Hamilton equations - Aetherwisp

The Hamilton equations are a set of equations that derive the motion of a physical system from its Hamiltonian $H$ :

\dot{q}_{i}=\frac{ \partial H }{ \partial p_{i} },\quad \dot{p}_{i}=-\frac{ \partial H }{ \partial q_{i} }

$q_{i}$ are the generalized coordinates and $p_{i}$ are their conjugate momenta. Just like the Lagrange equation, in a system of $n$ degrees of freedom, there are $2n$ Hamilton equations. Solving all of them determines the trajectory of the system. The $2n$ equations can be written in compact form as

\dot{\mathbf{x}}=\mathrm{E}\nabla_{\mathbf{x}}H(\mathbf{x},t)=\{ \mathbf{x},H \}

where $\mathbf{x}=(\mathbf{q},\mathbf{p})$ are canonical coordinates and $\mathrm{E}$ is the $2n$ -dimensional standard symplectic matrix. $\nabla_{\mathbf{x}}$ represents the Gradient as computed in canonical coordinates, which is the to say the phase space gradient. The curly braces are the Poisson brackets.

A system whose dynamics are governed by the Hamilton equations is called a Hamiltonian system.

Derivation from the Lagrange equation#

The Lagrange equation is a second order differential equation of the form $\ddot{\mathbf{q}}=f(q,\dot{q},t)$ , which can be also written in as a linear system of first order equations:

\begin{cases} \dot{\mathbf{q}}=\boldsymbol{\eta} \\ \dot{\boldsymbol{\eta}}=f(q,\eta,t) \end{cases}

Typically, we have $n$ Lagrange equations, or $2n$ first order equivalents. In terms of the Lagrangian, each equation reads

\frac{d}{dt} \underbrace{ \frac{ \partial L }{ \partial \dot{q}_{i} } }_{ p_{i} } - \frac{ \partial L }{ \partial q_{i} } =0

where $i=1,\ldots,n$ and $p_{i}$ are the conjugate momenta. If the determinant of the Jacobian of this quantity is nonzero, so

\det\left( \frac{ \partial }{ \partial \dot{q}_{j} } \frac{ \partial L }{ \partial \dot{q}_{i} } \right)\neq0

we can invert it to extract $\dot{q}_{k}$ as

\dot{q}_{k}=u_{k}(p,q,t)

for some function $u_{k}$ . This comes at the cost of turning $p\equiv(p_{1},\ldots,p_{n})$ into a variable. To motivate this, let's look at an example.

> which can be inverted as > $$\dot{\mathbf{q}}=\mathrm{a}^{-1}(\mathbf{p}-\mathbf{b}(q,t))

In this case, the functions $u_{k}$ are packaged as the inverse of the kinetic matrix. In fact, all mechanical systems follow this shape.

With this said, we can make the following proposition.

> is equivalent to the **Hamilton equations** > $$\begin{cases} > \dot{q}_{i}=\frac{ \partial H }{ \partial p_{i} }(p,q,t) \\ > \dot{p}_{i}=-\frac{ \partial H }{ \partial q_{i} } (p,q,t) > \end{cases}$$ > where > $$H(p,q,t)\equiv\left.{(\mathbf{p}\cdot \dot{\mathbf{q}}-L(q,\dot{q},t))}\right|_{\dot{q}_{i}=u_{i}(p,q,t)}

is the Hamiltonian. The following is also true:

> [!quote]- Proof > Start from the definition of $H(p,q,t)$ and take the [[Differential|total derivative]]: > $$dH=\sum_{l=1}^{n} \left( \frac{ \partial H }{ \partial p_{l} } dp_{l}+\frac{ \partial H }{ \partial q_{l} } dq_{l} \right)+\frac{ \partial H }{ \partial t } dt\tag{1}

