Elliptic functions in parametric equations

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SUMMARY

The discussion centers on solving a problem involving elliptic functions in parametric equations, specifically using the functions cn(λ), sn(λ), and am(λ) with k² = 1/2. The user seeks to find the distance traveled by a point moving in three-dimensional space, where the speed vector is parallel to the vector (-4, 3, √2). The velocity vector is expressed as v = dp/dt = (-4·sn(λ)·dn(λ), 3·cn(λ)·dn(λ), dn(λ))·(dλ/dt), with the goal of determining the relationship between the parameters to solve for the distance.

PREREQUISITES
  • Understanding of elliptic functions, specifically cn, sn, and am functions.
  • Familiarity with parametric equations and their derivatives.
  • Knowledge of vector calculus, particularly velocity and speed vectors.
  • Basic understanding of the complete elliptic integral of the first kind.
NEXT STEPS
  • Study the properties and applications of elliptic functions, focusing on cn, sn, and dn.
  • Learn how to compute the distance traveled along parametric curves.
  • Explore vector calculus techniques for analyzing speed and direction in three dimensions.
  • Investigate the relationship between elliptic integrals and parametric equations in motion.
USEFUL FOR

Mathematics students, physicists, and engineers who are working with parametric equations and elliptic functions, particularly in the context of motion analysis and vector calculus.

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Hey guys so I had trouble with this problem, I hope it's not too long and you can help me out.

Homework Statement


A point moves following (starting at λ = 0 and increasing):
x = 4·cn(λ)
y = 3·sn(λ)
z= am(λ)
Find the distance the point has traveled when the speed vector is parallel to (-4, 3, √2).

Homework Equations


The elliptic functions are of the first kind with k2 = 1/2.

The Attempt at a Solution


I thought of the following:
v = dp/dt = (-4·sn(λ)·dn(λ), 3·cn(λ)·dn(λ), dn(λ))·(dλ/dt) = c·(-4, 3, √2) (with c a constant real number)
But then I don't know what to do next...

Thank you so much for your help!
 
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Anyone could at least give me a clue on what my next move should be? Thanks.
 

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