Lagrangian L(x,λ) = f(x) - λg(x)

How do we find the path an object naturally takes?

Imagine you're throwing a ball; it doesn't just go straight up but follows a curved path. Why does it curve?

Think of the ball's path as a journey it takes, with different forces acting on it. The Lagrangian helps us figure out this path by balancing the ball's energy and the forces acting on it.

Example

If a ball is thrown with a certain speed and angle, the Lagrangian helps predict its curved trajectory.

Remember this

The Lagrangian formula (L(x,λ) = f(x) - λg(x)) calculates the path by balancing energy and forces.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

Related concepts

Normalization (machine learning)

L2 normalization equation: x_i' = x_i / ||x||_2

Chain rule

Chain rule formula: h'(x) = z'(y(x)) * y'(x)

Hessian matrix

The Hessian matrix is denoted by H or ∇²

Variational autoencoder

ELBO formula in variational inference

Taylor series

Taylor series formula: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! +

Cross-entropy

Cross-entropy loss equation: H(p, q) = -Σ(p(x) * log(q(x)))

Swipe through 100 ML concepts daily

Open Pocket Polymath