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 \(t \in \mathbb{R}\).

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.

Proof:

Will be added later.

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