EQ-NEWTON-II · Classical Mechanics
Newton's Second Law
F - a*m = 0
Variables
variable
F
net (vector) force on the body, summed over all contact and field interactions
- Object
- rigid_body
- Property
- Force
- Context
- inertial_frame
- Constraint
- net_external
variable
a
second time derivative of the centre-of-mass position
- Object
- rigid_body
- Property
- Acceleration
- Context
- inertial_frame
variable
m
inertial mass of the body (strictly positive, time-independent)
- Object
- rigid_body
- Property
- Mass
- Context
- inertial_frame
Axioms
algebraic classical commutative_factors constant_coefficients deterministic linear non_relativistic
Assumptions
- m is strictly positive (m > 0)
- m is constant in time (not a rocket; not a variable-mass system)
- the reference frame is inertial (non-accelerating, non-rotating to leading order)
- |v| << c: classical, non-relativistic regime
- The body is treated as a point particle, or F is the net force on its centre of mass
Derivation
- Axiomatic in Newton (Principia, 1687, Lex II); not derived from anything deeper in that framework.
- Equivalent to the Euler–Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0 for L = (1/2) m v² − V(x), which yields m a = −∂V/∂x = F.
- Also obtainable as the flat-spacetime, low-energy limit of the geodesic equation in general relativity.
References
- Goldstein, Classical Mechanics, 3rd ed., §1.1, Eq. (1.3)
- Landau & Lifshitz, Mechanics, §5, Eq. (5.3)
- Newton, Philosophiae Naturalis Principia Mathematica, Lex II (1687)