# What is a dynamical variable

I see the phrase 'dynamical variable' in several of my texts and they don't really define it. A google search doesn't yield much either.

I have gathered that a dynamical variable can be represented in terms of position and momentum, and examples are velocity and kinetic energy, but what is the definition?

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I know no rigorous definition, but not every dynamic variable can be represented in terms of position and momentum. Spin, for instance. – Siyuan Ren Jan 20 '12 at 4:49
Well, then please tell me a colloquial definition. – StuartHa Jan 20 '12 at 5:15
Physical observable intrinsic to a system (so time is not a dynamical variable). This is just my understanding, no guarantee. – Siyuan Ren Jan 20 '12 at 5:18

In classical mechanics, it is true that the dynamical variables can be represented in terms of position and momentum. The terminology « dynamical variable » in fact comes from classical Hamiltonian mechanics. For example, parts of a mechanical system in Hamiltonian or Lagrangian mechanics.

Quantum Mechanics borrowed much of this philosophy from the Hamiltonian formulation of classical mechanics (which is related to the Lagrangian formulation of dynamics, and even Quantum Field Theory borrowed much from Lagrangian Dynamics). But in Quantum Mechanics things have to be more abstract since we have many non-classical quantities such as spin.

In Classical Mechanics, the state of the system is given by the three position coordinates and the three momenta along the coordinate axes of each particle. If there are $n$ particles, this is 6$n$ coordinates for the entire state. These are themselves variables, but any function of these coordinates is also a « dynamic variable » of the system.

In Quantum Mechanics one has a kind of analogy and there is a formal definition in this case: if the space of all states of the quantum system is $H$, a Hilbert space, the dynamic variables are the self-adjoint (Hermitian) operators on $H$. These are no longer functions of the states, but operators on the states. But a function can be thought of as a non-commutative kind of function, or, rather, the non-commutative generalisation of a function, so the passage from classical mechanics to quantum mechanics has often been expressed as the passage from commutative dynamical variables to non-commutative dynamical variables.

The physical meaning, either way, of a dynamical variable is that it is any physical quantity of the state which can be measured. A synonym for « dynamical variable » is « observable ».

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A dynamical variable is a mathematical variable describing a physical system that depends on time; the dependence of systems in Nature on time is what is referred to as "dynamics".

In various theories, like classical mechanics or quantum mechanics, dynamical variables are functions of $x,p$, or may depend on the spin, or become operators (everywhere in quantum mechanics) but these context-dependent things have nothing to do with the adjective "dynamical".

Dynamical variables should be contrasted with non-dynamical ones, like the total charge of the Universe, which don't depend on time. In a similar way, "dynamical equations" are those that include time derivatives while "non-dynamical equations" (e.g. constraints) are those that don't contain time derivatives.

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In the popular book Quantum Reality (which explores the various physical interpretations of QM - whether the reality behind QM's formalism is contextual, observer-created, many-worlds, etc), "static variable" means one that is precise, objectively knowable, unchanging and does not depend on the state of knowledge of some other variable. Mass, charge and spin magnitude (not spin orientation) are static variables. They can be known to arbitrary precision, are persistent (don't change from measurement to measurement) and do not depend for their precision or knowability on the measured value of some conjugate attribute. They do not have a non-commutative relationship with another variable, the measurement of which blurs out the possible knowledge one can have of the first variable. Static variables, in short are not subject to the uncertainty relation, which, after all, holds only between conjugate variables (Fourier duals). "Dynamic variables", then, are those which can change or take on a range of magnitudes, those which depend for their precision on the degree to which the state of a conjugate variable is known, etc. In short, dynamic variables are those with a canonical conjugate, those to which the uncertainty relation applies.

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I think that- Dynamical Variables are those kind of variable which are changes with time. Like position, momentum, etc are some examples. Mass isn't a dynamical variable because it doesn't change with time.

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Did you read Lubos's answer? – John McVirgo Jun 21 '12 at 14:02