port from mathematics-physics notes
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# Equations of Hamilton
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## The Hamiltonian
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> *Definition 1*: let $\mathcal{L}: (\mathbf{q},\mathbf{q}',t) \mapsto \mathcal{L}(\mathbf{q},\mathbf{q}',t)$ be the Lagrangian of the system, suppose that the generalized momenta $\mathbf{p}$ are defined in terms of the active variables $\mathbf{q}'$ and the passive variables $(\mathbf{q},t)$ such that
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>
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> $$
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> \mathbf{p} = \nabla_{\mathbf{q}'}\mathcal{L}(\mathbf{q},\mathbf{q}',t),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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We may now pose that there exists a function that meets the inverse, which can be obtained with Legendre transforms.
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> *Theorem 1*: there exists a function $\mathcal{H}: (\mathbf{q},\mathbf{p},t) \mapsto \mathcal{H}(\mathbf{q},\mathbf{p},t)$ such that
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>
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> $$
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> \mathbf{q}' = \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t),
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> $$
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>
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> for all $t \in \mathbb{R}$. Where $\mathcal{H}$ is the Hamiltonian of the system and is related to the Lagrangian $\mathcal{L}$ by
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>
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> $$
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> \mathcal{H}(\mathbf{q},\mathbf{p},t) = \langle \mathbf{q'}, \mathbf{p} \rangle - \mathcal{L}(\mathbf{q},\mathbf{q}',t),
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> $$
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>
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> for all $t \in \mathbb{R}$ with $\mathcal{L}$ and $\mathcal{H}$ the Legendre transforms of each other.
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??? note "*Proof*:"
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Will be added later.
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## The equations of Hamilton
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> *Corollary 1*: the partial derivatives of $\mathcal{L}$ and $\mathcal{H}$ with respect to the passive variables are related by
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>
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> $$
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> \begin{align*}
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> \nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \nabla_{\mathbf{q}} \mathcal{L}(\mathbf{q},\mathbf{q}',t), \\
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> \partial_t \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \partial_t \mathcal{L}(\mathbf{q},\mathbf{q}',t),
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> \end{align*}
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> $$
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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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Obtaining the equations of Hamilton
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$$
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\begin{align*}
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\mathbf{p}' &= -\nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t), \\
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\mathbf{q}' &= \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t),
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\end{align*}
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$$
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for all $t \in \mathbb{R}$.
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> *Proposition 1*: when the Hamiltonian $\mathcal{H}$ has no explicit time dependence it is a constant of motion.
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??? note "*Proof*:"
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Will be added later.
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To put it differently; a Hamiltonian of a conservative autonomous system is conserved.
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> *Theorem 2*: for conservative autonomous systems, the Hamiltonian $\mathcal{H}$ may be expressed as
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>
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> $$
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> \mathcal{H}(\mathbf{q},\mathbf{p}) = T(\mathbf{q},\mathbf{p}) + V(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$ with $T: (\mathbf{q},\mathbf{p}) \mapsto T(\mathbf{q},\mathbf{p})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ the kinetic and potential energy of the system.
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??? note "*Proof*:"
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Will be added later.
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It may be observed that the Hamiltonian $\mathcal{H}$ and [generalised energy](/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations/#the-generalized-energy) $h$ are identical. Note however that $\mathcal{H}$ must be expressed in $(\mathbf{q},\mathbf{p},t)$ which is not the case for $h$.
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> *Proposition 2*: a coordinate $q_j$ is cyclic if
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>
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> $$
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> \partial_{q_j} \mathcal{H}(\mathbf{q},\mathbf{p},t) = 0,
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> $$
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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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> *Proposition 3*: the Hamiltonian is seperable if there exists two mutually independent subsystems.
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??? note "*Proof*:"
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Will be added later.
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## Poisson brackets
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> *Definition 2*: let $G: (\mathbf{q},\mathbf{p},t) \mapsto G(\mathbf{q},\mathbf{p},t)$ be an arbitrary observable, its time derivative may be given by
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>
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> $$
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> \begin{align*}
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> d_t G(\mathbf{q},\mathbf{p},t) &= \sum_{j=1}^f \Big(\partial_{q_j} G q_j' + \partial_{p_j} G p_j' \Big) + \partial_t G, \\
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> &= \sum_{j=1}^f \Big(\partial_{q_j} G \partial_{p_j} \mathcal{H} - \partial_{p_j} G \partial_{q_j} \mathcal{H} \Big) + \partial_t G, \\
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> &\overset{\mathrm{def}}= \{G, \mathcal{H}\} + \partial_t G.
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> \end{align*}
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> $$
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>
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> for all $t \in \mathbb{R}$ with $\mathcal{H}$ the Hamiltonian and $\{G, \mathcal{H}\}$ the Poisson bracket of $G$ and $\mathcal{H}$.
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The Poisson bracket may simplify expressions; it has distinct properties that are true for any observables. The following theorem demonstrates the usefulness even more.
