[NOTES/CM-02001] Limitations of Newtonian Mechanics

Use of Cartesian coordinates
One must always write EOM in Cartesian form and then change variables if necessary to a new coordinates such as $(r,\theta,\phi)$. For example must write
\[ {m}\ddot{x}=F_x,\quad {m}\ddot{y}=F_y,\quad {m}\ddot{z}=F_z,\] where ${F_x},{F_y}, {F_z}$\]
are the components of the force.
It would be wrong to write. \[m\ddot{r}=F_r, \qquad {m}\ddot\theta=F_\theta, \text{etc.}\] where $F_r, F_\theta$ are radial or $r,\theta$ components of the force.

Constraints
For many systems the coordinates and velocities must satisfy constraint relations.
For example for a particle moving on the surface of a sphere the constraint relation \[x^2+y^2+z^2=R^2$\]
must be imposed separately on the solutions of EOM. Several different types of constraints are possible.

1. $z=f(x,y)$ particle moves on a surface
2. $f(x,y,z,\dot{x},\dot{y},\dot{z})=0$
3. For gas molecules in a cubical container the position coordinates satisfy \[-L\leq x\leq L,\qquad -L\leq y\leq L \qquad -L\leq z\leq L\]

Constraint relations involving only coordinates, and possibly time, are called holonomic constraints.
These are given by expressions of the form \[f_j(\vec{x}_1,\vec{x}_2,\dots \vec{x}_N,t)$ = 0, \qquad\qquad $j=1 \dots m \]

Need to include forces due to constraints
In addition to the applied forces or the external forces, one must also include forces of reaction or forces of constraints
while setting up the EOM. The forces of constraints become known only after the full solution has been obtained.
For example, for the pendulum we have take into account of the tension of the string which can be computed only after EOM are solved.

Need to specify all components of forces on all bodies in the system.
For N- particle system one needs to know  \(N\) forces, I.e.3N components to set up the equations of motion.