målHamiltonian

MålHamiltonian is a term occasionally used in control theory and optimization to refer to a Hamiltonian augmented with a goal-oriented penalty. The compound word combines 'mål' (goal or target in several Scandinavian languages) with Hamiltonian, signaling an adaptation of the classical framework to emphasize reaching a desired state. It is not a standard concept in mainstream physics and appears only in a limited set of interdisciplinary writings and tutorials.

Formal definition: Let q ∈ R^n denote generalized coordinates, p ∈ R^n the conjugate momenta, and H(q, p, t) the traditional Hamiltonian. A målHamiltonian is defined as H_m(q, p, t) = H(q, p, t) + λ G(q, p, t), where λ ≥ 0 is a weight and G is a nonnegative function measuring deviation from a target state (q*, p*, t*). Common choices for G include quadratic penalties G = (q − q*)^T K_q (q − q*) + (p − p*)^T K_p (p − p*) or a squared norm of a constraint violation.

Dynamics and interpretation: The system evolves according to Hamilton's equations derived from H_m, so dq/dt = ∂H_m/∂p and dp/dt = −∂H_m/∂q. The penalty term generally breaks exact energy conservation and tends to steer trajectories toward the specified goal, effectively blending conservative motion with goal-directed shaping.

Relation to other methods: The construction is analogous to adding a soft constraint, similar in spirit to potential shaping in control theory, regularization in optimization, or barrier-like terms in constrained problems. It provides a framework for goal-oriented planning within the Hamiltonian formalism, and can be extended to time-varying targets and stochastic settings.

Applications and examples: In simple systems, a målHamiltonian with a quadratic goal penalty yields a harmonic oscillator attracted toward a target state. In robotics and autonomous systems, it can be used to encode goal-reaching behavior while preserving structure of Hamiltonian dynamics. The concept remains niche and is not widely standardized.

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