1.5 KiB
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 pointmassm \in \mathbb{R}
with positionx: 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 pointmassm \in \mathbb{R}
with positionx: 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 positionx: 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)
andV: 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.