Relation Between the Partial Derivatives of the Kinetic Energy in the Lagrangian and Hamiltonian Formalisms of Dynamics

TL;DR

This paper explores the relationship between the partial derivatives of kinetic energy in Lagrangian and Hamiltonian formalisms, revealing a sign difference and identifying conditions where derivatives vanish.

Contribution

It establishes an exact relation between the partial derivatives of kinetic energy in both formalisms for conservative systems and introduces a form of kinetic energy with zero derivative.

Findings

Partial derivatives differ by a sign in Lagrangian and Hamiltonian formalisms.

An exact relation between the derivatives is derived for conservative systems.

A form of kinetic energy with zero partial derivative is identified.

Abstract

The partial derivative of the kinetic energy of a dynamical system with respect to a generalized coordinate as it appears in the Lagrangian formalism is not equal to the derivative of the kinetic energy with respect to the same coordinate in the Hamiltonian formalism but differs by a sign. We find another exact relation between the two partial derivatives in the case of a conservative system. We also identify another form of kinetic energy whose partial derivative with respect to a generalized coordinate vanishes identically.