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> *Theorem 3*: let $f: (\mathbf{q}, \mathbf{p}, t) \mapsto f(\mathbf{q}, \mathbf{p}, t)$ and $g: (\mathbf{q}, \mathbf{p}, t) \mapsto f(\mathbf{q}, \mathbf{p}, t)$ be two integrals of Hamilton's equations given by
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>
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> $$
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> \begin{align*}
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> f(\mathbf{q}, \mathbf{p}, t) = c_1, \\
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> g(\mathbf{q}, \mathbf{p}, t) = c_2,
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> \end{align*}
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> $$
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>
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> for all $t \in \mathbb{R}$ with $c_{1,2} \in \mathbb{R}$. Then
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>
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> $$
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> \{f,g\} = c_3
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> $$
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>
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> with $c_3 \in \mathbb{R}$ for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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# Hamiltonian formalism of mechanics
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The Hamiltonian formalism of mechanics is based on the definitions posed by [Lagrangian mechanics](/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism) and the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/).
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Where the Lagrangian formalism used the [principle of virtual work](/en/physics/mechanics/lagrangian-mechanics/lagrange-equations/#principle-of-virtual-work) to derive the Lagrangian equations of motion, the Hamiltonian formalism will derive the Lagrangian equations with the stationary action principle. A derivative of Fermat's principle of least time.
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In Hamilton's formulation the stationary action principle is referred to as Hamilton's principle.
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## Hamilton's principle
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> *Principle 1*: of all the kinematically possible motions that take a mechanical system from one given configuration to another within a time interval $T \subset \mathbb{R}$, the actual motion is the stationary point of the time integral of the Lagrangian $\mathcal{L}$ of the system. Let $S$ be the functional of the trajectories of the system, then
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>
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> $$
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> S = \int_T \mathcal{L} dt,
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> $$
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>
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> has stationary points.
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The functional $S$ is often referred to as the action of the system. With this principle the equations of Lagrange can be derived.
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> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the equations of Lagrange are given by
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>
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> $$
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> \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) - d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) = 0,
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Let the redefined generalized coordinates $\mathbf{q}: (t,a) \mapsto \mathbf{q}(t,a)$ be given by
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$$
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\mathbf{q}(t,a) = \mathbf{\hat q}(t) + a \varepsilon(t),
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$$
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with $\mathbf{\hat q}: t \mapsto \mathbf{\hat q}(t)$ the generalized coordinates of the system and $\varepsilon: t \mapsto \varepsilon(t)$ a smooth differentiable function.
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Let $S: a \mapsto S(a)$ be the action of the system and let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian of the system, according to Hamilton's principle
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$$
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S(a) = \int_T \mathcal{L}(\mathbf{q}, \mathbf{q'})dt,
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$$
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for all $a \in \mathbb{R}$. To determine the stationary points we must have that $S'(0) = 0$. We have that $S'$ is given by
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$$
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\begin{align*}
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S'(a) &= \int_T \partial_a \mathcal{L}(\mathbf{q}, \mathbf{q'})dt, \\
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&= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \partial_a q_j + \partial_{q_j'} \mathcal{L} \partial_a q_j'\bigg)dt, \\
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&= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) + \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j\bigg)dt. \\
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\end{align*}
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$$
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Partial integration may be used for the second part:
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$$
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\begin{align*}
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\int_T \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j dt &= \Big[\partial_{q_j'} \mathcal{L} \partial_a q_j \Big]_T - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt, \\
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&= \Big[\partial_{q_j'} \mathcal{L} \varepsilon_j(t) \Big]_T - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt.
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\end{align*}
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$$
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Choose $\varepsilon_j$ such that
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$$
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\Big[\partial_{q_j'} \mathcal{L} \varepsilon_j(t) \Big]_T = 0.
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$$
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Obtains
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$$
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\int_T \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j dt = - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt.
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$$
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The general expression of $S'$ may now be given by
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$$
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\begin{align*}
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S'(a) &= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) - \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})\bigg)dt, \\
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&= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) - \varepsilon_j(t) d_t (\partial_{q_j'} \mathcal{L})\bigg)dt, \\
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&= \sum_{j=1}^f \int_T \varepsilon_j(t) \Big(\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L})\Big)dt.
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\end{align*}
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$$
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Then
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$$
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S'(0) = \sum_{j=1}^f \int_T \varepsilon_j(t) \Big(\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L})\Big)dt = 0,
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$$
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since $\varepsilon_j$ can be chosen arbitrary this implies that
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$$
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\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L}) = 0.
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$$
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# The equations of Lagrange
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## Principle of virtual work
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> *Definition 1*: a virtual displacement is a displacement at a fixed moment in time that is consistent with the constraints at that moment.
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The following principle addresses the problem that the constraint forces are generally unknown.
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> *Principle 1*: let $\mathbf{\delta x}_i \in \mathbb{R}^m$ be a virtual displacement and let $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ be the total force excluding the constraint forces. Then
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>
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> $$
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> \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}) - m_i \mathbf{x}_i''(\mathbf{q}), \mathbf{\delta x}_i \Big\rangle = 0,
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> $$
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>
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> is true for sklerenomic constraints and all $t \in \mathbb{R}$.
