Number of degrees of freedom:

The number of degrees of freedom of a system refers to the number of independent parameters required to describe the configuration of the system. In the case of two particles moving on a space curve with a constant distance between them, we need to determine the number of independent parameters required to describe their positions.

Particles moving on a space curve:

When two particles are moving on a space curve, their positions can be described using three coordinates each, as we are dealing with three-dimensional space. Therefore, the total number of coordinates required to describe the positions of both particles is 6.

Constant distance between particles:

Given that the distance between the two particles is constant, we can express this constraint mathematically as an equation involving their positions. Let's assume the positions of the two particles as r₁ and r₂. The distance between them can be calculated using the Euclidean distance formula:

|r₁ - r₂| = d

where d is the constant distance between the particles.

Imposing the constraint:

To impose the constraint of constant distance, we substitute the positions of the particles in the equation and solve for one of the positions in terms of the other. Let's say we solve for r₁ in terms of r₂:

|r₁ - r₂| = d

Squaring both sides of the equation:

|r₁ - r₂|² = d²

Expanding and simplifying the equation:

(r₁ - r₂)·(r₁ - r₂) = d²

r₁·r₁ - 2r₁·r₂ + r₂·r₂ = d²

Simplifying further:

|r₁|² - 2r₁·r₂ + |r₂|² = d²

Since the distance between the particles is constant, d² is also constant. Therefore, the equation above represents a constraint on the positions of the particles.

Deducing the number of degrees of freedom:

From the equation above, we can observe that there is only one independent parameter, r₂, required to describe the positions of both particles. Once we choose a value for r₂, the value of r₁ is determined by the constraint equation.

Therefore, the number of degrees of freedom in this system is 1.