Force

Forced is defined with Newton’s Second Law for a nonrelativistic inertial reference frame as,

\[F \equiv ma\]

More general and appropriate is to write this in vectors and time differential dot notation,

\[ \boxed{\boldsymbol{F} = m\boldsymbol{\ddot r}} \]

Force in Polar Coordinate

Force in polar coordinates uses \(\boldsymbol{a}\) in polar coordinates which is,

\[ a = -r\dot \phi^2 \boldsymbol{\hat r} + r\ddot \phi \boldsymbol{\hat \phi} \]

\[ a_\text{c} \equiv r\dot \phi^2 = r\omega^2, \qquad a_\perp \equiv r\ddot \phi = r\alpha\]

\[\boxed{\boldsymbol{F} = m\left[a_c \boldsymbol{\hat r} + a_\perp \boldsymbol{\hat \phi} \right]} \]

Preferably you may write it as,

\[ \boldsymbol{F} = m \langle \ddot r - r\dot\phi^2 , r\ddot\phi + 2\dot r \dot\phi \rangle \]

Force and Momentum

Another form of the second law shows \(F\) in terms of momentum \(p\) given that mass is constant,

\[\begin{split} \begin{align} p &\equiv mv\\ \dot p &= ma \end{align} \end{split}\]

\[ \boxed{F = \dot p} \tag{$\dot m = 0$}\]