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Which implies that the constraint forces do not do any (net) virtual work.
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## The equations of Lagrange
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> *Theorem 1*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ be the kinetic energy of the system. For holonomic constraints we have that
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>
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> $$
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> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}') \Big) - \partial_{q_j} T(\mathbf{q},\mathbf{q}') = Q_j(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$. With $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ the generalized forces of type I given by
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>
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> $$
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> Q_j(\mathbf{q}) = \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}), \partial_j \mathbf{x}_i(\mathbf{q}) \Big\rangle,
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> $$
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>
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> for all $t \in \mathbb{R}$ with $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ the total force excluding the constraint forces.
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??? note "*Proof*:"
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Will be added later.
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Obtaining the equations of Lagrange. Note that the position of each point mass $\mathbf{x}_i$ is defined in the [Lagrangian formalism](lagrangian-formalism.md#generalizations).
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### Conservative systems
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For conservative systems we may express the force $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ in terms of a potential energy $V: X \mapsto V(X)$ by
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$$
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\mathbf{F}_i(\mathbf{q}) = -\nabla_i V(X),
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$$
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for $X: \mathbf{q} \mapsto X(\mathbf{q}) \overset{\mathrm{def}}= \{\mathbf{x}_i(\mathbf{q})\}_{i=1}^n$.
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> *Lemma 1*: for a conservative holonomic system the generalized forces of type I $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ may be expressed in terms of the potential energy $V: \mathbf{q} \mapsto V(\mathbf{q})$ by
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>
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> $$
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> Q_j(\mathbf{q}) = -\partial_{q_j} V(\mathbf{q}),
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> $$
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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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The equation of Lagrange may now be rewritten, which obtains the following lemma.
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> *Lemma 2*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ be the kinetic and potential energy of the system. The Lagrange equations for conservative systems are given by
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>
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> $$
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> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}')\Big) - \partial_{q_j}T(\mathbf{q},\mathbf{q}') = - \partial_{q_j} V(\mathbf{q}),
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> $$
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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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> *Definition 2*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ be the kinetic and potential energy of the system. The Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is defined as
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>
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> $$
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> \mathcal{L}(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') - V(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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With this definition we may write the Lagrange equations in a more formal way.
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> *Theorem 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the equations of Lagrange for conservative holonomic systems are given by
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>
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> $$
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> d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) - \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
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> $$
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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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# Lagrange generalizations
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## The generalized momentum and force
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> *Definition 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized momentum** $p_j: (\mathbf{q}, \mathbf{q}') \mapsto p_j(\mathbf{q},\mathbf{q}')$ is defined as
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>
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> $$
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> p_j(\mathbf{q},\mathbf{q}') = \partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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The generalized momentum may also be referred to as the canonical or conjugated momentum. Recall that $j \in \mathbb{N}[j\leq f]$.
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> *Definition 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized force of type II** $F_j: (\mathbf{q}, \mathbf{q}') \mapsto F_j(\mathbf{q},\mathbf{q}')$ is defined as
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>
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> $$
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> F_j(\mathbf{q},\mathbf{q}') = \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'})
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> $$
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>
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> for all $t \in \mathbb{R}$.
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We may also write $\mathbf{p} = \{p_j\}_{j=1}^f$ and $\mathbf{F} = \{F_j\}_{j=1}^f$.
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## The generalized energy
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> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the generalized energy $h: (\mathbf{q}, \mathbf{q'},\mathbf{p}) \mapsto h(\mathbf{q}, \mathbf{q'},\mathbf{p})$ is given by
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>
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> $$
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> h(\mathbf{q}, \mathbf{q'}, \mathbf{p}) = \sum_{j=1}^f \big(p_j q_j' \big) - \mathcal{L}(\mathbf{q}, \mathbf{q'}),
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> $$
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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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A generalization of the concept of energy.
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* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'},t) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'},t)$ is explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'},t) \neq 0$ and the generalized energy $h$ is not conserved.
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* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is not explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0$ and the generalized energy $h$ is conserved.
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|
||||
> *Theorem 2*: for autonomous systems with only conservative forces the generalized energy $h: (\mathbf{q}, \mathbf{q'}) \mapsto h(\mathbf{q}, \mathbf{q'})$ is conserved and is given by
|
||||
>
|
||||
> $$
|
||||
> h(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') + V(\mathbf{q}) \overset{\mathrm{def}}= E,
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ the kinetic and potential energy of the system and $E \in \mathbb{R}$ the total energy of the system.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
In this case the generalized energy $h$ is conserved and is equal to the total energy $E$ of the system.
|
||||
|
||||
## Conservation of generalized momentum
|
||||
|
||||
> *Definition 3*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, a coordinate $q_j$ is **cyclic** if
|
||||
>
|
||||
> $$
|
||||
> \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
Therefore the Lagrangian is independent of a cyclic coordinate.
