Degree of Freedom and Generalized Coordinates in Classical Mechanics

The concept of degrees of freedom is crucial in describing and analyzing the behavior of physical systems. It’s used to determine the minimum number of coordinates needed to specify the complete state of a system.

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Degree of Freedom

The degree of freedom of a mechanical system is the minimum number of independent variables without violating constraints of the motion required to completely describe its motion.

In a mechanical system, each degree of freedom corresponds to a possible way in which the system can change its configuration. For example:

Particle in 3D Space: A single particle that can move freely in three-dimensional space has three degrees of freedom, one for each spatial dimension (x, y, and z).

Rigid Body: A rigid body that can rotate and translate in 3D space has six degrees of freedom: three for translation and three for rotation.

Simple Pendulum: A simple pendulum has one degree of freedom, which is the angle it makes with the vertical.

Free Particle

When a single free particle moves in space, it has 3 degrees of freedom. e.g. a bird flying in the sky.

2 Free Particles

When a system of 2 free particle moves in space, it has 3.2=6 degrees of freedom.

N Free particles

When a system of N free particle moves in space, it has 3N degrees of freedom.

N Free Particles with k Constraints

A system of N free particles subjected to k constraints, has $f=3N-k$ degrees of freedom.

A Bead Sliding on a Wire

Here, No. of free particles (N) = 1

No. of constraints (k) = 2

1^st constraint = the particle cannot move along y-axis

2^nd constraint = the particle cannot move along z-axis

Degree of freedom = 3N – k = 3.1 – 2 = 1

Motion of a Simple Pendulum

Here, No. of free particles (N) = 1

No. of constraints (k) = 2

1^st constraint = length of the string is constant

2^nd constraint = the bob moves in a plane

Degree of freedom = 3N – k = 3.1 – 2 = 1

Motion of a Double Pendulum

Here, No. of free particles (N) = 2

No. of constraints (k) = 4

1^st constraint = length of the 1^st string is constant

2^nd constraint = length of the 2^nd string is constant

3^rd constraint = 1^st bob moves in a plane

4^th constraint = 2^nd bob also moves in plane

Degree of freedom = 3N – k = 3.2 – 4 = 2

Motion of Three Particles Connected by Three Straight Lines

Here, No. of free particles (N) = 3

No. of constraints (k) = 3

1^st constraint = length between 1^st and 2^nd particle is constant

2^nd constraint = length between 2^nd and 3^rd particle is constant

3^rd constraint = length between 3^rd and 1^st particle is constant

Degree of freedom = 3N – k = 3.3 – 3 = 6

Sl. No.	No. of particles	Degree of freedom
1	For 1 particle	3
2	For 2 particles	6
3	For N particles	3N
4	When a single particle moves in space	3
5	When a single particle moves in plane	2
6	When a single particle moves in straight line	1

Generalized Coordinates

The minimum number of independent coordinates or variables which is required to describe the motion of a dynamical system is known as generalized coordinates.

For a system of N particles and for k constraints and d dimension, the number of independent coordinates (f) =dN-k. These ‘f’ number of minimum independent coordinates required to describe configuration and motion of a mechanical system are called generalized coordinates and are denoted by $q_i (i=1,2,3,\dots,f)$ .

Generalized coordinates can be any set of parameters that equivalently specify a point in space.

We can express cartesian coordinates $r_i$ in terms of generalized coordinates in the form

$\overrightarrow{r_i}=\overrightarrow{r_i}((q_1, q_2, q_3, ... , q_f, t)$

Simple Pendulum

A simple pendulum consists of a point mass called bob suspended at the lower end of a massless and inextensible string of constant length (l) fixed at its upper end to a fixed rigid support.

Here, No. of free particles (N) = 1

No. of constraints (k) = 2

1^st constraint = length of the string is constant

2^nd constraint = the bob moves in a plane

Generalized Coordinates = 3N – k = 3.1 – 2 = 1

∴ Generalized Coordinate is given by θ.

Advantages of Generalized Coordinates

Generalized coordinates are not limited to Cartesian coordinates. They allow for the use of alternative coordinate systems that may be more suitable for describing the configuration of a specific system.
Generalized coordinates provide a natural and convenient way to handle constraints in classical mechanics. By utilizing appropriate generalized coordinates, the constraints can be expressed as equations, simplifying the analysis and allowing for the incorporation of constraints directly into the equations of motion.
Generalized coordinates enable a more concise and elegant representation of complex systems. By appropriately choosing the generalized coordinates, the degrees of freedom and independent variables necessary to describe the system can be significantly reduced.
Generalized coordinates are closely tied to the concept of energy and the Lagrangian formulation in classical mechanics. The Lagrangian function, which is expressed in terms of generalized coordinates and their derivatives, simplifies the derivation of equations of motion using the principle of least action, providing a powerful and systematic approach to solving problems in classical mechanics.
Generalized coordinates allow for system-specific descriptions that are tailored to the unique properties and geometry of the system under study.
Many physical systems naturally possess non-Cartesian characteristics. Using generalized coordinates allows for a seamless transition between different coordinate systems, facilitating the analysis and understanding of systems with curved or non-rectangular geometries.