58 lines
1.5 KiB
Markdown
58 lines
1.5 KiB
Markdown
# Energy
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## Potential energy
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> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](../../mathematical-physics/vector-analysis/vector-operators/#potentials)
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>
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> $$
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> \nabla \times \mathbf{F} = 0,
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> $$
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>
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> obtaining a scalar potential $V$ such that
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>
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> $$
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> \mathbf{F} = - \nabla V,
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> $$
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>
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> referred to as the potential energy.
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## Kinetic energy
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> *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
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>
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> $$
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> T(t) - T(0) = \int_0^t \langle \mathbf{F}(x), dx \rangle,
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> $$
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>
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> for all $t \in \mathbb{R}$.
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<br>
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> *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
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>
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> $$
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> T(t) - T(0) = \frac{1}{2} m \|x'(t)\|^2 - \frac{1}{2} m \|x'(0)\|^2,
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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## Energy conservation
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> *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
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>
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> $$
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> T(x) + V(x) = T(0) + V(0) \overset{\mathrm{def}} = E,
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> $$
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>
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> for all x, with $T: x \mapsto T(x)$ and $V: x \mapsto V(x)$ the kinetic and potential energy of the point mass.
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??? note "*Proof*:"
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Will be added later.
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Obtaining conservation of energy with $E \in \mathbb{R}$ the total (constant) energy of the system.
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