|
||||
|
||||
> *Proposition 1*: the generalized momentum $p_j$ corresponding to a cyclic coordinate $q_j$ is conserved.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
## Seperable systems
|
||||
|
||||
> *Proposition 2*: the Lagrangian is seperable if there exists two mutually independent subsystems.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
Obtaining a decoupled set of partial differential equations.
|
||||
|
||||
## Invariances
|
||||
|
||||
> *Proposition 3*: the Lagrangian is invariant for Gauge transformations and therefore **not unique**.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
There can exist multiple Lagrangians that may lead to the same equation of motion.
|
||||
|
||||
According to the theorem of Noether, the invariance of a closed system with respect to continuous transformations implies that corresponding conservation laws exist.
|
||||
|
|
@ -0,0 +1,75 @@
|
|||
# Lagrangian formalism of mechanics
|
||||
|
||||
The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/).
|
||||
|
||||
## Configuration of a system
|
||||
|
||||
Considering a system of $n \in \mathbb{R}$ point masses $m_i \in \mathbb{R}$ with positions $\mathbf{x}_i \in \mathbb{R}^m$ in dimension $m \in \mathbb{N}$, for $i \in \mathbb{N}[i \leq n]$.
|
||||
|
||||
> *Definition 1*: the set of positions $\{\mathbf{x}_i\}_{i=1}^n$ is defined as the configuration of the system.
|
||||
|
||||
Obtaining a $n m$ dimensional configuration space of the system.
|
||||
|
||||
> *Definition 2*: let $N = nm$, the set of time dependent coordinates $\{q_i: t \mapsto q_i(t)\}_{i=1}^N$ at a time $t \in \mathbb{R}$ is a point in the $N$ dimensional configuration space of the system.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 3*: let the generalized coordinates be a minimal set of coordinates which are sufficient to specify the configuration of a system completely and uniquely.
|
||||
|
||||
The minimum required number of generalized coordinates is called the number of degrees of freedom of the system.
|
||||
|
||||
## Classification of constraints
|
||||
|
||||
> *Definition 4*: geometric constraints define the range of the positions $\{\mathbf{x}_i\}_{i=1}^n$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 5*: holonomic constraints are defined as constraints that can be formulated as an equation of generalized coordinates and time.
|
||||
|
||||
Let $g: (q_1, \dots, q_N, t) \mapsto g(q_1, \dots, q_N, t) = 0$ is an example of a holonomic constraint.
|
||||
|
||||
> *Definition 6*: a constraint that depends on velocities is defined as a kinematic constraint.
|
||||
|
||||
If the kinematic constrain is integrable and can be formulated as a holonomic constraint it is referred to as a integrable kinematic constraint.
|
||||
|
||||
> *Definition 7*: a constraint that explicitly depends on time is defined as a rheonomic constraint. Otherwise the constraint is defined as a sklerenomic constraint.
|
||||
|
||||
If a system of $n$ point masses is subject to $k$ indepent holonomic constraints, then these $k$ equations can be used to eliminate $k$ of the $N$ coordinates. Therefore there remain $f \overset{\mathrm{def}}= N - k$ "independent" generalized coordinates.
|
||||
|
||||
## Generalizations
|
||||
|
||||
> *Definition 8*: the set of generalized velocities $\{q_i'\}_{i=1}^N$ at a time $t \in \mathbb{R}$ is the velocity at a point along its trajectory through configuration space.
|
||||
|
||||
The position of each point mass may be given by
|
||||
|
||||
$$
|
||||
\mathbf{x}_i: \mathbf{q} \mapsto \mathbf{x}_i(\mathbf{q}),
|
||||
$$
|
||||
|
||||
with $\mathbf{q} = \{q_i\}_{i=1}^f$ generalized coordinates.
|
||||
|
||||
Therefore the velocity of each point mass is given by
|
||||
|
||||
$$
|
||||
\mathbf{x}_i'(\mathbf{q}) = \sum_{r=1}^f \partial_r \mathbf{x}_i(\mathbf{q}) q_r',
|
||||
$$
|
||||
|
||||
for all $t \in \mathbb{R}$ (inexplicitly).
|
||||
|
||||
> *Theorem 1*: the total kinetic energy $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q}')$ of the system is given by
|
||||
>
|
||||
> $$
|
||||
> T(\mathbf{q}, \mathbf{q}') = \sum_{r,s=1}^f a_{rs}(\mathbf{q}) q_r' q_s',
|
||||
> $$
|
||||
>
|
||||
> with
|
||||
>
|
||||
> $$
|
||||
> a_{rs}(\mathbf{q}) = \sum_{i=1}^n \frac{1}{2} m_i \Big\langle \partial_r \mathbf{x}_i(\mathbf{q}), \partial_s \mathbf{x}_i(\mathbf{q}) \Big\rangle,
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
|
@ -0,0 +1,58 @@
|
|||
# Energy
|
||||
|
||||
## Potential energy
|
||||
|
||||
> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](../../mathematical-physics/vector-analysis/vector-operators/#potentials)
|
||||
>
|
||||
> $$
|
||||
> \nabla \times \mathbf{F} = 0,
|
||||
> $$
|
||||
>
|
||||
> 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 $t \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *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.
