How is velocity defined for different systems?

Question

I'm currently a beginner learning classical mechanics, and I just learned about the Lagrangian formalism. I completely understand the theory behind it, but a lot of times, I'm not able to calculate the velocity correctly for the kinetic energy when doing real exercises. I also have some doubts about defining potential energy.

Let's take an example: Imagine a point mass m hanging from the ceiling near the Earth (meaning there will be GPE), attached to a rigid massless stick of length l.

We will be using polar coordinates: in this case, we will take l as the radius and we will define an angle $\theta$ that is used as a coordinate along with l. Kinetic energy is $T=\frac12mv^2$, and Potential energy is $K=mgl\cos\theta$.

• Are there any general rules that will make defining velocity easy (including systems like pulleys, pendulums, toruses or wheels rotating)? If possible, please show me how can we apply these rules.

Answer 1

The concrete answer to this is that in the lagrangian formalism, for systems where the action is written like $$S=\int L(q_1,…q_n,\dot{q_1},…\dot{q_n},t) dt $$ where $t$ is time, for each generalized coordinate $q_i$ there is a corresponding “velocity” $\dot{q_i} = \frac{dq_i}{dt}$.

A concrete example is $$L(x,y,\dot{x}, \dot{y},t) = \frac{1}{2} m(v_x^2 +v_y^2)+k(x^2+y^2)$$ where the $q_1=x$ and so $\dot{x} = \frac{dx}{dt}$ and $q_2=y$ and so $\dot{y} = \frac{dy}{dt}$.

Answer 2

I am not 100% sure what you mean by "defining" velocity and potential energy, however, if you mean calculating them in terms of generalized coordinates, then a good procedure for doing this is as follows:

First write the kinetic energy and potential energy in terms of a coordinate system you are familiar with like the cartesian one perhaps, then after you have decided on a set of generalized coordinates $x,y,z$; proceed to transform the cartesian coordinates into the new ones $x=x(q_1,...q_n), ...z=z(q_1,...q_n)$, calculate the time derivative of each one (you will need to use the chain rule), $$\dot x={d\over dt}x(q_1,...,q_n),...\dot z={d\over dt}z(q_1,...,q_n).$$ Finally, substitute these quantities into the Lagrangian $L(x,y,z,\dot x,\dot y,\dot z,t)$, wherever appropriate.

Remember to choose generalized coordinates in such a way as to make your problem easier to solve rather than more difficult. This choice is usually indicated by the geometry of the problem, for instance, if the motion is constrained to a doughnut use toroidal coordinates.

Hope this helps, good luck!