Why are independent variables treated differently in kinetic energy calculations across problems?

Question

In two different problems involving Lagrangian mechanics, I am confused about how independent variables are treated in the kinetic energy calculations. Specifically, in one case, an independent variable is treated as a "parameter" in part of the calculation, while in another case, all independent variables are treated explicitly as variables. Here are the two cases:

Case 1: The Spring Pendulum

Consider a pendulum made of a spring with a mass m at its end. The spring lies in a straight line and has an equilibrium length ℓ. The length of the spring at any time is ℓ + x(t), where x(t) is the radial displacement. The angle θ(t) specifies the orientation of the spring with respect to the vertical. The independent variables are x(t) and θ(t).

The kinetic energy is split into radial and tangential parts:

T = (1/2)m [ẋ² + (ℓ + x)² θ̇²].

In the tangential velocity term (ℓ + x)θ̇, ℓ + x (an independent variable) is treated as an instantaneous "parameter" when calculating the tangential contribution to the kinetic energy, even though x(t) varies with time.

Case 2: Two Hinged Massless Sticks

Two massless sticks of length 2r, each with a mass m fixed at its middle, are hinged at one end. One stick is vertical, while the other is tilted at a small angle θ₂(t) with respect to the vertical. The independent variables are the angles θ₁(t) (for the lower stick) and θ₂(t) (for the upper stick).

The position of the top mass is given by:

(2r sin θ₁ − r sin θ₂, 2r cos θ₁ + r cos θ₂).

Differentiating these coordinates with respect to time gives the velocity components, and the kinetic energy becomes:

T = (1/2)m [(2r cos θ₁ θ̇₁ − r cos θ₂ θ̇₂)² + (−2r sin θ₁ θ̇₁ − r sin θ₂ θ̇₂)²].

Here, both θ₁ and θ₂ are treated explicitly as variables.

My Question:

In the spring pendulum problem, ℓ + x (an independent variable) is treated as a parameter in the tangential kinetic energy term, while in the hinged sticks problem, all independent variables (θ₁ and θ₂) are treated explicitly as variables when calculating the kinetic energy.

What is the general principle that determines whether an independent variable should be treated as a parameter or explicitly as a variable in such problems?

Answer 1

you start always with the position vector to the masses .

Case I

$$\vec P_m=\begin{bmatrix} X \\ Y \\ \end{bmatrix}=\left[ \begin {array}{c} \left( l+x \right) \sin \left( \phi \right) \\ - \left( l+x \right) \cos \left( \phi \right) \end {array} \right] $$ the generalized coordinates are $~x(t)~,\phi(t)~$

from here the velocity $$v^2=\dot X^2+\dot Y^2 $$ and the kinetic energy $$T=\frac m2\,v^2+\frac k2\,(l+x)^2$$

Case II

Position to $~m_1$

$$\vec P_1=\begin{bmatrix} X_1 \\ Y_1 \\ \end{bmatrix}=\left[ \begin {array}{c} r\sin \left( \phi_{{1}} \right) \\ r\cos \left( \phi_{{1}} \right) \end {array} \right] $$

Position to $~m_2$ $$\vec P_2=\begin{bmatrix} X_2 \\ Y_2 \\ \end{bmatrix}=2\,\vec P_1+\left[ \begin {array}{c} r\sin \left( \phi_{{2}} \right) \\ r\cos \left( \phi_{{2}} \right) \end {array} \right] $$

the generalized coordinates are $~\phi_1(t)~,\phi_2(t)~$

thus the velocities are

$$v_1^2=\dot X_1^2+\dot Y_1^2 \\ v_2^2=\dot X_2^2+\dot Y_2^2$$

and the kinetic energy

$$T=\frac m2\,(v_1^2+v_2^2)$$

What is the general principle that determines whether an independent variable should be treated as a parameter or explicitly as a variable in such problems?

the only independent variable are the generalized coordinates , those are implicit time dependent.