|
||||
|
||||
|
|
@ -0,0 +1,30 @@
|
|||
# Momentum
|
||||
|
||||
> *Definition 1*: the **momentum** $\mathbf{p}$ of a particle is defined as the product of the mass and velocity of the particle
|
||||
>
|
||||
> $$
|
||||
> \mathbf{p} = m \mathbf{v},
|
||||
> $$
|
||||
>
|
||||
> with $m$ the mass of the particle and $\mathbf{v}$ the velocity of the particle.
|
||||
|
||||
For the case that $\mathbf{v}: t \to \mathbf{v}(t) \implies \mathbf{v}'(t) = \mathbf{a}(t)$ we have the following theorem.
|
||||
|
||||
> *Theorem 1*: let $\mathbf{v}$, $\mathbf{a}$ be the velocity and acceleration of a particle respectively, if we have
|
||||
>
|
||||
> $$
|
||||
> \mathbf{v}: t \to \mathbf{v}(t) \implies \forall t \in \mathbb{R}: \mathbf{v}'(t) = \mathbf{a}(t),
|
||||
> $$
|
||||
>
|
||||
> then
|
||||
>
|
||||
> $$
|
||||
> \mathbf{p}'(t) = \mathbf{F}(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
|
|
@ -0,0 +1,128 @@
|
|||
# Newtonian formalism of mechanics
|
||||
|
||||
## Fundamental assumptions
|
||||
|
||||
> *Postulate 1*: there exists an absolute space in which the axioms of Euclidean geometry hold.
|
||||
|
||||
The properties of space are constant, immutable and entirely independent of the presence of objects and of all dynamical processes that occur within it.
|
||||
|
||||
> *Postulate 2*: there exists an absolute time, entirely independent.
|
||||
|
||||
From postulate 1 and 2 we obtain the notion that simultaneity is absolute. In the sense that incidents that occur simultaneously in one reference system, occur simultaneously in all reference systems, independent of their mutual dynamic states or relations.
|
||||
|
||||
The definition of a reference system will follow in the next section.
|
||||
|
||||
> *Principle of relativity*: all physical axioms are of identical form in all **inertial** reference systems.
|
||||
|
||||
It follows from the principle of relativity that the notion of absolute velocity does not exist.
|
||||
|
||||
> *Postulate 3*: space and time are continuous, homogeneous and isotropic.
|
||||
|
||||
Implying that there is no fundamental limit to the precision of measurements of spatial positions, velocities and time intervals. There are no special locations or instances in time all positions and times are equivalent. The properties of space and time are invariant under translations. And there are no special directions, all directions are equivalent. The properties of space and time are invariant under rotations and reflections.
|
||||
|
||||
## Galilean transformations
|
||||
|
||||
> *Definition 1*: a **reference system** is an abstract coordinate system whose origin, orientation, and scale are specified by a set of geometric points whose position is identified both mathematically and physically.
|
||||
|
||||
From the definition of a reference system and postulates 1, 2 and 3 the Galilean transformations may be posed, which may be used to transform between the coordinates of two reference systems.
|
||||
|
||||
> *Principle 1*: let $(\mathbf{x},t) \in \mathbb{R}^4$ be a general point in spacetime.
|
||||
>
|
||||
> A uniform motion with velocity $\mathbf{v}$ is given by
|
||||
>
|
||||
> $$
|
||||
> (\mathbf{x},t) \mapsto (\mathbf{x} + \mathbf{v}t,t),
|
||||
> $$
|
||||
>
|
||||
> for all $\mathbf{v}\in \mathbb{R}^3$.
|
||||
>
|
||||
> A translation by $(\mathbf{a},t)$ is given by
|
||||
>
|
||||
> $$
|
||||
> (\mathbf{x},t) \mapsto (\mathbf{x} + \mathbf{a},t + s),
|
||||
> $$
|
||||
>
|
||||
> for all $(\mathbf{a},t) \in \mathbb{R}^4$.
|
||||
>
|
||||
> A rotation by $R$ is given by
|
||||
>
|
||||
> $$
|
||||
> (\mathbf{x},t) \mapsto (R \mathbf{x},t),
|
||||
> $$
|
||||
>
|
||||
> for all orthogonal transformations $R: \mathbb{R}^3 \to \mathbb{R}^3$.
|
||||
|
||||
The Galilean transformations may form a Lie group.
|
||||
|
||||
## Axioms of Newton
|
||||
|
||||
> *Axiom 1*: in the absence of external forces, a particle moves with a constant speed along a straight line.
|
||||
>
|
||||
> *Axiom 2:* the net force on a particle is equal to the rate at which the particle's momentum changes with time.
|
||||
>
|
||||
> *Axiom 3:* if two particles exert forces onto each other, then the mutual forces have equal magnitudes but opposite directions.
|
||||
|
||||
From axiom 1 and the principle of relativity the definition of a inertial reference system may be posed.
|
||||
|
||||
> *Definition 2*: an **inertial reference system** is a reference system in which the first axiom of Newton holds.
