Kinetic and Potential Energy

Lagrangian construction from kinetic and potential energy. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Analytical mechanics does not begin by writing every force component. It first organizes the system through kinetic energy $T$ and potential energy $V$, expressed in suitable coordinates. This changes the way the model is read: motion and configuration become ingredients of a dynamical function.

The goal of this leaf is to build the Lagrangian $L$ without confusing it with mechanical energy $E$. Although all these quantities are measured in joules, they do not play the same role. $E$ is used for balance and conservation; $L$ is used to generate equations of motion in the next leaf.

The graph gives a simple reading. If height $h$ is fixed, gravitational potential energy remains constant while kinetic energy grows with speed $v$. This comparison shows where motion dominates and where configuration dominates.

🟢 Essential level

The essential idea is that a Lagrangian model does not use arbitrary energy terms. Its kinetic and potential parts must describe the same physical system. The kinetic part represents motion; the potential part represents configuration. If references, coordinates, or systems are mixed, the later difference loses physical meaning.

Kinetic energy grows quadratically with speed. Doubling speed does not double the motion energy; it multiplies it by four. Potential energy instead depends on position, height, or deformation measured from a declared reference.

The key didactic distinction is between two questions. If the question is how much mechanical energy is stored or conserved, use the sum. If the question is how to construct the function used by Lagrangian dynamics, use the difference between motion and configuration.

A practical way to consolidate Kinetic and Potential Energy is to separate the physical system, the independent variables, and the mathematical representation before calculating. Lagrangian construction from kinetic and potential energy. The student should be able to explain what information the analytical description preserves, what it replaces from the Newtonian treatment, and which observable consequence would reveal an inconsistent choice.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

Elementary translational kinetic energy is:

Translational kinetic energy

In generalized coordinates, an effective linear form may be written through generalized velocity:

Kinetic energy in a generalized coordinate

In curvilinear or coupled coordinates, the advanced form uses the inertial metric matrix $g_{ij}$. Kinetic energy then depends not on an isolated scalar mass, but on the local geometry of the coordinates:

Kinetic energy with metric matrix

Gravitational potential energy in a uniform field is:

Gravitational potential energy

For a linear elastic model around equilibrium:

Elastic potential energy

For a natural mechanical system with conservative forces derived from a velocity-independent potential, the Lagrangian is built as:

Mechanical Lagrangian

Real student questions

• Why is $L$ not $E$ if both use $T$ and $V$?
  Because they combine the same ingredients with different signs and serve different tasks.

• Can $L$ be negative?
  Yes. It only means that $V$ exceeds $T$ in the chosen reference.

• When do I use elastic instead of gravitational potential energy?
  When configuration stores energy through deformation around equilibrium.

• What should I review if the result is unreasonable?
  Review units of $v$, reference for $h$, sign of $V$, and compatibility between $q_i$ and $\dot{q}_i$.

Final synthesis

Mastering this topic means choosing and combining energy terms without confusing their roles. $T$ describes motion, $V$ describes configuration, $E$ organizes conservation, and $L$ builds dynamics. That separation is the basis of serious Lagrangian reading.