Newtonian Mechanics Review & Constraints
Newtonian Mechanics Review & Constraints
Classical mechanics begins with Newton's three laws and the equation $\mathbf{F}=m\ddot{\mathbf{r}}$. While complete in principle, this vector equation becomes cumbersome when particles are confined to move along curves, surfaces, or linkages — the constrained systems that dominate engineering and physics. The constraint forces (normal forces, tension in rods, reaction forces at joints) are often unknown and unwanted; they merely enforce geometric restrictions. The entire programme of Lagrangian and Hamiltonian mechanics can be understood as a systematic strategy for eliminating constraint forces by choosing coordinates adapted to the constraints.
A holonomic constraint is one expressible as an algebraic equation among coordinates and possibly time: $f(q_1,\ldots,q_n,t)=0$. Rolling without slipping on a line, a bead on a wire, and a pendulum bob on a rigid rod are all holonomic. Non-holonomic constraints involve velocities in ways that cannot be integrated into a pure position relation — a ball rolling without slipping on a plane is the classic example because $\dot{x}=a\dot{\phi}$ cannot be reduced to a relation among $x$ and $\phi$ alone. The number of degrees of freedom equals the total coordinates minus the number of independent holonomic constraints.
D'Alembert's principle is the conceptual bridge from Newton to Lagrange. It states that the virtual work done by constraint forces vanishes for any virtual displacement consistent with the constraints: $\sum_i(\mathbf{F}_i-m_i\ddot{\mathbf{r}}_i)\cdot\delta\mathbf{r}_i=0$. By restricting to virtual displacements satisfying the constraints, constraint forces drop out entirely, leaving only applied forces and inertial terms — the single insight that generates the Euler-Lagrange equations.
Definition — Newton's Second Law
For $N$ particles: $\mathbf{F}_i^{\text{applied}}+\mathbf{f}_i^{\text{constraint}}=m_i\ddot{\mathbf{r}}_i$. The total force splits into an applied part and a constraint part.
Theorem — D'Alembert's Principle
For virtual displacements $\delta\mathbf{r}_i$ consistent with the constraints, $$\sum_{i=1}^N\bigl(\mathbf{F}_i^{\text{applied}}-m_i\ddot{\mathbf{r}}_i\bigr)\cdot\delta\mathbf{r}_i=0.$$ Constraint forces perform no virtual work and are therefore absent from this equation.
Example 1: Simple Pendulum
A bob of mass $m$ on a rigid rod of length $l$ satisfies the holonomic constraint $x^2+y^2=l^2$. Choosing $\theta$ as the single generalised coordinate eliminates the unknown tension $T$ entirely, yielding $$ml^2\ddot{\theta}=-mgl\sin\theta.$$ The constraint force (tension) never appears.
Example 2: Bead on a Rotating Hoop
A bead on a hoop of radius $R$ rotating at $\Omega$ about a vertical diameter has one degree of freedom $\theta$. After eliminating the normal force via D'Alembert: $$mR\ddot{\theta}=mR\Omega^2\sin\theta\cos\theta-mg\sin\theta.$$ A non-trivial equilibrium $\cos\theta_0=g/(R\Omega^2)$ bifurcates into existence for $\Omega^2>g/R$.
Example 3: Atwood's Machine
Masses $m_1,m_2$ connected over a massless pulley obey the constraint $x_1+x_2=\text{const}$. One degree of freedom gives $$a=\frac{(m_1-m_2)g}{m_1+m_2},\quad T=\frac{2m_1m_2g}{m_1+m_2}.$$ The Lagrangian approach would yield $a$ directly without solving for $T$.
Example 4: Polar Coordinates & Kinematic Terms
Newton's law in plane polars $(r,\theta)$: $$m(\ddot{r}-r\dot{\theta}^2)=F_r,\qquad m(r\ddot{\theta}+2\dot{r}\dot{\theta})=F_\theta.$$ The centripetal term $-mr\dot{\theta}^2$ and Coriolis term $2m\dot{r}\dot{\theta}$ are kinematic artefacts of curvilinear coordinates, not real forces. The Lagrangian formalism absorbs them automatically.
Practice Problems
- A particle moves on a sphere of radius $R$. How many holonomic constraints apply, and what is the number of degrees of freedom?
- Classify $\dot{x}\sin\theta-\dot{y}\cos\theta=0$ as holonomic or non-holonomic and justify your answer.
- For the simple pendulum, show using D'Alembert's principle that the rod tension $T$ does no virtual work for a virtual displacement $\delta\theta$.
- A double pendulum (two rigid rods, two bobs) moves in a plane. Write its holonomic constraints in Cartesian coordinates and state its number of degrees of freedom.
- A mass $m$ slides frictionlessly on a wedge of angle $\alpha$ which is itself free to slide on a horizontal surface. Derive the equations of motion using Newton's laws and identify all constraint forces.
Show Answer Key
1. The constraint is $x^2+y^2+z^2=R^2$ (one holonomic constraint). Three Cartesian coordinates minus one constraint gives 2 degrees of freedom (e.g., $\theta, \phi$ on the sphere).
2. Non-holonomic. The constraint $\dot{x}\sin\theta - \dot{y}\cos\theta = 0$ involves velocities and cannot be integrated to a relation among coordinates alone (it constrains rolling without slipping direction, as in a vertical disk or ice skate).
3. Virtual displacement: $\delta\mathbf{r} = L\delta\theta\,\hat{\theta}$ (tangential). Tension $\mathbf{T}$ is along the rod (radial direction). $\mathbf{T}\cdot\delta\mathbf{r} = T\hat{r}\cdot L\delta\theta\hat{\theta} = 0$ since $\hat{r}\perp\hat{\theta}$. Constraint forces do no virtual work, as required by D'Alembert's principle.
4. Two rods of lengths $l_1, l_2$: $(x_1,y_1)$ with $x_1^2+y_1^2=l_1^2$ and $(x_2,y_2)$ with $(x_2-x_1)^2+(y_2-y_1)^2=l_2^2$. Four Cartesian coordinates, two constraints → 2 degrees of freedom ($\theta_1, \theta_2$).
5. Let $X$ = wedge displacement, $x, y$ = mass position relative to ground. Constraints: $y = -(x-X)\tan\alpha$ (mass slides on wedge face). Newton's laws for mass: $m\ddot{x} = N\sin\alpha$, $m\ddot{y} = N\cos\alpha - mg$. For wedge: $M\ddot{X} = -N\sin\alpha$. Constraint force $N$ (normal) is eliminated by substituting the constraint into the equations.