|
||||
|
||||
This implies that a inertial reference system is reference system not undergoing any acceleration. Therefore we may postulate the following.
|
||||
|
||||
> *Postulate 4*: inertial reference systems exist.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 3*: considering two particles $i \in \{1,2\}$ which exert forces onto each other having accelerations $\mathbf{a}_i$. Since by the 2nd and 3rd axiom we have that $\mathbf{a}_1 = - \mathbf{a}_2$ and that the ratio of their magnitudes is a constant we define the ratio of the inertial masses by
|
||||
>
|
||||
> $$
|
||||
> \frac{m_1}{m_2} = \frac{\|\mathbf{a}_2\|}{\|\mathbf{a}_1\|}.
|
||||
> $$
|
||||
|
||||
A particle with a mass can be considered as a point mass, which is defined below.
|
||||
|
||||
> *Definition 4*: a point mass is defined as a point in space and time appointed with a mass.
|
||||
|
||||
## Forces
|
||||
|
||||
> *Definition 5*: a force $\mathbf{F}$ is defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{F} = m \mathbf{a},
|
||||
> $$
|
||||
>
|
||||
> with $m \in \mathbb{R}$ the inertial mass and $\mathbf{a}$ the acceleration of the particle.
|
||||
|
||||
Definition 5 also implies the equation of motion, for a constant force a second order ordinary differential equation of the position.
|
||||
|
||||
> *Proposition 1*: in the case that a force only depends on position, the equation of motion is invariant to time inversion and time translation.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
This implies that for a moving a particle in a force field it can not be deduced at what point in time it occured and whether it is moving forward or backward in time.
|
||||
|
||||
> *Definition 6*: a central force $\mathbf{F}$ representing the interaction between two point masses at positions $\mathbf{x}_1$ and $\mathbf{x}_2$ is defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{F} = F(\mathbf{x}_1,\mathbf{x}_2) \frac{\mathbf{x}_2 - \mathbf{x}_1}{\|\mathbf{x}_2 - \mathbf{x}_1\|} \overset{\mathrm{def}} = F(\mathbf{x}_1,\mathbf{x}_2) \mathbf{e}_r,
|
||||
> $$
|
||||
>
|
||||
> with $F: (\mathbf{x}_1,\mathbf{x}_2) \mapsto F(\mathbf{x}_1,\mathbf{x}_2)$ the magnitude.
|
||||
|
||||
Which for a isotropic central force depends only on the distance between the pointmasses $\|\mathbf{x}_2 - \mathbf{x}_1\|$.
|
||||
|
||||
### Gravitational force of Newton
|
||||
|
||||
> *Postulate 5*: the force $\mathbf{F}$ between two particles described by their positions $\mathbf{x}_{1,2}: t \mapsto \mathbf{x}_{1,2}(t)$ is given by
|
||||
>
|
||||
> $$
|
||||
> \mathbf{F} = G \frac{m_1 m_2}{\|\mathbf{x}_2 - \mathbf{x}_1\|^2} \mathbf{e}_r,
|
||||
> $$
|
||||
>
|
||||
> with $m_{1,2} \in \mathbb{R}$ the gravitational mass of both particles and $G \in \mathbb{R}$ the gravitational constant.
|
||||
|
||||
According to the observation of Galilei; all object fall with equal speed (in the absence of air friction), which implies that the ratio of inertial and gravitational mass is a constant for any kind of matter.
|
||||
|
||||
> *Principle 2*: the inertial and gravitational mass of a particle are equal.
|
||||
|
||||
|
|
@ -0,0 +1,216 @@
|
|||
# Particle systems
|
||||
|
||||
For a system of particles we have the mutual forces among the selected particles referred to as internal forces, otherwise external forces. If there are no external forces, the system is called closed, otherwise open.
|
||||
|
||||
> *Definition 1*: the internal interaction forces $\mathbf{F}_i$ in a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ may be approximated by pairwise interaction forces given by
|
||||
>
|
||||
> $$
|
||||
> \mathbf{F}_i (\mathbf{x}_i) = \sum_{j=1}^n \mathbf{F}_{ij}(\mathbf{x}_i, \mathbf{x}_j) \epsilon_{ij},
|
||||
> $$
|
||||
>
|
||||
> for all $\mathbf{x}_i$ with $\mathbf{F}_{ij}$ the pairwise interaction force between particle $i$ and $j$.
|
||||
|
||||
For high density systems this approximation diverges.
|
||||
|
||||
## Systems with conservative internal forces
|
||||
|
||||
Considering a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ and mass $m_i \in \mathbb{R}$ with conservative external forces $\mathbf{F}_i$. For each particle an equation of motion can be formulated using the pairwise interaction approximation (definition 1), obtaining
|
||||
|
||||
$$
|
||||
m_i \mathbf{x}_i''(t) = \mathbf{F}_i(\mathbf{x}_i(t)) + \sum_{j=1}^n \mathbf{F}_{ij}(\mathbf{x}_i, \mathbf{x}_j) \epsilon_{ij},
|
||||
$$
|
||||
|
||||
for all $t \in \mathbb{R}$ with $\mathbf{F}_{ij}$ the pairwise interaction force.
