In component design, topology optimization provides a significantly higher degree of design freedom compared to classical approaches such as shape or sizing optimization. Its objective is to determine an optimal material distribution within a given design domain with respect to prescribed performance criteria. Among the available approaches, the level set method is well established and implicitly represents the component boundary by means of a higher-dimensional function whose zero level defines the structural boundary. While single-objective optimization problems lead to a unique optimal solution, multi-objective optimization involves several competing objectives. As a result, the solution cannot be reduced to a single design but instead forms a set of non-dominated solutions, known as the Pareto frontier. To address this class of problems, a unified variational framework is presented for deriving the equation of motion of the level set function in multi-objective topology optimization. In this framework, the level set function is interpreted as a generalized coordinate of a fictitious physical matter. By assigning kinetic and potential energy to this fictitious matter, Hamilton's principle is employed to derive the governing equation, which results in a damped wave equation. A dynamic update scheme for the weightings induces a controllable trajectory in the space of objective functionals, enabling enhanced exploration of the Pareto frontier, as demonstrated by numerical examples. The Pareto frontier is further interpreted as an imbedded manifold, allowing local refinement through simplex-based strategies.