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Contents
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{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 \EOM. The \EOM{} 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,(Eq 04), 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.