|
||||
|
||||
> *Definition 2*: the total mass $M$ of the system is defined as
|
||||
>
|
||||
> $$
|
||||
> M = \sum_{i=1}^n m_i.
|
||||
> $$
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 3*: the center of mass $\mathbf{R}: t \mapsto \mathbf{R}(t)$ of the system is defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{R}(t) = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{x}_i(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 4*: the total momentum $\mathbf{P}$ and angular momentum $\mathbf{J}$ of the system are defined as
|
||||
>
|
||||
> $$
|
||||
> \begin{align*}
|
||||
> \mathbf{P} &= \sum_{i=1}^n \mathbf{p}_i, \\
|
||||
> \mathbf{J} &= \sum_{i=1}^n \mathbf{x}_i \times \mathbf{p}_i,
|
||||
> \end{align*}
|
||||
> $$
|
||||
>
|
||||
> with $\mathbf{p}_i$ the momentum of each particle.
|
||||
|
||||
We have for $\mathbf{P}: t \mapsto \mathbf{P}(t)$ the total momentum equivalently given by
|
||||
|
||||
$$
|
||||
\mathbf{P}(t) = M \mathbf{R}'(t),
|
||||
$$
|
||||
|
||||
for all $t \in \mathbb{R}$ with $\mathbf{R}: t \mapsto \mathbf{R}(t)$ the center of mass.
|
||||
|
||||
> *Definition 5*: the total external force $\mathbf{F}$ and torque $\mathbf{\Gamma}$ of the system are defined as
|
||||
>
|
||||
> $$
|
||||
> \begin{align*}
|
||||
> \mathbf{F} &= \sum_{i=1}^n \mathbf{F}_i, \\
|
||||
> \mathbf{\Gamma} &= \sum_{i=1}^n \mathbf{x}_i \times \mathbf{F}_i,
|
||||
> \end{align*}
|
||||
> $$
|
||||
>
|
||||
> with $\mathbf{F}_i$ the conservative external force.
|
||||
|
||||
<br>
|
||||
|
||||
> *Proposition 1*: the total momentum $\mathbf{P}: t \mapsto \mathbf{P}(t)$ is related to the total external force $\mathbf{F}: t \mapsto \mathbf{F}(t)$ by
|
||||
>
|
||||
> $$
|
||||
> \mathbf{P}'(t) = \mathbf{F}(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be adder later.
|
||||
|
||||
> *Proposition 2*: the total angular momentum $\mathbf{J}: t \mapsto \mathbf{J}(t)$ is related to the total external torque $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ by
|
||||
>
|
||||
> $$
|
||||
> \mathbf{J}'(t) = \mathbf{\Gamma}(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ if the internal forces are central forces.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be adder later.
|
||||
|
||||
### Orbital and spin angular momentum
|
||||
|
||||
Considering internal position vectors $\mathbf{r}_i$ relative to the center of mass $\mathbf{r}_i = \mathbf{x}_i - \mathbf{R}$. I propose that the total angular momentum $\mathbf{J}$ can be expressed as a superposition of the orbital $\mathbf{L}$ and spin $\mathbf{S}$ angular momentum components given by
|
||||
|
||||
$$
|
||||
\mathbf{J} = \mathbf{L} + \mathbf{S}.
|
||||
$$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
> *Definition 6*: the orbital angular momentum $\mathbf{L}$ of the system is defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{L} = \mathbf{R} \times \mathbf{P},
|
||||
> $$
|
||||
>
|
||||
> with $\mathbf{R}$ the center of mass and $\mathbf{P}$ the total momentum of the system.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 7*: the spin angular momentum $\mathbf{S}: t \mapsto \mathbf{S}(t)$ of the system is defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{S}(t) = \sum_{i=1}^n \mathbf{r}_i(t) \times m_i \mathbf{r}'_i(t)
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $\mathbb{r}_i$ the internal position.
|
||||
|
||||
Analoguosly the orbital and spin torque may be defined.
|
||||
|
||||
> *Definition 8*: the orbital and spin torque $\mathbf{\Gamma}_{o,s}$ of the system are defined as
|
||||
>
|
||||
> $$
|
||||
> \begin{align*}
|
||||
> \mathbf{\Gamma}_o &= \mathbf{R} \times \mathbf{F}, \\
|
||||
> \mathbf{\Gamma}_s &= \sum_{i=1}^n \mathbf{r}_i \times \mathbf{F}_i,
|
||||
> \end{align*}
|
||||
> $$
|
||||
>
|
||||
> with $\mathbf{R}$ the center of mass, $\mathbf{r}_i$ the internal position and $\mathbf{F}_i$ the conservative external force.
|
||||
|
||||
Similarly, the total torque $\mathbf{\Gamma}$ of the system is the superposition of the orbital and spin torque $\mathbf{\Gamma}_{o,s}$ given by
|
||||
|
||||
$$
|
||||
\mathbf{\Gamma} = \mathbf{\Gamma}_o + \mathbf{\Gamma}_s.
