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Constraint Dynamics

Numerous physical systems lead to a Hamiltonian system of ordinary differential equations:
$\displaystyle \dot q$ $\textstyle =$ $\displaystyle p$ (1)
$\displaystyle \dot p$ $\textstyle =$ $\displaystyle -\nabla_q V(q),$ (2)

where $p$ and $q$ are the momenta and positions of the particles, respectively. However, such a set of equations is altered when the motion of the bodies (in our case, unit-mass particles) is restricted by a constraint of the form
$\displaystyle g\left(q\right)$ $\textstyle =$ $\displaystyle 0.$ (3)

New equations of motion reflect the addition of a second ``force'' from the constraint acting upon the particles. For the general constrained system, determining the trajectories of motion now involves solving constrained dynamical equations of the form:

$\displaystyle \dot q$ $\textstyle =$ $\displaystyle p,$ (4)
$\displaystyle \dot p$ $\textstyle =$ $\displaystyle -\nabla_{q} V(q)+ g'\left(q\right)^T \lambda,$ (5)
$\displaystyle g\left(q\right)$ $\textstyle =$ $\displaystyle 0.$ (6)

Here, $g^{\prime}$ is the Jacobian of the vector of constraints and $\lambda$ is the vector of Lagrange multipliers associated with the constraints.

Solutions are said to lie on the solution manifold, $\mathcal{M}$, if the constraint $g(q) = 0$ is satisfied. If one differentiates the $g(q) = 0$ constraint with respect to time, another, ``hidden'' constraint is obtained. When the constraint represents a surface, the hidden constraint confines particle velocities to a plane tangent to the surface, providing a fourth and final equation of motion:

$\displaystyle g^{\prime}(q)p$ $\textstyle =$ $\displaystyle 0.$ (7)

The position and momentum pairs which satisfy the two constraints are said to lie in the tangent bundle. The exact solution of the constrained system consists of position and momentum pairs that are members of this tangent bundle.

We can model the lattice (and traditional) Newton's Cradle using this formulation of constrained differential equations. In this model, the constraints describe the surfaces of spheres in three-dimensional space:

$\displaystyle g_i\left(q\right)$ $\textstyle =$ $\displaystyle \frac{1}{2} \left( \left\vert q_i - b_i \right\vert^2 - L^2 \right),$ (8)

where $q_i$ and $b_i$ are the positions of the $i$th ball and hook respectively. For the lattice Newton's Cradle, $q_i$ and $b_i$ are vectors in $\mathcal{R}^3$, and $L$ is the length of the pendulum rods. Introducing this constraint equation into equations (4-6), our system is reduced to:
$\displaystyle \dot q_i$ $\textstyle =$ $\displaystyle p_i,$ (9)
$\displaystyle \dot p_i$ $\textstyle =$ $\displaystyle -\nabla_{q_i} V(q)+ \left( q_i - b_i \right) \lambda_i,$ (10)
$\displaystyle 0$ $\textstyle =$ $\displaystyle \frac{1}{2} \left( \left\vert q_i - b_i \right\vert^2 - L^2 \right), \quad \quad \quad \quad i=1,...,N.$ (11)

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Next: Discretization Up: Mathematical Formulation Previous: Mathematical Formulation