When I was reading the calculus of Variations chapter in Classical Mechanics by John R. Taylor, he mentioned that cart rolling back and forth on cycloid-shaped track are exactly isochronous (period perfectly independent of amplitude). As I was trying to solve this by the original integral, I find it hard to compute and the normal Casio type calculator cannot solve it as well, also I sought help from websites like Integral Calculator, the result function is not independent of the starting angle I set. Can someone help me, great great thanks!
2026-03-30 04:57:34.1774846654
How to prove the brachistochrone to be Isochronous
52 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in CALCULUS
- Equality of Mixed Partial Derivatives - Simple proof is Confusing
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- Proving the differentiability of the following function of two variables
- If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Number of roots of the e
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- How to prove $\frac 10 \notin \mathbb R $
- Proving that: $||x|^{s/2}-|y|^{s/2}|\le 2|x-y|^{s/2}$
Related Questions in CALCULUS-OF-VARIATIONS
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Weak formulation of Robin boundary condition problem
- Why is the index of a harmonic map finite?
- Variational Formulation - inhomogeneous Neumann boundary
- Relationship between Training Neural Networks and Calculus of Variations
- How to prove a Minimal Surface minimizes Surface Tension
- Derive the Euler–Lagrange equation for a functional a single variable with higher derivatives.
- Does the covariant derivative commute with the variational derivative?
- Derivative of a functional w.r.t. a single point?
- calculus of variations with double integral textbook?
Related Questions in CLASSICAL-MECHANICS
- An underdetermined system derived for rotated coordinate system
- Bouncing ball optimization
- Circular Motion Question - fully algebraic
- How can I solve this pair of differential equations?
- How to solve $-\frac{1}{2}gt^2\sin \theta=x$ when $x$ equals $0$
- Find the acceleration and tension in pulley setup
- Derive first-order time derivatives in a second-order dynamic system
- Phase curves of a spherical pendulum
- Velocity dependent force with arbitrary power
- An explanation for mathematicians of the three-body problem using a simple example, and the moons of Saturn
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?