---
date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 21:41:45
title: Hamiltonian
aliases: [Hamiltonians, contraction]

---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [symplectic topology](Unsorted/symplectic.md)

---

# Hamiltonian

- For $(X, \omega)$ symplectic, the Poisson bracket is $\ts{f,g} \da -\omega(X_f, X_g)$ where $X_f$ is the Hamiltonian vector field of $f$ satisfying $\iota_{X_f}\omega = df$.
- Contraction: $\iota_{X_f}\omega(\wait) = \omega(X_f, \wait)$.

![](attachments/Pasted%20image%2020220512180947.png)

![](attachments/Pasted%20image%2020220501004705.png)

**Definition (Hamiltonian)**:
A smooth function $H: M \to \RR$ will be referred to as an energy functional or a *Hamiltonian*.
If we have $H: M\cross I \to \RR$, we'll refer to this as a *time-dependent Hamiltonian*, i.e. the time slices $H_t: M \to \RR$ given by $H_t(p) = H(p, t)$ are Hamiltonians. ^1c3cf2

> *Remark:*
> If $(M, \omega)$ is a [interior product](interior product).

# Hamiltonian Vector Field

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**Definition (Hamiltonian vector field):**
Given a smooth functional $H: (M, \omega) \to \RR$, the associated *Hamiltonian vector field* is the unique field $X_H$ satisfying $\omega(X_H, \wait) = dH$.

Remark: Conservation of energy
Since $\omega$ is alternating,
$$
X_H(H) = dH(X_H) = \omega(X_H, X_H) = 0
$$