The following are the core principles that define the _Hamiltonian_ [formulation](Mathematical%20formulations%20of%20classical%20mechanics.md) of classical [mechanics.](Mechanics%20(index).md) These are ordered in such a way where each principle builds upon prior principles in this list. 1. The _state_ of a system in classical mechanics is completely described by points, $(q_i,p_i)$ in a $2d$ dimensional space referred to as a [phase space.](Phase%20space.md#Position-momentum%20phase%20space) ^14f211 2. Each _[observable](Observable%20(classical%20mechanics).md)_ (i.e. measurable quantity) is given by a [function](Analysis%20(index).md#Functions) in phase space, $f(q_i,p_i).$ ^53ab52 3. The total [energy](Mechanics%20(index).md#Energy) of a system is given by an [observable,](Hamiltonian%20mechanics.md#^53ab52) $\mathcal{H}(q_i,p_i),$ referred to as the [Hamiltonian.](Hamiltonians.md) ^950f4f 4. The [state](Hamiltonian%20mechanics.md#^14f211) changes in time in accordance with [Hamilton equations of motion](Hamilton%20equations%20of%20motion.md) for a points in phase space, $(q_i,p_i)$ such that  and  where $\mathcal{H}(q_i,p_i)$ is the [Hamiltonian.](Hamiltonian%20mechanics.md#^950f4f) ^efbe60 %%Consider also precisely defining what states and observables are in classical physics%% --- # Recommended reading These principles are often given in contrast with postulates in [quantum mechanics](Quantum%20Mechanics%20(index).md), which require a distinct shift in how measurement is conceptualized along with being based on modified dynamical equations. In addition, the first [formulation of quantum mechanics](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md) introduced to students tends to be a modification of [Hamiltonian mechanics.](Hamiltonian%20mechanics.md) Thus we find these principles listed in the following introductions to quantum mechanics: * [Woit, Peter. _Quantum Theory, Groups and Representations: An Introduction_, Springer, 2017](Woit,%20Peter.%20Quantum%20Theory,%20Groups%20and%20Representations%20An%20Introduction,%20Springer,%202017.md)(pgs 164-165) Here these [principles](Hamiltonian%20mechanics.md) are presented as a set of axioms in order to motivate an introduction to Poisson brackets and symplectic geometries. However, these aren't axioms in a purely mathematical sense and the [equations of motion](Hamiltonian%20mechanics.md#^efbe60) are also defined by more fundamental principles. * [Shankar, R., _Principles of Quantum Mechanics_, Plenum Press, 2nd edition, 1994.](Shankar,%20R.,%20Principles%20of%20Quantum%20Mechanics,%20Plenum%20Press,%202nd%20edition,%201994..md), pgs. 115-116. Here postulates are introduced for classical mechanics in order to contrast them with quantum mechanics in a side-by-side comparison. #Mechanics #Mechanics/MathematicalFoundations