diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index f77896a..4067fc7 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -135,7 +135,7 @@ nav: - 'Oscillations': physics/mechanics/lagrangian-mechanics/applications/oscillations.md - 'Hamiltonian mechanics': - 'Hamiltonian formalism': physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md - - "Hamilton's equations": physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md + - "Hamilton's equations": physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md # - 'Relativistic mechanics': # - 'Quantum mechanics': - 'Electromagnetism': diff --git a/docs/en/physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md b/docs/en/physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md new file mode 100644 index 0000000..df34e08 --- /dev/null +++ b/docs/en/physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md @@ -0,0 +1,136 @@ +# Equations of Hamilton + +## The Hamiltonian + +> *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 +> +> $$ +> \mathbf{p} = \nabla_{\mathbf{q}'}\mathcal{L}(\mathbf{q},\mathbf{q}',t), +> $$ +> +> for all $t \in \mathbb{R}$. + +We may now pose that there exists a function that meets the inverse, which can be obtained with Legendre transforms. + +> *Theorem 1*: there exists a function $\mathcal{H}: (\mathbf{q},\mathbf{p},t) \mapsto \mathcal{H}(\mathbf{q},\mathbf{p},t)$ such that +> +> $$ +> \mathbf{q}' = \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t), +> $$ +> +> for all $t \in \mathbb{R}$. Where $\mathcal{H}$ is the Hamiltonian of the system and is related to the Lagrangian $\mathcal{L}$ by +> +> $$ +> \mathcal{H}(\mathbf{q},\mathbf{p},t) = \langle \mathbf{q'}, \mathbf{p} \rangle - \mathcal{L}(\mathbf{q},\mathbf{q}',t), +> $$ +> +> for all $t \in \mathbb{R}$ with $\mathcal{L}$ and $\mathcal{H}$ the Legendre transforms of each other. + +??? note "*Proof*:" + + Will be added later. + +## The equations of Hamilton + +> *Corollary 1*: the partial derivatives of $\mathcal{L}$ and $\mathcal{H}$ with respect to the passive variables are related by +> +> $$ +> \begin{align*} +> \nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \nabla_{\mathbf{q}} \mathcal{L}(\mathbf{q},\mathbf{q}',t), \\ +> \partial_t \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \partial_t \mathcal{L}(\mathbf{q},\mathbf{q}',t), +> \end{align*} +> $$ +> +> for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. + +Obtaining the equations of Hamilton + +$$ +\begin{align*} + \mathbf{p}' &= -\nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t), \\ + \mathbf{q}' &= \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t), +\end{align*} +$$ + +for all $t \in \mathbb{R}$. + +> *Proposition 1*: when the Hamiltonian $\mathcal{H}$ has no explicit time dependence it is a constant of motion. + +??? note "*Proof*:" + + Will be added later. + +To put it differently; a Hamiltonian of a conservative autonomous system is conserved. + +> *Theorem 2*: for conservative autonomous systems, the Hamiltonian $\mathcal{H}$ may be expressed as +> +> $$ +> \mathcal{H}(\mathbf{q},\mathbf{p}) = T(\mathbf{q},\mathbf{p}) + V(\mathbf{q}), +> $$ +> +> 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. + +??? note "*Proof*:" + + Will be added later. + +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 $h$. + +> *Proposition 2*: a coordinate $q_j$ is cyclic if +> +> $$ +> \partial_{q_j} \mathcal{H}(\mathbf{q},\mathbf{p},t) = 0, +> $$ +> +> for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. + +> *Proposition 3*: the Hamiltonian is seperable if there exists two mutually independent subsystems. + +??? note "*Proof*:" + + Will be added later. + +## Poisson brackets + +> *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 +> +> $$ +> \begin{align*} +> d_t G(\mathbf{q},\mathbf{p},t) &= \sum_{j=1}^f \Big(\partial_{q_j} G q_j' + \partial_{p_j} G p_j' \Big) + \partial_t G, \\ +> &= \sum_{j=1}^f \Big(\partial_{q_j} G \partial_{p_j} \mathcal{H} - \partial_{p_j} G \partial_{q_j} \mathcal{H} \Big) + \partial_t G, \\ +> &\overset{\mathrm{def}}= \{G, \mathcal{H}\} + \partial_t G. +> \end{align*} +> $$ +> +> for all $t \in \mathbb{R}$ with $\mathcal{H}$ the Hamiltonian and $\{G, \mathcal{H}\}$ the Poisson bracket of $G$ and $\mathcal{H}$. + +The Poisson bracket may simplify expressions; it has distinct properties that are true for any observables. The following theorem demonstrates the usefulness even more. + +> *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 +> +> $$ +> \begin{align*} +> f(\mathbf{q}, \mathbf{p}, t) = c_1, \\ +> g(\mathbf{q}, \mathbf{p}, t) = c_2, +> \end{align*} +> $$ +> +> for all $t \in \mathbb{R}$ with $c_{1,2} \in \mathbb{R}$. Then +> +> $$ +> \{f,g\} = c_3 +> $$ +> +> with $c_3 \in \mathbb{R}$ for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file diff --git a/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md b/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md index 8184d18..4b28727 100644 --- a/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md +++ b/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md @@ -4,7 +4,7 @@ The Hamiltonian formalism of mechanics is based on the definitions posed by [Lag 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. -In Hamilton's formulation the principle is referred to as Hamilton's principle. +In Hamilton's formulation the stationary action principle is referred to as Hamilton's principle. ## Hamilton's principle diff --git a/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md b/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md deleted file mode 100644 index 975b3f4..0000000 --- a/docs/en/physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md +++ /dev/null @@ -1,2 +0,0 @@ -# Hamilton's equations -