## Kinematic Equations in Mechanics

Kinematic equations are constrained equations found in dynamics of body.
In the mechanics, the constraint that is incurred on is to have constant
acceleration on a body with mass *m*.

$$v={v}_{0}+at$$ | A1 |

$$x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$$ | A2 |

$${v}^{2}={v}_{0}^{2}+2a\left(x-{x}_{0}\right)$$ | A3 |

$$\overline{v}=\frac{v+{v}_{0}}{2}$$ | A4 |

## Determinant

Determinant is used in eigenvalue equations and other important. The simplest case is 2 x 2 determinant,$$\left|\begin{array}{cc}{a}_{1}& {a}_{2}\\ {a}_{3}& {a}_{4}\end{array}\right|={a}_{1}{a}_{4}-{a}_{2}{a}_{3}$$ | A5 |

$$\left|\begin{array}{ccc}{a}_{1}& {a}_{2}& {a}_{3}\\ {a}_{4}& {a}_{5}& {a}_{6}\\ {a}_{7}& {a}_{8}& {a}_{9}\end{array}\right|={a}_{1}\left({a}_{5}{a}_{9}-{a}_{6}{a}_{8}\right)-{a}_{2}\left({a}_{4}{a}_{9}-{a}_{6}{a}_{7}\right)+{a}_{3}\left({a}_{4}{a}_{8}-{a}_{5}{a}_{7}\right)$$ | A6 |

## Lagrangian Dynamics

Setting up equations of motion in terms of force can be rather nuisance. It is particularly true when there are many two-body interactions. The general procedure is to set up Lagrangian equations of motion to make them into matrix form.

The Lagrangian function is defined by

$$L\left({q}_{i},{\dot{q}}_{i},t\right)=T\left({q}_{i},{\dot{q}}_{i},t\right)-V\left({q}_{i}\right)$$ | A7 |

*T*is kinetic energy and

*V*is potential energy. The variables are in terms of

*i*coordinate

^{th}*q*, and ${\dot{q}}_{i}$ is the velocity associated with

_{i}*q*(

_{i}*q*with a dot and a double-dot on top indicates first derivative in time and second derivative, respecitively), and

*t*is time.

$$T\left({q}_{i},{\dot{q}}_{i},t\right)=\frac{1}{2}m{\dot{q}}_{i}^{2}$$ | A8 |

$$V\left({q}_{i}\right)=\frac{1}{2}k{\left({q}_{i}-{q}_{0}\right)}^{2}$$ | A9 |

Then, the dynamical equation, called *Lagrange's equation*, is obtained
by the following.

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_{i}}\right)-\frac{\partial L}{\partial {q}_{i}}=0$$ | A10 |

_{2}, as shown below.

**Figure A1. CO**

_{2}molecule and its coordinatesThe Lagrangian is written as

$$L\left({q}_{i},{\dot{q}}_{i},t\right)=\frac{1}{2}{m}_{O}{\dot{x}}_{1}^{2}+\frac{1}{2}{m}_{C}{\dot{x}}_{2}^{2}+\frac{1}{2}{m}_{O}{\dot{x}}_{3}^{2}-\frac{1}{2}k{\left({x}_{2}-{x}_{1}\right)}^{2}-\frac{1}{2}k{\left({x}_{3}-{x}_{2}\right)}^{2}$$ | A11 |

We now need to subject Eqn. A11 to Eqn. A10 for each of *i ^{th}*
component. The first term on the left-hand side of A10 is

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_{i}}\right)={\displaystyle \sum _{i}{m}_{i}{\ddot{x}}_{i}}$$ | A12 |

*m*is the mass associated with

_{i}*x*coordinate. You can easily see it that by taking the derivative of the first three terms of the right-hand side of Eqn. A11, and subsequently taking the time-derivative.

_{i}

The second term on the left-hand side of A10 is

$$\frac{\partial L}{\partial {q}_{i}}=-k\left({x}_{2}-{x}_{1}\right)+k\left({x}_{2}-{x}_{1}\right)-k\left({x}_{3}-{x}_{2}\right)$$ | A13 |

*i*component,

$$\begin{array}{l}{m}_{O}{\ddot{x}}_{1}-k\left({x}_{2}-{x}_{1}\right)=0\\ {m}_{C}{\ddot{x}}_{2}+k\left({x}_{2}-{x}_{1}\right)-k\left({x}_{3}-{x}_{2}\right)=0\\ {m}_{O}{\ddot{x}}_{3}-k\left({x}_{3}-{x}_{2}\right)=0\end{array}$$ | A14 |