The structure of physical theories

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Classical theory of a physical system consists of components listed in Table below :-1 & Description of physical system of interes

 At first we shall briefly explain each item in the table and later discuss them by means of examples of different physical systems.

Physical states: By state of a physical system one means ways of specifying complete information about the system at a time t

Dynamical variables: The dynamical variables of a classical system are functions of state of the system and can be computed when the state has been specified.

Laws of motion: Not only we are interested in knowing about a system at a given time, we also want to know how the system changes with time. In order to describe behaviour of a system under time evolution one needs to know the laws of motion. Several different forms of the laws of motion are available for mechanical systems.

• Newton's laws
• Lagrangian equations of motion
• Hamilton's equations
• Poisson bracket formalism

When applicable, all the above formalism are equivalent.

Interactions: A classical description is completed by specifying the forces of the interactions of the system. It should be remarked that while the laws of motion are general and are applicable to a wide variety of physical systems, the nature of forces or the explicit form of interactions differs from system to system. The interactions are specified by by giving explicit expressions for forces acting on the bodies in the system.

Classical Systems: Some examples of classical systems of interest are

• System of point particles moving in a force field.
• One or more point particle moving on a surface of a sphere.
• Rigid body
• Vibrating string or a spring
• Electromagnetic waves
• Charged particles interacting with electromagnetic fields

System of point particles:  You are all familiar with the newtonian mechanics from your school days. A complete specification of state of a particle requires three position coordinates and three velocities. The dynamical variables are experimentally measurable quantities such as energy, momentum and angular momentum. They are functions of coordinates and velocities. The equations of motion are given by the second law. For a system of point particles, the behaviour of each particle is governed by the equation of motion  \begin{equation} m_\alpha \DD[\vec{r}_\alpha]{t} = \vec{F}_\alpha \label{EQ01} \end{equation} Note that the \EOM {} are a set of second order differential equations in time. Therefore one needs to know the values of position and velocities at a time in order $t_0$ to be able to predict the state of the system at a later time. One also needs to have information about all the forces acting on the particle. The Newton's laws require that the equations of motion be set up using the Cartesian coordinates to describe the particle. For a system consisting of several particles one needs to knows all the forces, including the forces of constraint. In order to set up equations of motion in a non Cartesian system of coordinates one has to start from the Cartesian system and take into account of the constraints. In general finding solution may require a change variables from Cartesian coordinates to a new set of coordinates. Thus, for example, for a bead sliding on a sphere one should change from Cartesian coordinates to polar coordinates.

Waves: The state of a vibrating string is described completely by specifying the displacement and velocity of the string at each point. The vibrations are also governed by the Newton’s Laws which can be used to derive the wave equation giving the propagation of waves in a medium.

Charged particles and radiation: The systems consisting of charged particles interacting with electromagnetic fields are very important. These are governed by the Maxwell’s equations and the Lorentz force equation. The state is described by specifying position and momenta of the charged particles and the electromagnetic fields, or the scalar and vector potentials, at all points in the space