \times \vec{F} = 0$ (in a simply connected region), which guarantees the existence of a potential $U$ with $\vec{F} = -\nabla U$. Non-conservative forces (friction, drag) dissipate mechanical energy into heat; see Thermodynamics.

Static equilibrium is the $a = 0$ special case of dynamics. It applies to structures (beams, trusses, frames) where all accelerations vanish. The two conditions (force balance and moment balance) give up to 6 scalar equations in 3-D (3 force + 3 moment), which determine unknown reaction forces.

For a 2-D problem: 3 equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum \tau_O = 0$).

Example: A uniform beam of mass $m$ and length $L$ rests on two supports at its ends. Each support carries $mg/2$.

Reduction: The center of mass moves uniformly (no external force). In the center-of-mass frame, the problem is equivalent to a single particle of mass $\mu$ in the potential $V(r)$.

Effective potential: Because angular momentum $L = \mu r^2 \dot{\theta}$ is conserved (central force → zero torque), the radial equation becomes:

The term $L^2/(2\mu r^2)$ acts as a centrifugal barrier. The effective potential is:

Kepler’s laws follow for $V(r) = -GM\mu/r$:

1. Orbits are conic sections (ellipse, parabola, hyperbola).
2. Equal areas are swept in equal times (angular momentum conservation).
3. $T^2 \propto a^3$ (period squared proportional to semi-major axis cubed).

The force law itself (Newton’s law of gravitation) is treated in Gravity; this section covers the orbital mechanics that follows from it.

Hamilton’s principle is the variational foundation of all of analytical mechanics. The Euler–Lagrange equations are not independent postulates; they are the Euler–Lagrange equations of calculus of variations applied to the functional $S$.

Derivation sketch: Let $q_i(t) \to q_i(t) + \epsilon\,\eta_i(t)$ with $\eta_i(t_1) = \eta_i(t_2) = 0$. Requiring $dS/d\epsilon|_{\epsilon=0} = 0$:

Integrating the second term by parts and discarding the boundary term (since $\eta_i$ vanishes at endpoints):

Since $\eta_i$ is arbitrary, the integrand must vanish, giving the Euler–Lagrange equations.

This variational viewpoint explains why the equations work: the true trajectory is the one that extremizes the action, just as a light ray extremizes optical path length (Fermat’s principle). It also generalizes naturally to fields (classical field theory) and to quantum mechanics (path integrals).

The Lagrangian $L = T - V$ encodes the dynamics in a scalar. The Euler–Lagrange equations are equivalent to Newton’s second law but work naturally with constraints (e.g., a pendulum’s fixed length) and in any coordinate system.

Connection to Newton’s second law: For a single particle in Cartesian coordinates, $T = \tfrac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2)$ and $V = V(x,y,z)$. The Euler–Lagrange equation for coordinate $x$:

which is exactly $m\ddot{x} = -\partial V/\partial x = F_x$. The same holds for $y$ and $z$, recovering $\vec{F} = m\vec{a}$. The two formulations are the same physics; the Lagrangian form simply packages the three component equations into a coordinate-free variational statement.

Noether’s theorem: Every continuous symmetry of $L$ yields a conserved quantity.

• Time-translation symmetry → energy conservation.
• Space-translation symmetry → momentum conservation.
• Rotation symmetry → angular momentum conservation.

This is the deep reason conservation laws exist: they are consequences of the symmetries of space and time, not independent postulates.

Phase space is the $2n$-dimensional space of $(q_1, \ldots, q_n, p_1, \ldots, p_n)$. Hamilton’s equations describe a flow in this space. Liouville’s theorem states that the phase-space volume is preserved under this flow, which is the mechanical foundation of Statistical Mechanics.

The Hamiltonian $H$ equals the total energy $T + V$ when the potential does not depend on velocities and the coordinate transformation is time-independent.

Forces along the incline: $mg\sin\theta$. Newton’s second law: $ma = mg\sin\theta$, so $a = g\sin\theta$.

Speed after distance $s$: $v^2 = 2as = 2gs\sin\theta$, so $v = \sqrt{2gs\sin\theta}$.

Energy check: $mgh = \tfrac{1}{2}mv^2$ where $h = s\sin\theta$. Same result.