Physical Systems

Physical System

Definition: Physical System

A physical system is any set of physical objects which we have chosen to model and examine using physics.

Definition: Environment

The environment of a physical system are all physical objects which are not part of the system.

Definition: Isolated System

A physical system is isolated if the objects inside it cannot interact with the objects in the environment.

Parametrization

We are rarely interested in all properties of a given physical system. Much more often we are considered with just a subset of its properties which are relevant to solving a given problem. However, to use mathematics and physics in order to solve problems, we need a way to quantify these properties mathematically by using numbers, vectors, etc.

Definition: Parameter

A parameter of a physical system is any one of its characteristics which can be quantified mathematically.

Example

Imagine a physical system which consists of a room full of people. The average age of these people is a parameter of the system because we can quantify it using a number.

Example

Imagine a physical system which consists of a person staring at a wall. The direction from which they are looking at a wall is a parameter of the system because we can quantify it using a vector.

Definition: Independent Parameters

A set of $n$ parameters is independent if the value of each parameter does not dependent on the values of the other $n-1$ parameters.

Parameters can be pretty arbitrary, but we are usually only interested in those which are useful for physical predictions and obey certain laws such as position, momentum, temperature, pressure, etc. Moreover, we are usually interested only in parameters which are numbers because each vector parameter can be broken down into numbers which are its components.

Definition: Parametrization

A parametrization $(P_1,\dots ,P_n)$ of a physical system is any set of parameters $P_1,\dots ,P_n$ which we have chosen to examine and consider relevant.

Definition: Degrees of Freedom

The number of degrees of freedom is the smallest number $k\le n$ of parameters (without loss of generality $P_1,\dots ,P_k$) whose values need to be known in order to determine the values of all parameters $P_1,\dots ,P_n$, i.e. there exist $n$ functions $f_1,\dots ,f_n$ such that

$$P_i=f_i(P_1,\dots ,P_k)i\in \{1,\dots ,n\}$$

Definition: Equivalence of Parametrizations

Two parametrizations $(P_1,\dots ,P_n)$ and $(P_1^{\prime },\dots ,P_m^{\prime })$ of a physical system are equivalent if it is possible to determine the values of $P_1,\dots ,P_n$ given the values of $P_1^{\prime },\dots ,P_m^{\prime }$ and vice versa, i.e. there exists a bijection $f$ such that

$$f(P_1,\dots ,P_n)=(P_1^{\prime },\dots ,P_m^{\prime })f^{-1}(P_1^{\prime },\dots ,P_m^{\prime })=(P_1,\dots ,P_n)$$

Definition: Physical State

Given a parametrization $(P_1,\dots ,P_n)$ of a physical system, we call any $n$-tuple $(v_1,\dots ,v_n)$ of values for $P_1,\dots ,P_n$ a (physical) state of the system.

Definition: Phase Space

The set of all physical states is known as phase space.