|
||||
$$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
> *Proposition 3*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the orbital angular momentum and let $\mathbf{S}: t \mapsto \mathbf{S}(t)$ be the spin angular momentum. Then we have
|
||||
>
|
||||
> $$
|
||||
> \begin{align*}
|
||||
> \mathbf{L}'(t) &= \mathbf{\Gamma}_o(t), \\
|
||||
> \mathbf{S}'(t) &= \mathbf{\Gamma}_s(t),
|
||||
> \end{align*}
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $\mathbf{\Gamma}_o: t \mapsto \mathbf{\Gamma}_o(t)$ and $\mathbf{\Gamma}_s: t \mapsto \mathbf{\Gamma}_s(t)$ the orbital and spin torque.
|
||||
|
||||
### Energy
|
||||
|
||||
> *Definition 9*: the total kinetic energy $T$ of the system is defined as
|
||||
>
|
||||
> $$
|
||||
> T = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{x}_i'\|^2,
|
||||
> $$
|
||||
>
|
||||
> with $\mathbf{x}_i$ the position of each particle.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition 10*: the orbital and internal kinetic energy $T_{o,r}$ of the system are defined as
|
||||
>
|
||||
> $$
|
||||
> \begin{align*}
|
||||
> T_o = \frac{1}{2} M \|\mathbf{R}\|^2, \\
|
||||
> T_r = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{r}_i'\|^2,
|
||||
> \end{align*}
|
||||
> $$
|
||||
>
|
||||
> with $M$ the total mass, $\mathbf{R}$ the center of mass and $\mathbf{r}$ the internal position of each particle.
|
||||
|
||||
<br>
|
||||
|
||||
> *Proposition 4*: the total kinetic energy $T$ of the system is a superposition of the orbital and internal kinetic energy given by
|
||||
>
|
||||
> $$
|
||||
> T = T_o + T_r.
|
||||
> $$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
> *Proposition 5*: the dynamics of the orbital and kinetic energy $T_o: t \mapsto T_o(t)$ is decoupled
|
||||
>
|
||||
> $$
|
||||
> T_o'(t) = \langle \mathbf{F}, \mathbf{R}'(t) \rangle,
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $\mathbf{F}$ the total external force and $\mathbf{R}$ the center of mass.
|
||||
>
|
||||
> The dynamics of the internal kinetic energy $T_r: t \mapsto T_r(t)$ is not decoupled
|
||||
>
|
||||
> $$
|
||||
> T_r'(t) = \sum_{i=1}^n \langle \mathbf{f}_i, \mathbf{r}_i'(t) \rangle,
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $\mathbf{f}_i$ the sum of both external and internal forces for each particle.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
|
@ -0,0 +1,47 @@
|
|||
# Rotation
|
||||
|
||||
Rotation is always viewed with respect to the axis of rotation, therefore in the following definitions the origin of the position is always implies to be the axis of rotation.
|
||||
|
||||
## Angular momentum
|
||||
|
||||
> *Definition 1*: the angular momentum $L$ of a point mass with position $\mathbf{r}$ and a momentum $\mathbf{p}$ is defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{L} = \mathbf{r} \times \mathbf{p},
|
||||
> $$
|
||||
>
|
||||
> for all $\mathbf{r}$ and $\mathbf{p}$.
|
||||
|
||||
## Torque
|
||||
|
||||
> *Definition 2*: the torque $\mathbf{\Gamma}$ acting on a point mass with position $\mathbf{r}$ for a force $\mathbf{F}$ os defined as
|
||||
>
|
||||
> $$
|
||||
> \mathbf{\Gamma} = \mathbf{r} \times \mathbf{F},
|
||||
> $$
|
||||
>
|
||||
> for all $\mathbf{r}$ and $\mathbf{F}$.
|
||||
|
||||
The torque is related to the angular momentum by the following proposition.
|
||||
|
||||
> *Proposition 1*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the angular momentum of a point mass, then it holds that
|
||||
>
|
||||
> $$
|
||||
> \mathbf{L}'(t) = \mathbf{\Gamma}(t),
|
||||
> $$
|
||||
>
|
||||
> for a constant $\mathbf{r}$ and all $t \in \mathbb{R}$ with $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ the torque acting on the point mass.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the angular momentum of a point mass and suppose $\mathbf{r}$ is constant, then
|
||||
|
||||
$$
|
||||
\mathbf{L}'(t) \overset{\mathrm{def}} = d_t (\mathbf{r} \times \mathbf{p}(t)) = \mathbf{r} \times \mathbf{p}'(t),
|
||||
$$
|
||||
|
||||
by [proposition](momentum.md) we have $\mathbf{p}'(t) = \mathbf{F}(t)$, therefore
|
||||
|
||||
$$
|
||||
\mathbf{L}'(t) = \mathbf{r} \times \mathbf{F}(t) \overset{\mathrm{def}} = \mathbf{\Gamma}(t).
|
||||
$$
|
||||
Loading…
Add table
Add a link
Reference in a new issue