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I am a bit confused about the number of degrees of freedom of a simple pendulum. Is there

  1. only angular displacement, or

  2. angular displacement and angular velocity?

Also, I am a bit confused on the number of equations of motion required to describe this system, is it the same number as the number of degrees of freedom?

Qmechanic
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Ee Kin Chan
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2 Answers2

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Actually it depends on context. When we speak of "degrees of freedom" we usually refer to so-called "configuration space" which is the minimum number of variables needed to define the position(s) of the system. In the case of the pendulum it would be ONE. However, there is also an analogous set of dimensions called "state space." This is the (minimum) number of variables needed to describe the system configuration along with the derivatives -- say if you wanted to simulate it. In the case of the pendulum that dimension number would be TWO. For holonomic systems, the number of dimensions of the state space is twice that of the configuration space. But for nonholonomic systems, there can be restrictions in the state space that don't equate to the configuration space.

Holonomic systems that can be described by ODEs will have N FIRST order governing differential equations, where N is equal to the dimension of the state space. Note that the typical way to write the governing equation of a simple pendulum only involves one equation but it is SECOND order. It can be trivially reduced to two first order equations by defining a second variable equal to the derivative of the position angle.

Mariano G
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Degrees of freedom is defined as the number of coordinates we need for analysing the motion of that particle. In a simple pendulum there will be $1$ degree of freedom, which would be its angle, if you're making its equations with rectangular coordinates then still there will be $1$ degree of freedom as $x \& y$ are not independant.

Amit Verma
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