mathematics-physics-wiki/docs/en/physics/mechanics/newtonian-mechanics/energy.md

Energy

Potential energy

Definition 1: a force field \mathbf{F} is conservative if it is irrotational

\nabla \times \mathbf{F} = 0,

obtaining a scalar potential V such that

\mathbf{F} = - \nabla V,

referred to as the potential energy.

Kinetic energy

Definition 2: the kinetic energy T: t \mapsto T(t) of a pointmass m \in \mathbb{R} with position x: t \mapsto x(t) subject to a force \mathbf{F}: x \mapsto \mathbf{F}(x) is defined as

T(t) - T(0) = \int_0^t \langle \mathbf{F}(x), dx \rangle,

for all t \in \mathbb{R}.

Proposition 1: the kinetic energy T: t \mapsto T(t) of a pointmass m \in \mathbb{R} with position x: t \mapsto x(t) subject to a force \mathbf{F}: x \mapsto \mathbf{F}(x) is given by

T(t) - T(0) = \frac{1}{2} m |x'(t)|^2 - \frac{1}{2} m |x'(0)|^2,

for all t \in \mathbb{R}.

??? note "Proof:"

Will be added later.

Energy conservation

Theorem 1: for a pointmass m \in \mathbb{R} with position x: t \mapsto x(t) subject to a force \mathbf{F}: x \mapsto \mathbf{F}(x) we have that

T(x) + V(x) = T(0) + V(0) \overset{\mathrm{def}} = E,

for all x, with T: x \mapsto T(x) and V: x \mapsto V(x) the kinetic and potential energy of the point mass.

??? note "Proof:"

Will be added later.

Obtaining conservation of energy with E \in \mathbb{R} the total (constant) energy of the system.