Now also take the differential with respect to the definition $H=\mathbf{p}\cdot \dot{\mathbf{q}}-L$ evaluated in $\dot{\mathbf{q}}=\mathbf{u}(p,q,t)$ :
$dH&=\sum_{l=1}^{n} (p_{l}du_{l}+u_{l}dp_{l})-\sum_{l=1}^{n} \underbrace{ \frac{ \partial L }{ \partial \dot{q}_{l} }(q,u,t) }_{ p_{l} }du_{l}-\sum_{l=1}^{n} \frac{ \partial L }{ \partial q_{l} } dq_{l}-\frac{ \partial L }{ \partial t } dt \\ &=\cancel{ \sum_{l=1}^{n} p_{l}du_{l} }+\sum_{l=1}^{n}u_{l}dp_{l}-\cancel{ \sum_{l=1}^{n} p_{l}du_{l} }-\sum_{l=1}^{n} \frac{ \partial L }{ \partial q_{l} } dq_{l}-\frac{ \partial L }{ \partial t } dt \tag{2} \end{align}$

> By comparing terms with the same differentials in both $(1)$ and $(2)$ we get > $$u_{i}=\dot{q}_{i}=\frac{ \partial H }{ \partial p_{i} },\quad \frac{ \partial L }{ \partial q_{i} }=\dot{p}_{i} =-\frac{ \partial H }{ \partial q_{i} } ,\quad \frac{ \partial L }{ \partial t }=-\frac{ \partial H }{ \partial t }

which proves our point.

We may also write the Hamilton equations as a first order differential equation. See the equations as as vectors:

\begin{pmatrix} \dot{q}_1 \\ \vdots \\ \dot{q}_n \\ \dot{p}_1 \\ \vdots \\ \dot{p}_n \end{pmatrix} = \begin{pmatrix} \frac{\partial H}{\partial p_1} \\ \vdots \\ \frac{\partial H}{\partial p_n} \\ -\frac{\partial H}{\partial q_1} \\ \vdots \\ -\frac{\partial H}{\partial q_n} \end{pmatrix}

This can be expressed compactly as:

\dot{\mathbf{x}} = \underbrace{ \begin{pmatrix} 0 & -\mathrm{I}_{n} \\ \mathrm{I}_{n} & 0 \end{pmatrix} }_{\mathrm{E}} \underbrace{ \begin{pmatrix} \frac{\partial H}{\partial p} \\ \frac{\partial H}{\partial q} \end{pmatrix} }_{\nabla_{\mathbf{x}} H}

and so

\boxed{\dot{\mathbf{x}}=\mathrm{E}\nabla_{x}H(\mathbf{x},t)}

Derivation from the least-action principle#

Above we found the Hamilton equations from the Lagrange equation, which itself was derived from Newton's second law. However, the Lagrange equation can also be found from the least action principle: the motion is the one that minimizes the action. The Hamilton equations can too be derived from the least action principle, and it involves minimizing the action Functional:

S[\mathbf{p},\mathbf{q}]=\int_{t_{1}}^{t_{2}}\left( \sum_{i=1}^{n} p_{i}(t)\dot{q}_{i}(t)-H(p(t),q(t),t) \right)dt

This is because the Lagrangian can be expressed in terms of the Hamiltonian ( $L=\mathbf{p}\cdot \dot{\mathbf{q}}-H$ ), so a simple substitution gives an equivalent definition for $H$ instead of $L$ .

In differential geometry#

Recall how the set of possible positions of a system is its configuration space $Q$ . When we add the possible velocities, as we do in the Lagrangian case, we get the so-called tangent bundle $TQ$ of the configuration space. Here in the Hamiltonian case, we add the momentum instead. It can be found that momenta are cotangent vectors to a given point in the configuration space (like how velocities are tangent vectors), and thus the set of all momenta for a given configuration $P$ forms a cotangent space $T^{*}_{P}Q$ (instead of a tangent space $T_{P}Q$ ) and the union of all of these spaces makes a cotangent bundle $T^{*}Q$ (instead of a tangent bundle $TQ$ ). This cotangent bundle is known as the phase space.

Changing coordinates#

As usual, we may want to change system of coordinates to make our life easier, such as going from Cartesian coordinates to spherical coordinates using some Coordinate transformation. However, we need to be a little careful when dealing with the Hamilton equations. Say our original coordinates satisfy the Hamilton equations. Then, we transform to a different set of coordinates. The new ones are not guaranteed to satisfy them. We say that the new ones are not guaranteed to preserve the form of the Hamilton equations.

Since changing coordinates is nevertheless very useful, we want some way of identifying which transformations do preserve the form. These transformations are called canonical transformations.