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Classical theory of a physical system consists of components listed below.
- States of the physical system; `Co-ordinates
- Dynamical Variables
- Laws of Motion
- Forces, Interactions
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.
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
- Hamiltons 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.
The Newtonian formulation has limitations which make it unsuitable for description of several physical systems. Many different formalisms exist which generalise the Newtonian formalism. We mention a few of these here which are useful for systems with a few degrees of freedom. Apart from Newtonian mechanics, other formulations of mechanics are
- Lagrangian formulation
- Hamiltonian and Poisson brackets
- Hamilton Jacobi formulation
Each of the above formalism will be described briefly.
Lagrangian form of classical dynamics
In the Lagrangian approach the state of a system is described by a set of generalized coordinates and velocities. The generalized coordinates are not restricted to be Cartesian. They are a set of independent variables \(q_k\) needed to specify the system completely. The knowledge of these variables \(q_k\), called generalized coordinates, and their time derivatives allows us to compute all dynamical variables of the system. The dynamical laws or the equations of motion are given in terms of a single function of generalized coordinates and momenta, $ {\cal L}(q,\dot{q}, t)$, called Lagrangian of the system. Knowing the Lagrangian, the equations of motion are given by
\begin{equation} \dd{t}\left( \pp[ \cal L]{\dot{q_k} } \right) - \pp[\cal L] {q_k} = 0, \qquad k=1,2,\ldots \label{A:EQ01} \end{equation}
The Lagrangian formalism offers distinct advantages over the Newtonian formalism.
Hamiltonian form of classical dynamics
In the Hamiltonian approach to the classical mechanics, the state of a system at time $t$ is described by giving the values of generalised coordinates and momenta $q_k, p_k, (k=1,\ldots,n)$ at that time. The canonical momentum $p_k$ is defined as derivative of the Lagrangian of the system w.r.t. the generalised velocity $\dot{q}_k$:
\begin{equation} p_k = \pp[L]{\dot{q}_k} \Label{EQ02}. \end{equation}
The interaction is specified by giving Hamiltonian $H(q,p)$ which determines the equations of motion. The equations of motion in the Hamiltonian approach take the form
\begin{equation} \dot{q}_k = \pp[H]{p_k}, \qquad \dot{p}_k=-\pp[H]{q_k}, \qquad k=1,\ldots,n. \Label{EQ03} \end{equation}
Poisson bracket formalism
For two functions $F(q,p), G(q,p)$ of canonical variables, the Poisson bracket $[F,G]_\text{PB}$ is defined as
\begin{equation} [F,G]_\text{PB} = \sum_k \left( \pp[F]{q_k} \pp[G]{p_k} - \pp[F]{p_k} \pp[G]{q_k} \right). \Label{EQ04} \end{equation}
The Hamilton's equations,\eqref{EQ03}, written in terms of Poisson brackets assume the form
\begin{equation} \dot{q}_k = [q_k,H]_\text{PB}, \qquad \dot{p}_k=[p_k,H]_\text{PB}.\Label{EQ05} \end{equation}
In general the time evolution of any dynamical variable is given by
\begin{equation} \dd[F]{t} = [F,H]_\text{PB}. \Label{EQ06} \end{equation}
The classical mechanics has been formulated in several different ways. We mention the Newtonian, the Lagrangian, the Hamiltonian and the Poisson bracket formulations. The Hamiltonian form of mechanics turns out to be the most convenient and suitable for making a transition to quantum mechanics; the Schrodinger and Heisenberg formulations of quantum mechanics requiring an understanding of the Hamiltonian and Poisson bracket formulations. Frequently it is asked if Lagrangian formulation has a role in the quantum theory ? The answer is in affirmative and the Lagrangian plays an essential role in the Feynman path integral approach to quantum mechanics.
For systems consisting of a large number of particles, such as gases, the classical mechanics, in the form used for point particles, , is not very useful. One needs to use statistical methods. While thermodynamics and statistical mechanics were successful in describing the behaviour of a large number of systems very closely, there were some notable disagreements with experiments.
When it comes to the structure of thermodynamics and statistical mechanics, they too have the same structure that is outlined above for mechanics. Except that most of the time we talk about equilibrium situations. Therefore, we do not quite discuss time evolution. A discussion of time evolution of statistical systems falls under the non-equilibrium statistical mechanics and that is very hard subject.
