$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}${}$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}${}$\newcommand{\ket}[1]{|#1\rangle}$ {}$\newcommand{\bra}[1]{\langle #1|}$

Learning Goals |

In this lesson you will learn

- about constraints, degrees of freedom and definition of generalized coordinates
- use of D' Alembert's principle to eliminate the forces of constraints
- use of independence of generalized coordinates to arrive at a new form of equations of motion
- For systems with forces derivable from a potential, or a generalized potential, the equations of motion take the Euler Lagrange form: \begin{equation} \dd{t}\Bigr(\pp[L]{q\dot{q}_k}\Bigr) - \pp[L]{q_k}=0. \end{equation}
- For a system of particles with forces derived from a potential energy function \(V({\mathbf q})\), the Lagrangian is given by \(L=T-V\).

### Recall and Discuss

- Newton's equations of motion
- Holonomic constraints are those given by equations of the form \[\phi_k(\vec{r}_1, \ldots, \vec{r}_N, t)=0, k=1,\ldots \] where \(\phi_k(\vec{r}_1, \ldots, \vec{r}_N, t)\) are functions of the position vectors \(\vec{r}_\alpha\) of the particles making up the system. Our discussion will be concerned exclusively with holonomic constraints.
- The forces of constraints have the property that the total work done by all forces of constraints on a system under a small virtual displacement is zero.

\(\S1\) Constraints, Degrees of freedom, Generalized coordinates

\(\S3.1\) Constraints

\(\S3.2\) Degrees of freedom

\(\S3.3\) Generalized coordinates

\(\S2\) Euler Lagrange Form of Equations of Motion\(\S4.1\) Start with Newton's Second Law

\(\S4.2\) Separate out the forces due to constraint

\(\S4.3\) Eliminating forces of constraints

\(\S4.4\) Express total work done in terms of generalised coordinates

\(\S4.5\) Bring in kinetic Energy

\(\S4.6\) Define generalized force

\(\S4.7\) The variations in generalized coordinates are independent

\(\S3\) Lagrangian for Common Systems

\(\S5.1\) Case I : Conservative forces

\(\S5.2\) Case II : Generalized Potential Dependent on generalized velocities

### End Notes

\(\odot\) Point Transformations:

Lagrangian, \(L{'}\), in terms of new coordinates \(\mathbf Q_k\) can be obtained by expressing the coordinates \(\mathbf q_k\) in terms of new coordinates. Thus \begin{equation} L {'} (\mathbf Q, \dot{\mathbf Q}, t) = L \big(\mathbf q(\mathbf Q),\dot{\mathbf q}(\mathbf Q, \dot{\mathbf Q}\big), t\big) \end{equation}

\(\odot\) Want to learn related skills? This will be critical later for problem solving Optional now, but recommended for problem solving later

- Simple Examples of constraints
- Counting the number of independent variables to be determined
- Examples of choosing generalized coordinates.

\(\odot\) Want to dig deeper? or What you missed here?

Most of this will not be required later . Here is a short list of what you can explore

- The constraints are classified according to whether or not they are scleronomic or rheonomic,

holonomic or nonholonomic, and conservative or nonconservative. - Principle of virtual work illustrated with examples
- d' Alembert's principle and generalized coordinates with an application.

To learn all this and more, see \cite{Calkin Ch II [2] ; Greiner Ch 14, 15 [5];

References |

- Landau, L. D. and Lifshitz, E. M.,
*Mechanics*, Volume 1 of Course of Theoretical Physics, Butterworth-Heinenann Linacre House, Jordan Hill, Oxford 3rd Ed.(1976) -
Calkin, M.G. Lagrangian and Hamiltonian Mechanics, World Scientific Publishing Co. Pte. Ltd. (1996);

This book has several interesting problems on d' Alembert's principle. -
Whitaker E. T. A

*Treatise on the Analytical Dynamics of Particles and Rigid Bodies*, Dover Publications (1944) -
Rana, N. C. and Jog P. C.,

*Mechanics*Tata McGraw Hill Publishing Co, New Delhi (1991). -
Greiner, Walter,

*Clasical Mechanics -- Systems of Particles and Hamiltonian Dynamics,*Springer New York (2003)

This book has a lot of details about types of constraints. It has examples of applications of principles of virtual work and d' Alembert's principle.