The motion of single particles

Inertial frames:

Newton's first law defines a class of inertial frames. Inertial frames are reference frames for which the trajectories for force-free motion are solutions to d²r/dt² = 0. With respect to inertial frames Newton's second law has the form
F = dp/dt. (r = coordinate, F = force, p = mv momentum)

Let F_ik be the force that particle i exerts on particle k. Newton's third law states that F_ik= -F_ki.
Newton's laws are well suited for the study of unconstrained mechanical systems. Constraints, such as requiring a particle to follow a given curve in space, tell us that there are external forces, but do not tell us what these forces are. The forces are only known in terms of their effect on the motion.

Conservation laws are very important tools in solving mechanics problems.

For a system of particles momentum is conserved if F_ext= 0; F_ext= 0 <--> P = constant.
Angular momentum (L = r × p) is conserved if the torque τ_ext= 0; τ_ext= 0 <--> L = constant.
Energy (E = T + U) is conserved if all forces are conservative; ∫F∙dr = 0 <--> T + U = constant.

Formulas:

Laws:
Newton's 2nd law:	F = dp/dt
Newton's third law:	F_ik= -F_ki

Forces:
Static and kinetic friction:	f_s ≤ μ_sN, f_k = μ_kN
Gravity:	F₁₂ = -Gm₁m₂r₁₂/r₁₂³
Coulomb's law	F₁₂ = k_eq₁q₂r₁₂/r₁₂³, k_e = 1/(4πε₀)
Hooke's law:	F = -kr, F_x= -kx

Concepts:
Uniform circular motion:	F = mv²/r
Work:	W = F∙d
Kinetic energy:	K = ½mv²
Work-kinetic energy theorem:	W_net= ΔK = ½m(v_f²- v_i²)
Elastic potential energy:	U = ½kx²
Gravitational potential energy:	U_f - U_i = -∫_r12f^r12iF₁₂∙dr₁₂ = -Gm₁m₂(1/r_12f - 1/r_12i)
Conservative systems:	E = K + U, F= -∇U
Power:	P = F·v or P = dW/dt
Momentum:	p = mv
Impulse:	I = Δp = F_avgΔt
Angular momentum:	L = r × p
Torque	τ = r × F
Angular momentum and torque:	dL/dt = d/dt(r×p) = r×dp/dt + dr/dt×p = r×dp/dt = r×F = τ dL = τ dt

Non-inertial frames:

In non-inertial frames fictitious forces appear.
Consider a particle moving with velocity v in a reference frame K which moves with velocity V(t) relative to the inertial frame K₀ and rotates with angular velocity Ω(t).
The equations of motion are

mdv/dt = -∂U/∂r - mdV/dt + mr × dΩ/dt - 2mΩ × v - mΩ × (Ω × r).

Here

-∂U/∂r = force as observed in an inertial frame
(If the force in an inertial frame is is not derivable from a potential, replace -∂U/∂r by F_inertial.)
-mdV/dt = fictitious force due to acceleration of frame
mr × dΩ/dt = fictitious force due to non-uniform rotation of frame
-2mΩ × v = Coriolis force
-mΩ × (Ω × r) = centrifugal force

For a uniformly rotating frame dΩ/dt = 0, dV/dt, and the equations of motion are
mdv/dt = F_inertial - 2mΩ × v - mΩ × (Ω × r).