# Degrees of freedom in a mechanical system

Here our professor told that the degree of freedom of the system is 2 as we just need 2 angles shown in the figure to completely specify the configuration of the system but this system with a given angle can have two configurations as shown, so I think that it should have total 3 parameters and degree of freedom should be 3.

• – Emilio Pisanty Apr 30 '19 at 7:56
• If the thetas 1 and 2 are fixed, none of the other angles can be adjusted. If there were a third degree of freedom, it could be adjusted without changing theta1 or theta2. Sure, you could let gravity help you go from "up" to "down", and flip it back to "up" if you move it fast enough that momentum can carry it. But the entire system follows what you do with the two attached pivots, none of the others operate independently. – Greg May 9 '20 at 0:14

Let us name the joints in the question as A,B,C,D and E (starting from left bottom of the figure). Now since there is angle $$\theta$$ in each joint, so we can have Total five (non-independent) coordinates in the system. Now consider triangle ABC, in this triangle the sum of all angles at joints must be equal to $$180^0$$, similarly in triangles BCD and DEF.

So we have total 3 constraints in 5 coordinates. so degree of freedom is 2.

• I know that but my question is that with the two given angles, we still can't completely know the configuration of the system. – Shivam Apr 30 '19 at 7:51
• As Wikipedia defines:- In physics, the degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration. – Shivam Apr 30 '19 at 7:53
• Let us assume you stretch both arms( with angles $\theta_1 and \theta_2$). You will notice then you stretch it completely, the angle at the middle of the system will be $180^0$, hence all angles depends on angle $\theta_1 and \theta_2$ – Shine kk Apr 30 '19 at 8:02

By 'degrees of freedom' we mean the number of continuous parameters needed to specify the state of the system; discrete parameters, like the binary choice you've outlined, do not count here.

Ultimately, this refers to the dimension of the manifold of the state space of the system. The two angles in your question are a sub-optimal coordinate chart for this manifold, which appears to "fold back", but there are better coordinate charts that use only two continuous parameters and don't involve the binary choice you've outlined.

• can you give a reference where it tells that we need 'continuous' parameter – Shivam Apr 30 '19 at 7:57
• @Shivam Pretty much any textbook on analytical mechanics uses this notation. – Emilio Pisanty Apr 30 '19 at 8:01
• If degrees of freedom refer to a number it doesn't make sense to say 2 or 3 degrees of freedom. Shouldn't we say just degrees of freedom is 2 or 3? – Antonios Sarikas Mar 8 '20 at 11:28