In mathematics and physics, Herglotz's variational principle, named after German mathematician and physicist Gustav Herglotz, is an extension of the Hamilton's principle, where the Lagrangian L explicitly involves the action formula_1 as an independent variable, and formula_1 itself is represented as the solution of an ordinary differential equation (ODE) whose right hand side is the Lagrangian formula_3, instead of an integration of formula_3. Herglotz's variational principle is known as the variational principle for nonconservative Lagrange equations and Hamilton equations.
Mathematical formulation.
Suppose there is a Lagrangian formula_5 of formula_6 variables, where formula_7 and formula_8 are formula_9 dimensional vectors, and formula_10 are scalar values. A time interval formula_11 is fixed. Given a time-parameterized curve formula_12, consider the ODE formula_13When formula_14 are all well-behaved functions, this equation allows a unique solution, and thus formula_15 is a well defined number which is determined by the curve formula_16. Herglotz's variation problem aims to minimize formula_17 over the family of curves formula_16 with fixed value formula_19 at formula_20 and fixed value formula_21 at formula_22, i.e. the problem formula_23Note that, when formula_3 does not explicitly depend on formula_1, i.e. formula_26, the above ODE system gives exactly formula_27, and thus formula_28, which degenerates to the classical Hamiltonian action. The resulting Euler-Lagrange-Herglotz equation is formula_29which involves an extra term formula_30 that can describe the dissipation of the system.
Derivation.
In order to solve this minimization problem, we impose a variation formula_31 on formula_32, and suppose formula_33 undergoes a variation formula_34 correspondingly, thenformula_35and since the initial condition is not changed, formula_36. The above equation a linear ODE for the function formula_37, and it can be solved by introducing an integrating factor formula_38, which is uniquely determined by the ODE formula_39By multiplying formula_40 on both sides of the equation of formula_41 and moving the term formula_42 to the left hand side, we get formula_43Note that, since formula_44, the left hand side equals to formula_45and therefore we can do an integration of the equation above from formula_20 to formula_47, yielding formula_48where the formula_49 so the left hand side actually only contains one term formula_50, and for the right hand side, we can perform the integration-by-part on the formula_51 term to remove the time derivative on formula_52:formula_53and when formula_54 is minimized, formula_55 for all formula_56, which indicates that the underlined term in the last line of the equation above has to be zero on the entire interval formula_57, this gives rise to the Euler-Lagrange-Herglotz equation.
Examples.
One simple one-dimensional (formula_58) example is given by the Lagrangian formula_59The corresponding Euler-Lagrange-Herglotz equation is given as formula_60which simplifies into formula_61This equation describes the damping motion of a particle in a potential field formula_62, where formula_63 is the damping coefficient.